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\title{Centralizers of Commuting Elements in Compact Lie Groups}

\author{{\bf Kristen A. Nairn}\\College of St. Benedict, MN, USA\\
knairn@csbsju.edu\\
MSC Classes: 22C05 (Primary); 17B20, 22E46 (Secondary) }

\date{}
\maketitle
\begin{abstract}
The moduli space for a flat $G$-bundle over the two-torus is completely determined by its holonomy representation.
When $G$ is compact, connected, and simply connected, we show that the moduli space is homeomorphic to a product
of two tori mod the action of the Weyl group, or equivalently to the conjugacy classes of commuting pairs of elements in $G$. Since the component group for a non-simply connected group is given by some finite dimensional subgroup in
the centralizer of an $n$-tuple, we use diagram automorphisms of the extended Dynkin diagram to prove properties of centralizers of pairs of elements in $G,$ followed by some explicit examples. We conclude by showing that as a result of a compact, connected, simply connected Lie group $G$ having a finite number of subgroups, each conjugate to the centralizer of any element in $G$, that there is a uniform bound on an irredundant chain of commuting elements.
\end{abstract}

\section{Introduction}
Classifying the moduli space of gauge equivalence classes of flat connections on
a principal $G-$bundle over a compact Riemann surface $\Sigma_g$ of genus $g$ is of
interest from various perspectives. For example, Atiyah-Bott \cite{AB82} proved that this moduli
space is equivalent to the finite dimensional representation space
$\{\rm{Hom}(\pi_1(\Sigma_g),G)\}/G$ by constructing a symplectic structure on the moduli space by symplectic reduction from the infinite-dimensional sympletic manifold of all connections.

If $A$ is a flat connection on a $K-$bundle over $T^3$ then the holonomy of $A$ is defined by the
conjugacy classes of commuting triples in $K.$ In topological field theory, vacua of Yang-Mills theory correspond to flat
$G$ bundles. Every nontrivial triple on $T^3$ equals an additional quantum vacuum state and determines
a distinct component of the moduli space $\mathcal{M}_G.$ If the triple
is of rank zero (rigid), then it is unique up to $G$-equivalence and
every element can be conjugated into the maximal torus $T$ for $G.$ If
the triple is not of rank zero, then there is an entire family of triples
with elements lying on some smaller torus inside the centralizer
$Z_G(x,y,z).$ Kac-Smilga \cite{KS00} proved that computing the number of quantum vacuum states over $T^3$ is equivalent to classifying commuting triples in a simple, compact, simply connected Lie group $G.$ Witten \cite{W98} proved that the number of extra quantum vacuum states for a flat principal $G$-bundle over a
spatial $3$-torus $T^3$ is the topological invariant called the {\sl Witten Index} which is equal to $g$, the dual Coxeter number of the Lie group $G.$

Our primary motivation comes from Borel-Friedman-Morgan \cite{BFM02}. Given $G$ is a compact, connected, semisimple Lie group, they proved that principal $G$-bundles $\zeta$ with flat connections over a maximal two torus $T^2$ are
classified up to restricted
gauge equivalence by classifying commuting pairs of elements in the simply connected covering $\tilde{G}$ of $G$ that commute up to the center.
The first invariant is the nontrivial characteristic
class $[w]\in H^2(T^2,\pi_1(G))=\pi_1(G).$ By identifying $\pi_1(G)$ with a subgroup of the center
$\mathcal{C}G$ we fix a topological type of the bundle by $w(\zeta)=c\in\mathcal{C}G.$ Since the
characteristic class is completely defined by the holonomy representation
$\rho\colon \pi_1(T^2)\cong\z\times\z\to G$ where the images of $\rho$
commute then for any lifts $\tilde{x},\tilde{y}\in\tilde{G}$, we have $[\tilde{x},\tilde{y}]=[w]=c.$
Elements with this property are called $c-$pairs or "almost commuting".

\begin{D}
A pair of elements $x,y\in G$ {\bf commutes} if $[x,y]=1.$
A {\bf c-pair} in the simply connected covering $\tilde{G}$
of $G$ is a pair of elements $(x,y)$ where $x,y\in G$ such that $[x,y]=1$ and
$[\tilde{x},\tilde{y}]=c\in\tilde{\mathcal{C}G}.$
\end{D}

To understand why a flat bundle is determined by its holonomy representation note the following.
Let $G$ be a compact, connected and not necessarily simply connected Lie group and $\pi\colon \tilde{G}\to G$
the universal covering map.
Certainly the choice of a lift $\tilde{x}$ is unique up to an element
in $\rm{Ker}(\pi)\cong\pi_1(G)$ which is identified as a subgroup of the
center of the simply connected covering. Extending this for a
$c$-pair: for $k\in\rm{Ker}(\pi),\ [\tilde{x},\tilde{y}]=
[k\tilde{x},k\tilde{y}]=c$ because $k\in \rm{Ker}(\pi)$ commutes with
every element in $\tilde{G}$ and is also invariant under the choice
of $x,y.$ We may define conjugation by $\tilde{g}\in \tilde{G}$ to be
$\tilde{g}[\tilde{x},\tilde{y}]\tilde{g}^{-1}=
[\tilde{g}\tilde{x}\tilde{g}^{-1},\tilde{g}\tilde{y}\tilde{g}^{-1}]$
satisfying $\pi(\tilde{g}\tilde{x}\tilde{g}^{-1})=g\pi(\tilde{x})g^{-1}=
gxg^{-1}.$ This lift is independent of the choice of
$c\in\mathcal{C}\tilde{G}$ and thus our $c$-pair is well-defined.

For completeness, we recall some definitions found in \cite{Bourbaki}
on Dynkin diagrams and root/coroot systems that we will use throughout the paper.
Let $\Phi$ be a reduced irreducible root system for a compact connected Lie group $G,$ and
let $\Delta=\{a_1,\ldots,a_n\}$ be a choice of simple roots for $G.$
Let $d$ be the highest root of $\Phi$ with respect to $\Delta.$ Set $\tilde{a}=-d$ and let
$\tilde{\Delta}=\Delta\cup \{\tilde{a}\}$ be the extended set of simple roots. Then
$\Delta^\vee$ is the set of coroots $a^\vee$ inverse to each root $a\in\Delta.$
If we define $A$ to be the unique alcove containing the origin in the positive Weyl chamber
associated to $\Delta$ then there is a bijection between the walls of $A$ and $\tilde{\Delta}.$
Therefore $\tilde{\Delta}$ is the set of nodes for the extended Dynkin diagram $\tilde{D}(G).$
For each element $c\in\mathcal{C}G$ the differential $w_c\in \mathcal{W}$ of the action of the
center on the alcove is a linear map normalizing $\tilde{\Delta}\subset t$ and the action of
$w_c$ on the nodes of $\tilde{D}(G)$ is a diagram automorphism. Given
a maximial torus $T\subset G$, denote Lie$(T)=\mathfrak{h}$ and the exponential map
identifies $T$ with $\mathfrak{h}/Q^\vee$
where $Q^\vee=\sum\z a_i^\vee$ is the lattice associated to the
coroots dual to a choice of simple roots $a_i\in\Delta$ for $G.$ Denote the affine Weyl group by $W_{aff}.$
The alcove is defined over the maximal torus $T\subseteq G$
as $A= \mathfrak{h}/W_{aff}(\Phi) \subseteq \mathfrak{h}$ where
$W_{aff}(\Phi)$ acts simply transitively on the set of alcoves in the vector space $V;$ thus
there is an induced action of the center $\mathcal{C}G$ on $A.$
x

\section{Component Group of the Centralizer of Commuting Pairs}
The work in \cite{BFM02} gave an explicit characterization of the moduli space of $c$-pairs in terms of the extended
coroot diagram of a simply connected group $G$ and the action of the Weyl group on that diagram.
This beneficial relationship between the root/coroot system and holonomy plays a crucial role.
Throughout, we assume that $G$ is a compact connected Lie group, unless otherwise defined.
Note that if $G$ is disconnected, then there is the appearance of a
$c-"1-$chain" coming from a component which may be a finite group of a certain order.
A group $G$ is {\sl reductive} if any representation is irreducible.
Notice that when a group is compact, it is equivalent to being reductive.
We will use the following theorem by Borel (\cite{B62}, Theorem 5)
\begin{theorem} \label{Borel}
Let $G$ be a compact, connected and simply connected Lie group. Then
$Z_G(x)$ is connected.
\end{theorem}
The following demonstrates the relationship between the conjugacy classes
for commuting pairs $(x,y)$ and flat $G$-bundles over $T^2:$
\begin{proposition}
Assume that $G$ is a compact, connected and simply connected Lie group.
For any maximal torus $S$, we have that
$\{\rm{Hom}(\pi_1(T^2,x),G)/G\}\to (S\times S)/\mathcal{W}$
is a homeomorphism.
\end{proposition}
\begin{proof}
Fix generators $(\gamma_x,\gamma_y)$ for $\pi_1(T^2,x).$
Notice that we have a representation $\rho\colon\pi_1(T^2)\to G$ where
$\rho(\gamma_x)=x,\ \rho(\gamma_y)=y$ and that these images define the
commutator in $G.$ In fact, the representation determines the commutator
in the following sense. Let $T$ be the maximal torus in $G.$ Then for
some $g\in G,\ gxg^{-1}\in T$ and $gyg^{-1}\in T$ since every element
in $G$ can be conjugated into the maximal torus. We
want to show that both $x,y\in T.$ To do this, define conjugation by
$g\in G$ for the pair $(x,y)$ by
$g(x,y)g^{-1}=(gxg^{-1},gyg^{-1})=(x',y')$
where $(x,y)\in T\times T$ and $(x',y')\in T'\times T'.$  The fact that
$G$ is simply connected implies that $Z_G(x)$ is connected
(~\ref{Borel}). Thus we may restrict to the connected component
of the identity $Z^0(x).$ Since
$x\in Z_G(x)$ we must show that $T\subseteq Z_G(x)$ because this
would imply that both $x,y\in T.$ By definition of the
representation, the image  $[x,y]=1$ so that $y\in T$ is conjugate to
$x$ which implies we may project $y$ to an element
$\zeta_y\in\mathcal{W}.$
If we conjugate the pair $(x,y)$ by
$(\zeta_y^{-1}g)\in G,$ then $(\zeta_y^{-1}g)(x,y)(\zeta_y^{-1}g)^{-1}=
\zeta_y^{-1}(x',y')\zeta\in T'\times T'.$ Thus we have shown that
$(\zeta_y^{-1}g)$ conjugates elements from $T\times T$ to $T'\times T'$
and $\zeta_y^{-1}g\in\mathcal{W}.$

To prove the converse, notice that $\mathcal{W}$ acts by simultaneous
conjugation
on $S\times S$ so that if $g\in \mathcal{W}$ is a reflection, then $gS\in
N_G(S)/S.$
Thus $\mathcal{W}\times S\to S$ by $(gS,t)\mapsto gtg^{-1}$ and thus we
have its
action on the pair $\mathcal{W}\times (S\times S)\to S\times S$ by
$(gS,t,h)\mapsto (gtg^{-1},ghg^{-1}).$ Define the commutator by
$[t,h]=[gtg^{-1},ghg^{-1}].$ Since elements in $S$ all commute,
if $\langle \gamma_x,\gamma_y\rangle$ generates $\pi_1(T^2)$ and
$\rho(\gamma_x,\gamma_y)=[gxg^{-1},gyg^{-1}]=1$ then the holonomy
determines the commutator and vice versa.
\end{proof}

The next corollary follows immediately because the fundamental group of the centralizer $Z(x_1)$
is trivial, and for a commuting $n$-tuple in a simply connected group,
the component group is contained in the fundamental group of the semisimple subgroup
$\pi_1(DZ(x_1)).$
\begin{corollary}
When $G$ is of type $A_n,\ C_n$ every commuting $n$-tuple can be
conjugated into the maximal torus $T$ in $G$ so that the moduli
space has the form $\mathfrak{M}_G =
\underset{n}{\underbrace{T\times\cdots\times T}}/\mathcal{W}.$
\end{corollary}
The corollary can also be seen directly as follows. If
$(x_1,\ldots,x_n)$ is a commuting $n$-tuple such that
$[\tilde{x_1},\tilde{x_2}]=c\in \mathcal{C}G$ and
$[\tilde{x_1},\tilde{x_i}]=1,\ \forall\  2<i\le n,$ choosing
$\tilde{x_1}$ in the alcove over the torus implies that
$\tilde{x_2}$ projects to a Weyl element and therefore conjugates
back into the maximal torus; every other element has trivial
commutator and thus can be conjugated to the maximal torus.
This also works when $(x_i,x_j)$
for $1\le i<j\le n$ is an arbritrary $n$-tuple because the lifts
$[\tilde{x_i},\tilde{x_j}]=c_{ij}\in\mathcal{C}G$ and for the cases
of type $A_n,\ C_n$ the center is generated by one cyclic element.
Hence only one pair in the $n$-tuple determines what happens to the
other elements in the $n$-tuple.

\begin{corollary}
When $G$ is simply connected, the component group $\pi_0(Z(S))$ is a
subgroup of $\z/{n_i}\z$ where $n_i\le 6$
and corresponds to the coroot integer for $x_1\in G$ which is
associated to the node in the extended Dynkin diagram $\tilde{D}(G).$
\end{corollary}

By [Lemma 3.1.5 in \cite{BFM02}], for $x_n\in Z(x_1,\ldots,x_{n-1}),\
{\rm Stab}_{W(\Phi(x_1,\ldots,x_{n-1}))}(x_n)\cong
{\rm Stab}_{\pi_1(DZ(x_1,\ldots,x_{n-1}))}(\tilde{x_n})).$
Thus the component group is a subgroup of the fundamental group
$\pi_1(Z(x_1,\ldots,x_{n-1})),$ which in turn is a finite subgroup
of the center of $\tilde{Z}(x_1,\ldots,x_{n-1}).$ If the
fundamental group of the centralizer $Z(x_1)$ is trivial, then
there are no further component groups for the remaining elements
in the $n$-tuple. So suppose that $\pi_1(DZ(x_1))\ne \{1\}.$
If $G$ is simply connected, then  $Z(x)$ is
connected and thus $\pi_0(Z(x_1,x_2))\subseteq \pi_1(DZ(x_1)).$
Even if $G$ is not simply connected but still connected, choosing
$x_2$ to lie in the connected component of the centralizer of
$Z(x_1)$ will yield the same result.

If $\pi_1(DZ(x_1))$ is not central then it defines a possibly
non-trivial diagonal subgroup $\mathcal{C}\tilde{Z}(x_1)$ which acts
simultaneously on the components of the centralizer $Z(x_1).$
If $x_2$ is chosen so that $\tilde{x_2}\in A$ does not lie in the
fixed point space under this diagonal action, the component
group is trivial. However, if $\tilde{x_2}$ lies somewhere in the
fixed space under the action of the center, there will be a
nontrivial component group $\pi_0(Z(x_1,x_2)).$ The structure of
the centralizer itself will differ depending on where the element
$x_2$ lies. Regardless of which compact, connected Lie group $G$
we are working with, whether or not there is a component group
from the choice of second element relies on the diagonal action
on $Z(x_1,x_2).$

\begin{proposition}
Let $G$ be simply connected and let $\Delta=\{a_1,\ldots,a_n\}$ be
a choice of simple roots. Let $\x=\{\tilde{a},a_1,\ldots,a_k\}, k\le n,$
be a choice of simple roots for $Z_G(x)$ and let
$\mathfrak{h}(x)\subseteq\mathfrak{h}$ be the real linear
span of the coroots dual to the roots in $\x.$ Then there
is an exact sequence $1\to Q^\vee(x)\to Q^\vee\cap\mathfrak{h}(x)\to\z/n_i\z \to 1.$
\end{proposition}

\begin{proof}
By definition of the fundamental group,
$n_i={\rm gcd}(g_{k+1},\ldots,g_n)$ knowing that
all the coroot integers for both the classical and exceptional
groups are less than or equal to six, $n_i\le 6.$ Dividing each of
the coroot integers in any group $G$ by $n_i$ we may define a new
integer $g_r'=g_r/{n_i}$ for $r > k.$ By definition, this element
will have order $n_i$ in the central subgroup
$Q^\vee\cap\mathfrak{h}/{Q^\vee(x)}$ and thus is a generator for the
cokernel. Hence
$Q^\vee\cap\mathfrak{h}/{Q^\vee(x)}\cong\z/n_i\z.$
\end{proof}

\begin{proposition}
For an arbitrary compact, connected simple group $G$ and for a
commuting $n$-tuple $\ov{x}=(x_1,\ldots,x_n),$ the component group of
the centralizer of the $n$-tuple can be defined in terms of the
roots as
$$\pi_0(Z_G(\ov{x}))=\frac{\rm{Stab}_{L/{Q^\vee}}(\ov{x})}
{\mathcal{W}(\Phi(\ov{x}))}.$$
\end{proposition}
\begin{proof}
If $G$ is not simply connected, then under complexification
$T_{\c}=\mathfrak{h}/L$, where $Q^\vee\subseteq L\subseteq P^\vee.$
If $\Phi(x)$ is a subset of roots which annihilate $x$ we must
determine how $\rm{Stab}_{\mathcal{W}}(x)$ is defined with respect
to this smaller subset of roots. If $x$ corresponds to some node $a_i$
in the extended Dynkin diagram such that $g_{a_i}\ne 1$ then
$\Phi(x)=\{a_k\in\tilde{\Delta}\mid k\ne 1\}.$ Since
$Q^\vee\subseteq L\subseteq P^\vee$ we have the nesting of tori
$\mathfrak{h}/{P^\vee}\subseteq \mathfrak{h}/L \subseteq
\mathfrak{h}/{Q^\vee}.$ Thus for the lift
$\mathfrak{h}/L\to \mathfrak{h}/{Q^\vee}$ sending $x\mapsto \tilde{x}$
its kernel consists of all the roots in $L$ not in $Q^\vee$ i.e.
$L/{Q^\vee}.$ Therefore the roots which annihilate $x$ are the same
as those annihilating $\tilde{x}.$

Let $\mathcal{W}(\Phi(x))$ be a subgroup in $\mathcal{W}$ defined
by a subroot system when viewed as characters which annihilate $x.$
The faithful action of $\mathcal{W}_{aff}$ on $\mathfrak{h}/{Q^\vee}$
yields a split exact sequence
$1\to P^\vee/{Q^\vee}\to \mathcal{W}_{aff}\to \mathcal{W}\to 1$
and since the kernel is central,
$\mathcal{W}_{aff}=P^\vee/{Q^\vee}\times \mathcal{W}$ is a direct product
because the action of the Weyl group is trivial on the center.
Restrict the Weyl group to $L\colon \ \mathcal{W}^L_{aff}=L/{Q^\vee}\times
\mathcal{W}.$ The torus action of $L/{Q^\vee}$ on $\mathfrak{h}/{Q^\vee}$
provides the quotient
$(\mathfrak{h}/{Q^\vee})/(L/{Q^\vee})=\mathfrak{h}/L.$  This implies that
$\rm{Stab}_{\mathcal{W}}(x)\subseteq L/{Q^\vee}\times\mathcal{W}$ and the
projection $\pi\colon
(\mathfrak{h}/{Q^\vee})\to (L/{Q^\vee})$ satisfies $\pi^{-1}(\tilde{x})=x$
as the unique lift to the alcove. Therefore we may define
$\rm{Stab}_{L/{Q^\vee}}(x)=\rm{Stab}_{\mathcal{W}_{aff}}(\tilde{x})$
in the sense that the roots which annihilate $\tilde{x}$ can be used
to define a subset $S\subseteq L/{Q^\vee},$ where
$S=\pi_0(Z_G(x_1,\ldots,x_n)).$ This allows for
a component group larger than the fundamental group and therefore it is
not necessarily cyclic. Since $S\subseteq \mathcal{C}G$ it induces a
well-defined cyclic permutation on the vertices in the alcove and its
fixed space $\mathfrak{h}^S$ may be something other than the barycenter.

$L$ is defined as follows. For $Z_G(x_1)$
the vector space is $\mathfrak{h}$ and its coroot lattice is the entire
$Q^\vee.$ Because we are considering commuting elements, we choose
$x_2\in Z_G(x_1)$ to lie in $\mathfrak{h}(x_1)/L_1$ where
$\mathfrak{h}(x_1)$ is the vector space associated to $DZ(x_1).$ Because
$Z(x_1)$ is not necessarily connected, the associated lattice is
$Q^\vee\subseteq L_1 \subseteq P^\vee.$ By induction,  the element
$x_n\in \mathfrak{h}(x_1\ldots,x_{n-1})/L_{n-1}$ and
$Q^\vee\subseteq L_1\subseteq\cdots\subseteq L_{n-1}\subseteq P^\vee$
so that $L=L_{n-1}$ as the associated lattice to the centralizer of the
prior $n-1$ elements. From the definition of these lattices,
when they are quotiented out by the coroot lattice, they will either
be a cyclic subgroup of the center whose order divides the order of the
center or will be the entire center. Thus we have the above conclusion
since $S\subseteq L/{Q^\vee}\subseteq P^\vee/{Q^\vee}.$
\end{proof}

\section{Properties of Centralizers}
In general, the component group of an ordered $n$-tuple is some subquotient of the Weyl group and lies in the
connected component $Z^0(x_1,\ldots,x_n).$
In order to determine the component group for a non-simply connected
group we note that the finite diagonal subgroup contained in the center of each
centralizer $Z(x_1,\ldots, x_k),$ for some $k,$ at some point
becomes the component group and therefore defines the singularities
in the moduli space. For the classical groups, $Z(x_1)$ will be a product of type
$A_n,\ B_n$ or $D_n$ and for the exceptional groups, $Z(x_1,x_2)$ will be
of type $A_n,\ D_n$ Therefore, it sufficies to consider
the diagonal group action of the fundamental group $\pi_1(DZ(x_1))$
on groups of these types. Since the fundamental group is a subgroup
of the center of the simply connected covering, for type $B_n$ we
only consider the $\z/2\z$ action on the alcove given by flipping two vertices;the action of any higher
order central cyclic group is trivial.

\begin{D}
Define the {\bf rank}
$rk(x_1,\ldots,x_n)$ of an $n$-tuple to be the rank of $Z(x_1,\ldots,x_n).$ An $n$-tuple has
{\bf rank zero} if and only if  $Z(x_1,\ldots,x_n)$ is a finite group.
A $c$-pair $(x,y)$ is in {\bf normal form} with respect to the maximal torus
$T$ in the alcove $A$ if $x\in T$ is the image under the exponential map of
$\tilde{x}\in \mathfrak{t}^c$ and $y\in N_G(T)$ projects to $w_c\in
\mathcal{W}.$ Note $w_c\in W$ is the differential action of $c\in\mathcal{C}G$
that, as a group of affine isometries of the Lie algebra $t$ of the maximal torus
$T$ normalizes the alcove $A.$
\end{D}

Let $\tilde{\Delta} =\{ \tilde{a},a_1,\ldots, a_n\}$ be the set of
extended simple roots for a Lie group $G.$ Any closed
subset of the extended simple roots for $G$ gives a subdiagram of the
extended diagram. We are interested in the subset of roots $\Delta_x$ that annihilate the $n$-tuple.
In particular, the simple root system for the centralizer $Z(x)$ for any $x\in G$ is defined as
by $\x=\{a\in \Delta\mid \a(x)\in\z\}$ which has an associated Weyl group  $\mathcal{W}(\Phi(x)).$ Let $\ov{x}=(x_1,\ldots,x_n)$ be a commuting $n$-tuple. Any element $x\in \pi_0(Z(\ov{x}))$ can be represented by $g\in
Z_G(\ov{x})$ since $g$ normalizes $Z_G(\ov{x})$ and therefore via conjugation defines a
map $\{x\}\times I \to G$ defining the path components of $G$.
Let $S_{\x}$ be the maximal torus in the centralizer
$Z(x_1,\ldots,x_n)$ generated by the roots in $\Delta_x$ given by its Lie algebra
$\mathfrak{s}= \underset{a\in\x}{\bigcap}{\rm Ker}(a).$
Since $G$ is reductive, there is a standard decomposition given by
$G=(\mathcal{C}G)^0\times_F DG$ where $(\mathcal{C}G)^0$ is a central
torus in the semisimple subgroup $DG$ of $G.$ and
$F=(\mathcal{C}G)^0\cap DG$ is a finite subgroup of the center of $DG.$
Thus we get a decomposition of the the centralizer into
$Z(S_{\x})= S_{\x}\times_{F_{\x}} DZ(S_{\x}).$

\begin{remark}
For $c\in\mathcal{C}G$, denote by $S_c$ the torus in $T$ fixed under the
action of the center.
The choice of roots $\Delta(c)=\{a\in \Delta\mid r_a\not\in {\bf Z}\}$
where $c=\rm{exp}(\lambda)$ for $\lambda=\underset{a\in\Delta}{\sum}
r_a a$ defining the fixed subtorus $S_c\subset T$ is independent of the
choice of lift $\lambda,$ let $\lambda'\underset{a\in\Delta}{\sum}
r_a'a'\in\mathfrak{t}$ be such that $\rm{exp}(\lambda')=c'.$
Then under $\rm{exp}\colon\mathfrak{t}\to T,$ the kernel of
this map is an integral lattice defined with respect to $T.$ Namely,
$\rm{Ker}(exp) =Q^\vee,$ thus for $\lambda-\lambda'\in \rm{Ker}(\pi)$
this implies that $r_a-r_a'\in {\bf Z}$ and hence $\Delta(c)=\Delta(c')$
if and only if $r_a-r_a'\equiv 0\pmod{\bf Z}$ which we have since
$r_a-r_a'\in Q^\vee.$ Therefore, $\Delta(c)$ depends only on the choice
$c\in\mathcal{C}G\cong P^\vee/Q^\vee$ and any two elements in the Lie
algebra $\mathfrak{t}$ differ by an element in the coroot lattice
$Q^\vee.$ By definition of a $c-$pair of rank zero, $c\in DZ(S_c).$
Thus the moduli space is precisely $\mathfrak{M}=(T\times T)/W(T,G).$
It certainly will not be true that for a general non-simply connected
group that every element of a commuting $n$-tuple can be put inside the
maximal torus.
\end{remark}

We show that the fundamental group of the centralizer is finite cyclic
by using diagram automorphisms.
\begin{proposition}\label{fundgp}
Under a cyclic permutation of the vertices in the extended diagram
of the type $A_n$ where the permutation is given by the fundamental
group $\pi_1(DZ(x_1))=\z_k,$ the quotient space has the form
$$A_n/{\mathbb Z}_k\cong \underset{k}{\times}A_{l-1}\times T^{k-1}\rtimes
{\mathbb Z}_k$$
where $n+1=kl$ and $\pi_1(DZ(x_1))\cong {\mathbb Z}_k,$ with $1\le k\le 6.$
\end{proposition}
\begin{proof}
Since any inner automorphism of type $A_n$ is dihedral, it is either
a rotation or a reflection. Consider the cyclic permutation $\tau$ given by
rotation. (note: this is an element of a group of affine automorphisms
of a vector space which normalizes the alcove of a root system on that
vector space. Such automorphisms are equivalent to diagram automorphisms
of the extended Dynkin diagram of the root system.)
If $\tau\in\z_k$ has order $n+1$ then the fixed point set
is simply the barycenter and thus $\mathfrak{t}^\tau=\{0\}.$ If $n$
is odd then $\tau$ may have order $k\mid(n+1).$ If $k=\frac{n+1}{2}$
then either the barycenter is the only fixed point or the fixed point
set is the join of type $A_{2k-1}$ or there is a rotation subgroup
of $\tau$ of order exactly $k$ which implies it is an involution of
the extended diagram which fixes two vertices and thus the quotient
coroot diagram is a product of type $A_1.$ The $\z_2$ action on the
alcove over $A_1$ is simply to switch the two vertices leaving the
barycenter fixed.

Specifically, if $n+1=kl$ then every node in the extended diagram included
in this $k$-orbit is nonzero which leaves the quotient coroot diagram
as the join of $k,\ (l-1)$-simplicies with the barycenter (since the
barycenter is the only fixed space under the action of the full center)
times the remaining torus and semidirect product with rotation group.
In terms of extended roots in the diagram, if
$\tilde{\Delta}=\{\tilde{a},a_1,\ldots,a_n\}$ is the set of simple roots
for $A_n$ then the quotient space $A_n/\z_k$ is defined by the elements
in the orbit,
$\tilde{\Delta}/{\z_k}=\{\tilde{a},a_l,a_{2l},\ldots,a_{(k-1)l}\}.$
Thus the gaps between the nodes are of length $(l-1).$
Therefore, the fixed space will be given by
$A_n/{\z_k}= (\underset{k}{\times}A_{l-1})\times T^{k-1}\rtimes\z_k.$
What we have shown is that in $A_n$ the
$\rm{Stab}_{\tau}(\underset{k}{\times}A_{l-1})=\z_k.$
The fact that $\pi_1(DZ(x_1))\cong \z_k$ where $1\le k\le 6,$ follows
directly from looking at the coroot integers for all the extended
Dynkin diagrams.
\end{proof}

\begin{proposition}
Let $G$ be a simple group of dimension $n.$ The centralizer
$Z(A_k\times_F T^{n-k})= Z_{A_k}\times_F T^{n-k}.$
\end{proposition}
\begin{proof}
Given $A_k\times_F T^{n-k},$ for any element $[A,t]=[A\zeta,\zeta^{-1}A]\in A_k\times_F T^{n-k}$
its centralizer is $Z([A,t])=\{[B,s]\colon [A,t][B,s]=[B,s][A,t]\}.$ This
implies that $[AB,ts]=[BA,st].$ But since $st=ta\in T,\ AB=BA.$
Therefore, $Z([A,t])=Z_{A_k}(A)\times_F T^{n-k}$ which is
connected and thus Proposition ~\ref{fundgp} applies. The conclusion
follows because the components in the almost direct product are
simply connected.
\end{proof}

\begin{proposition}\label{basic}
Given $G_1\times_F G_2$ where $G_1,G_2$ are subgroups of $G,\
F\subseteq \mathcal{C}G_1$ and $F\subseteq\mathcal{C}G_2$ and
$F\in(DG_1\cap DG_2).$ Then for $[a,b]\in G_1\times_F G_2,$
$$\{1\} \to F\to Z([a,b])\overset{\pi}{\rightarrow} Z_{G_1}(a)\times_F
Z_{G_2}(b)\rightarrow F.$$
\end{proposition}
\begin{proof}
Consider the map
$G_1\times_F G_2\overset{\pi}{\rightarrow} G_1/F\times G_2/F.$
The kernel is $\rm{ker}(\pi)=\{[c,d]\colon c,d\in F\}.$ Thus we
have the injective map
$\{1\}\to G_1\times_F G_2\overset{\pi}{\rightarrow} G_1/F\times G_2/F.$ By definition of the
centralizer of an element in $G_1\times_F G_2$
\begin{eqnarray*}
Z([a,b]) &=&\{[c,d]\colon [ac,bd]=[ca,db]\} \\
&=& \{\exists f_1,f_2\in F,\
[acf_1,bd]\sim [ac,f_1^{-1}bd]=[caf_2,db]\sim [ca,f_2^{-1}db]\} \\
&=&\{[c,d]\colon [a,c]=f,[b,d]=f^{-1},\ f=f_1f_2^{-1},\ [a,c][b,d]=1\}
\end{eqnarray*}
This demonstrates that the coker of $\pi$ is $F$ and that $\pi$
is not surjective. Therefore,
$\frac{\pi^{-1}(Z([a])\times Z([b]))}{F}=Z([a,b]).$
Note also that by the definition of the centralizer of
$[a,b]\in G_1\times_F G_2,$ that the generalized Stiefel-Whitney
class \cite{M00} is $w_2(a,c)= -w_2(b,d)\in F.$ Hence $w_2\colon H^2(T^*)\to {\bf Z}_n$
defines an obstruction.
\end{proof}

\begin{corollary}
Following propsition ~\ref{basic}, if
$G_1=T$ for some torus and $G_2$ is of type $A_r$ then
$$F\to Z_{A_r}(\tilde{A})\overset{\pi}{\rightarrow} Z_{A_r/F}([A])
\rightarrow F\to \{1 \}.$$
\end{corollary}
\begin{proof}
Given a sequence
$\{1\}\to T^k\hookrightarrow T^k\times_F A_r\overset{\pi}{\rightarrow}
A_r/F\,$ inside $Z_G([t,A])$ we have that
$[t,A][s,B]=[ts,AB]=[st,BA]=[s,B][t,A]$ and in
$Z_{A_r}([A])=\{[B]\colon [A,B]=[B,A]\}$ which implies that
$Z_G([t,A])\hookrightarrow \pi^{-1}(Z_{A_r}([A]))$ so that $AB=BA\zeta$
for $\zeta\in F.$ Thus they are equal up to an element in the finite
group. Therefore we have
$Z_G([t,A])\hookrightarrow \pi^{-1}(Z_{A_r}([A]))\to F.$
Suppose that $[A]\in A_r/F$ and consider its lift $\tilde{A}\in A_r$
arbitrary. Then
$$F\to Z_{A_r}(\tilde{A})\overset{\pi}{\rightarrow} Z_{A_r/F}([A])
\rightarrow F\to \{1 \}$$ because the kernel is
$\rm{Ker}(\pi)=F$ and from what we have already deduced, $AB=BA\zeta$
for $\zeta\in F$. Hence
$$Z_G([t,A])=\frac{\pi^{-1}(Z_{A_r}([A]))}{F}.$$
We used the simply connected component as follows. If we consider
$[B]$ such that there exists a $\tilde{B}$ with
$\tilde{A}\tilde{B}=\tilde{B}\tilde{A},$ then mulitiplication of the
equivalence classes is
$[s,\tilde{B}][t,\tilde{A}]=[st,\tilde{B}\tilde{A}=[ts,\tilde{A}\tilde{B}].$
Since $\pi_1(G)=\{ 1\}$ when we lift to the universal covering we can say that for
$[t,A]\in \tilde{G}=T^k\times_F G$ then $Z_{\tilde{G}}([t,A])=
T^k\times_F Z(A)$ and more importantly that $Z_{\tilde{G}}([t,A])$ is
connected.
\end{proof}

\begin{corollary}
Consider a subgroup in $G$ of the form $A_k\times_F A_r,$ for $r+k=n+1,$
then the centralizer of an element $[a,b]\in A_k\times_F A_r,$
for $r+k=n+1,$ is
$Z_G([a,b])=\frac{(Z_{A_k}(a)\times_{F'} Z_{A_r}(b))}{F},$ \ where \
$F'=\mathcal{C}DZ_{A_k}\cap\mathcal{C}DZ_{A_r}\supseteq F.$
\end{corollary}

\begin{corollary}
Consider a subgroup in $G$ of the form $A_k\times_F D_{n-k}.$
then the centralizer of an element $[a,b]\in A_k\times_F D_{n-k}$
is of the form
$$Z_G([a,d])=\frac{(Z_{A_k}(a)\times_{F'} Z_{AD_{n-k}}(d))}{F},$$ where
$F'=\mathcal{C}DZ_{A_k}\cap \mathcal{C}DZ_{D_{n-k}}\supseteq F.$
\end{corollary}

It does not necessarily follow that
$\pi_0(Z(x_1,\ldots,x_n)\subset\pi_1(DZ(x_1))$ because $DZ(x_1)$ is
not necessarily connected. The fact that for $G$ of type $D_n$ that
$\pi_1(D_n)=\mathcal{C}D_n\cong\z\times\z$ and that the
characteristic class for a principal $G$-bundle over $T^n$ lies in
$H^2(T^n;\pi_1(G))\cong\z\times\z$ means that there is a
possibility that the component group for an $n$-tuple inside $D_n$
will not be finite cyclic.

\section{Uniform Bound on Chains of Commuting Elements}

Assume that $G$ is a compact, connected Lie group.
Define a chain of elements $x_1,x_2,x_3,\ldots\in G$ where
$x_i\in Z_G(x_1,\ldots,x_{i-1})$ to be {\sl redundant} if for any
$x_i\in \mathcal{C}Z_G(x_1,\dots,x_{i-1})$ then
$Z_G(x_1,\ldots,x_i)=Z_G(x_1,\ldots,x_{i-1}).$ Otherwise, the
chain is said to be {\sl irredundant}.

Define an {\sl ordering} of the $n$-tuple by the property
$\rm{dim}\ Z(x,x)\ge \rm{dim}\ Z(x,x,y)$ for $x\ne y.$
Thus for a chain of commuting elements $x_1,x_2,\ldots\in G$
if we consider a decreasing chain of the centralizers of these
elements,
$$G\supset Z(x_1)\supset Z(x_1,x_2)\supset\cdots\supset
Z(x_1,x_2,\ldots,x_n)=Z(x_1,x_2,\ldots,x_n,t_1,\ldots t_k),$$
for $t_i\in CG$, the chain of strict inclusions will terminate
once the chain of elements becomes redundant, for any further
choices of $x_i\in G.$. This is equivalent to saying that the
chain of decreasing centralizers will terminate once  the
centralizer of an $n$-tuple $Z(x_1,\ldots,x_n)$
becomes abelian; this will occur if the centralizer is either the
maximal torus or a finite group. It is clear that if $x$ is a regular
element then $Z(x)=T$ and the chain terminates immediately.
Define the {\bf cardinality} $m$ of the centralizer
to be $m(Z(x_1,\ldots,x_n))=n$ where $n$ is the maximal length of an
irredundant chain.To compute $m(G)$ for $G,$ we determine the centralizer $Z_G(x_1,\ldots,x_n)$ for different
commuting $n$-tuples
in some numerical manner and relate the centralizer to certain nodes
in a subdiagram of the extended Dynkin diagram, where the subdiagram
is given by a closed subset $I\subseteq \tilde\Delta$ of roots which
annihilate all the elements in the commuting $n$-tuple.
Define a {\bf c-tuple} to be an $n$-tuple $(x_1,\ldots,x_n)$
such that $[x_1,x_2]=c$ and $[x_i,x_1]=[x_i,x_2]=1$ for $i>2.$

Fix a maximal torus $T$ in $G$ and let $A$ the alcove over the torus.
Then there is an affine linear transformation of the vector space
$\mathfrak{h}$ defined as follows:
$\phi\colon A\to A$ by $w_c(t-\zeta_{c^{-1}})$
where $t\in\mathfrak{h},\ w_c\in\mathcal{W}$ is the projection onto
the Weyl group from an element $y$ in a $c$-pair $(x,y)$ which normalizes
the centralier $Z(x),$ and $\rm{exp}(\zeta_{c^{-1}})=c^{-1}\in
\mathcal{C}G.$ The fixed space of this automorphism of the alcove
is the subspace $A^c=A\cap\mathfrak{h}^{w_c}.$ Since the alcove $A$
is the fundamental domain and clearly $A^c$ contains the barycenter,
then $A^c$ lies in the connected component of the identity.

\begin{lemma}
Let $G$ be a compact, connected Lie group of rank $n$ and
$A\in\mathfrak{h}$ be
the alcove over the maximal torus. Then for any $\tilde{x}\in A^c$ such
that $\rm{exp}(\tilde{x})=x\in T,$ it is sufficient to consider that
either $\tilde{x}$ is a vertex in the fixed subspace $A^c$ under the
central action of $c\in\mathcal{C}\tilde{G}$ or that $\tilde{x}$ lies on
an edge
whose vertices are both central elements in order to get a subgroup of
maximal rank in $G.$
\end{lemma}
\begin{proof}
The case when $G$ is of type $A_n$ is unique in that the fixed space
under the central action is the barycenter. Thus the only choice for
$\tilde{x}\in A^c$ is the barycenter and its centralizer is given by
$\tilde{Z}(x)=\mathfrak{h}\rtimes\mathcal{C}\tilde{G}.$

Every $c$-pair
is conjugate to one in normal form which means that $\tilde{x}$
corresponds to the barycenter in $A^c$ and $\tilde{y}$ projects to an
element in the Weyl group $\mathcal{W}.$ Since a $c$-tuple is defined
by a $c$-pair and then $n-2$ commuting elements, we may always choose
$\tilde{x}$ to be the barycenter in $A^c$ and therefore it will lie
in the connected component of the identity. Thus it follows from the
commuting case that the maximal subgroup always corresponds to choosing
$\tilde{x}$ as a vertex in $A^c.$

As a result of the commuting case, it's necessary to choose
$\tilde{x}\in A^c$ which is connected. However, when working with
$\langle\mathcal{C}\rangle$-tuples, we may choose $\tilde{x}$ to lie
either in $A^c$ or not. Elements in the connected component of the
identity of the centralizer follow from the case for commuting $n$-tuples.
 The center $\mathcal{C}\tilde{G}$ is a finite group,
thus the key is to always follow and extremal path when choosing elements
in the fixed space $A^c.$
\end{proof}

\begin{corollary}
Let $\ov{x}=(x_1,\ldots,x_n)$ be an $n$-tuple for a compact, connected Lie
group $G$. Assume that the $n$-tuple is
irreducible. Then it is sufficient to only consider the semisimple part
when determining the moduli space.
\end{corollary}
Take $Z_G(\ov{x})$ and look at $S_{max}\subseteq Z_G(\ov{x}).$ Then take
the centralizer of this small maximal torus
and consider its maximal torus $S\subseteq Z_G(S_{max}).$ Then
$Z_G(S_{max})=S_{max}$ which implies that $S_{max}$
is semisimple because it's a torus so communtes with the maximal torus $S$.
Thus there exists a $g\in Z_G(S_{max})$
such that $gS_{max}g^{-1}=S.$ But by definition  $S=S_{max}.$

The following theorem states that there exists a uniform upper bound
on irredundant chains of commuting elements in $G.$
\begin{theorem}\label{Weaktheorem} (Weak Theorem)
Given a compact Lie group $G$ there exists an integer $m=m(G)$
such that any irredundant chain has fewer than $m$ elements.
\end{theorem}

\begin{theorem}\label{Strongtheorem} (Strong Theorem)
Given $G$ a compact Lie group, there exists a finite number
of subgroups $H_1,\ldots,H_k\subset G$ such that for any $x\in G,$
$Z_G(x)$ is conjugate to $H_i$ for some $i.$
\end{theorem}
\begin{proof} (Strong Theorem)
Since $G$ is compact, there is a decomposition
$G=(\mathcal{C}G)^0\cdot_F DG$ where $F=(\mathcal{C}G)^0\cap DG$
is a finite group. Choosing $x\in (\mathcal{C}G)^0$ then $Z_G(x)=G$
hence it suffices to choose $x\in DG.$ If $x\in T$ is a non-central
element for a choice of a maximal torus $T\subset G,$ then since $T$ is
abelian and $x$ is regular, $Z(x)=T$ is the smallest possible
subgroup of maximal rank. We claim that there are a finite
number of closed subgroups between $G$ and $T$ such that each is
of the form $Z(x_i)$ for some $x_i\in G.$

When $\tilde{x}$ corresponds to a vertex
or an edge with two central vertices, the element yields a maximal rank
subgroup. Thus choose $x_1$ as image under the exponential map of
a non-central vertex. Then $H_1=Z(x_1)=DZ(x_1)\cdot\mathcal{C}^0Z(x_1).$
If $Z(x_1)$ is connected
then we may choose $\tilde{x_2}$ as a vertex or a "central edge" which
gives $H_2=Z(x_1,x_2)$ where the rank of the semisimple part of the
centralizer decreases by at least one. In other words, continuing to
choose vertices or "central edges" such that the rank decreases, we
are following a connected path and at some point there will be some
$H_k$ such that the rank of the semisimple part of $H_k$ is zero.
Thus $H_k$ is abelian and we have proven the claim.

If $Z(x_1)$ is not connected, then we may restrict to the connected
component of the identity which is still reductive and the above
analysis holds as long as there is no choice of element $x_j$ which
is fixed by the finite abelian subgroup of the center of
$H_{j-1}=Z(x_1,\ldots,x_{j-1}).$ Otherwise, if some $x_j\in
Z(x_1,\ldots,x_{j-1})$
is fixed by this diagonal subgroup $\Delta$ then
$H_j=Z(x_1\ldots,x_j)=DZ(x_1,\ldots,x_j)\cdot \mathcal{C}Z\rtimes\Delta$
is disconnected. If the semisimple part is not abelian, then we must
choose elements in the semisimple part until it becomes abelian. At
the point in which it does, say some $H_{k-1}$ where $1\le j < k-1,$
choosing $x_k\in\Delta$ makes $H_k$ into a finite abelian group and
thus we have again shown that there are only a finite number of
normal subgroups $H_1,\ldots,H_k$ in $G$ which are conjugate to the
centralizer of some number of elements.

$H_1=DZ(x)$ is a normal simple subgroup in $DG$ which is characteristic
in $DG$ because it lies in the center of $DG$ and the center is always
characteristic for any group. $H_1$ is also closed since it is a semisimple subgroup
in $G$ compact. If $x$ is a semisimple element, then it is
conjugate to an element in the torus $T$ which implies that the
connected component $Z^0(x)$ contains a torus $T'$ which is
conjugate to $T.$ In other words,
$Z^0(x)=DZ^0(x)\cdot\mathcal{C}Z^0(x)$ where the
semisimple part corresponds to a subsystem subgroup and the other
is a commuting torus. To consider the remaining cases, we use the algorithm by
Borel and de Siebenthal (Theorem 3 \cite{BS49}) which determines all the closed subsystems
of $\Phi(G)$ (all the roots of $G$) for closed
subgroups of maximal rank. Take the extended
Dynkin diagram $\tilde{D}(G)$ and remove a finite collection of
nodes, repeating the process with all the connected components of the
resultant graph. These diagrams correspond to subdiagrams of the
subroot systems and there is one conjugacy class of subsystem
subgroups for each such subsystem. In particular, if $G$ is connected,
compact then it possesses only finitely many conjugacy classes of subsystem
subgroups and hence the theorem is satisfied. It remains to prove what happens if $G$ is disconnected
because a direct product of connected subgroups will each
have a finite number of subgroups each conjugate to $Z(x_i)$ for some
$x_i\in G.$ Note that there does not necessarily have to be any ordering on
these $H_i.$ In fact, it would be virtually impossible to impose
such a requirement because the $H_i$ are not necessarily
subgroups of each other.
If $G$ is the semidirect product of a connected group $U$ and a discrete
group $V$ then the above applies to the connected part and the only
centralizer for any of the elements in the discrete subgroup is the
entire discrete group. Thus there is a still finite number. If
$G$ is disconnected, then there is a finite number of subgroups of
type $Z_G(x)$ when restricting to the connected component of the
identity.

If $G$ is a disconnected group where both the clopen sets $U,V$
are connected, then there is still a finite number. If $U$ is connected and
$V$ is finite and disconnected, then count the number of such $H_i$ in each
of the smaller connected sets, ultimately leaving a disconnected set with exactly one point whose
centralizer is itself, since it is totally disconnected.
If $U$ is also totally disconnected there are a finite number of points $u_i\in U$ because $G$ is compact and
thus $H_i$ will be conjugate to the  $u_i=Z(u_i).$
If $G=U\rtimes V$ where both $U,V$ are connected then there
exist $H_1,\ldots, H_l$ for $U$ and $K_1\ldots, K_r$ for $V$
such that each $H_i$ is conjugate to $Z(u_i)$ and $K_j$ is
conjugate to $Z(v_j),\ 1\le i\le l,\ 1\le j\le r.$
\end{proof}

For a simply connected, connected Lie group, the following
shows how theorem ~\ref{Strongtheorem} implies
theorem ~\ref{Weaktheorem}.

\begin{proposition}
If $G$ is a compact, connected, simply connected Lie group and
there are a finite number of subgroups $H_1,\ldots,H_k\subset G$
such that $Z_G(x)\cong H_i$ for some $i$ and for any $x\in G,$
then there is an integer $m$ such that any irredundant chain has
fewer than $m$ elements.
\end{proposition}
\begin{proof}
Since $G$ is connected, it is sufficient to choose $x\in T$
such that
$x={\rm exp}(\tilde{x})$ where $\tilde{x}$ is in the alcove $A.$
Since we are only dealing with commuting elements in the chain,
we are concerned with what happens in the walls of the alcove.
The proof is by induction. Define a pair of numbers
$(d,n)$ lexicographically where $d$ represents the dimension of
the subgroup and $n$ represents the number of components.
It is vacuously true for dimension one (we could choose $S^1$
if we wanted to make it slightly less trivial since $S^1=T^1$ is
the $1$-dimensional torus in the Lie group). Suppose that the
Weak theorem is true, i.e. that for every $H\subsetneqq G$ there
exists and $m$ for any irredundant chain. Note that if we let
$(x_1,\ldots,x_k)$ be an irredundant chain in $G$ such
that $Z_G(x_1)=H,\ x_1\not\in\mathcal{C}G,$ then since we have a
chain of commuting elements, $(x_2,\ldots,x_k)$ is an irredundant
chain in $H$ which, by induction, implies that $k\le m(H) + 1.$ Hence
any irredundant chain in $G$ has length
$m \le 1 + {\rm max}(m(H))$ where the maximum is over the
subgroups $H$ strictly contained in $G$ and where $H=Z_G(x)$ for
some element $x\in G.$

Now consider the connected component of the identity, $H^0\subset G^0.$
By the lexicographic ordering, ${\rm dim}(H^0)\le {\rm dim}(G^0)$
if either ${\rm dim}(H^0)<{\rm dim}(G^0)$ or their dimensions are
equal and $n\le n'.$ But
${\rm dim}(H^0)={\rm dim}(G^0)$ if and only if $H^0=G^0.$ So
$G^0\subset H\subsetneqq G$
which implies that the components $\pi_0(H)\underset{\neq}{<}\pi_0(G).$
Thus we have shown that the Strong theorem implies the Weak theorem.
\end{proof}

\def\cprime{$'$}
\def\p{{\mathbb P}}

config: arxiv


text:
\begin{document}
\title{\LARGE\bf Lifespan of solutions to a damped fourth-order wave equation with logarithmic nonlinearity}
\author{Yuzhu Han$^{\dag}$ \qquad Qi Li}
 \date{}
 \maketitle

\footnotetext{\hspace{-1.9mm}$^\dag$Corresponding author.\\
Email addresses: yzhan@jlu.edu.cn(Y. Han).

\thanks{
$^*$Supported by NSFC (11401252) and by The Education Department of Jilin Province (JJKH20190018KJ).}}
\begin{center}
{\noindent\it\small School of Mathematics, Jilin University,
 Changchun 130012, P.R. China}
\end{center}

\date{}
\maketitle

{\bf Abstract}\ This paper is devoted to the lifespan of solutions to a damped fourth-order wave equation
with logarithmic nonlinearity
$$u_{tt}+\Delta^2u-\Delta u-\omega\Delta u_t+\alpha(t)u_t=|u|^{p-2}u\ln|u|.$$
Finite time blow-up criteria for solutions at both lower and high initial energy levels are established,
and an upper bound for the blow-up time is given for each case.
Moreover, by constructing a new auxiliary functional and making full use of the strong damping term,
a lower bound for the blow-up time is also derived.

{\bf Keywords} Lifespan; Damped; Fourth-order wave equation;  Logarithmic nonlinearity; Initial energy.

{\bf AMS Mathematics Subject Classification 2010:} 35L35, 35L76.

\section{Introduction}
\setcounter{equation}{0}

In this paper, we are concerned with the following initial boundary value
problem for a damped fourth-order wave equation with logarithmic nonlinearity
\begin{equation}\label{1.1}
\begin{cases}
u_{tt}+\Delta^2u-\Delta u-\omega\Delta u_t+\alpha(t)u_t=|u|^{p-2}u\ln|u|, & (x,t)\in\Omega\times(0,T),\\
u(x,t)=\Delta u(x,t)=0, & (x,t)\in\partial \Omega\times(0,T),\\
u(x,0)=u_0(x), \quad u_t(x,0)=u_1(x), &x\in \Omega,
\end{cases}
\end{equation}
where $\Omega\subset \mathbb{R}^n(n\geq1)$ is a bounded domain with smooth boundary $\partial \Omega$,
$T\in(0,+\infty]$ is the maximal existence time of the solution $u(x,t)$, $\omega>0$,
$\alpha(t): [0,\infty)\rightarrow [0,\infty)$ is a nonincreasing bounded differentiable function,
and the exponent $p$ satisfies
$$(A)\qquad 2<p<2_*,$$
where $2_*=+\infty$ if $n\leq 4$ and $2_*=\dfrac{2n}{n-4}$ if $n\geq 5$.

Problems like \eqref{1.1} have their roots in many branches of physics such as nuclear physics, optics and geophysics.
They may also be used to describe some phenomena of granular materials such as the longitudinal motion
of an elastic-plastic bar. Interested reader may refer to \cite{Al,An-1,An-2,An-3,Guo-Li}
for more background of problems like \eqref{1.1}.
It is well known that the damping terms (both strong $\Delta u_t$ and weak $u_t$)
prevent solutions from blowing up while the nonlinear terms force solutions to blow up.
So it is of great interest to investigate how one dominates the other,
and much effort has been devoted to this direction during the past few years.
For example, Gazzola et al. \cite{Gazzola} investigated the following damped wave equation
\begin{equation}\label{sec}
 u_{tt}-\Delta u-\omega\Delta u_t+\mu u_t=|u|^{p-2}u
\end{equation}
in a bounded domain of $\mathbb{R}^n$, where $\omega\geq 0$, $\mu>-\omega\lambda_1$ and $p>2$.
By using the potential well method first proposed by Sattinger et al. \cite{Payne,Sattinger},
they obtained the existence of global and finite time blow-up solutions to \eqref{sec} for
initial data at different energy levels. As for the damped fourth-order wave equations,
Lin et al. \cite{Lin} considered the following hyperbolic equation with strong damping
\begin{equation}\label{fou-1}
 u_{tt}+\Delta^2u-\Delta u-\omega\Delta u_t=f(u)
\end{equation}
in a bounded domain of $\mathbb{R}^n$ with $\omega>0$.
Under certain conditions on the initial data and on the nonlinearity $f$,
they proved the existence of global weak solutions and global strong solutions
by using the classical potential well method. When the nonlinearity $f(u)$ grows super-linearly with respect to $u$ as $u$ tends to infinity,
the solutions to \eqref{fou-1} may blow up in finite time.
In 2018, Wu \cite{Wu} considered the following initial boundary value problem
\begin{equation}\label{fou-2}
\begin{cases}
u_{tt}+\Delta^2u-\Delta u-\omega\Delta u_t+\alpha(t)u_t=|u|^{p-2}u, & (x,t)\in\Omega\times(0,T),\\
u(x,t)=\Delta u(x,t)=0, & (x,t)\in\partial \Omega\times(0,T),\\
u(x,0)=u_0(x), \quad u_t(x,0)=u_1(x), &x\in \Omega,
\end{cases}
\end{equation}
where $\omega$ and $\alpha(t)$ fulfill the same conditions as that of problem \eqref{1.1} and
$p$ satisfies the so-called subcritical condition, i.e.,
\begin{equation*}
 p\in(2,\infty)\quad if\quad  n\leq 4;\quad p\in\big(2,\frac{2n-4}{n-4}\big)\quad  if \quad n\geq 5.
\end{equation*}
After showing that the unstable set is invariant under the flow of \eqref{fou-2},
he proved a blow-up result for problem \eqref{fou-2} with initial energy smaller than the depth of the
potential well, by applying concavity argument. Moreover, a lower bound for the blow-up time is derived.
Later, problem \eqref{fou-2} was reconsidered by Guo et al. \cite{Guo-Li} and
the results of \cite{Wu} were extended in two aspects. The first is that they obtained a blow-up result
for high initial energy, and the second is that lower bound for the blow-up time is also derived
for some supercritical $p$, with the help of inverse H\"{o}lder's inequality and interpolation inequality.

On the other hand, evolution equations with logarithmic nonlinearity have
also attracted more and more attention in recent years,
due to their wide applications to quantum field theory and other applied sciences.
Among the huge amount of interesting literature,
we only refer the interested reader to \cite{Cao1,Chen1,Chen2,Di,Han,Han-Cao-Sun,Ji1,Le1,Le2,Lian,Ma},
where qualitative properties of solutions to hyperbolic or parabolic equations with logarithmic nonlinearities were studied.
In particular, Di et al. \cite{Di} considered the following initial boundary value problem for a semilinear wave equation
with strong damping and logarithmic nonlinearity
\begin{equation}\label{sec-2}
\begin{cases}
u_{tt}-\Delta u-\Delta u_t=|u|^{p-2}u, & (x,t)\in\Omega\times(0,T),\\
u(x,t)=0, & (x,t)\in\partial \Omega\times(0,T),\\
u(x,0)=u_0(x), \quad u_t(x,0)=u_1(x), &x\in \Omega,
\end{cases}
\end{equation}
when $\Omega\subset \mathbb{R}^n(n\geq1)$ is a bounded domain with smooth boundary $\partial \Omega$,
$2<p<+\infty$ if $n=1,2$ and $2<p<\frac{2n}{n-2}$ if $n\geq 3$.
The existence of global or finite time blow-up solutions to problem \eqref{sec-2} with initial energy less
than or equal to the depth of the potential well was investigated by using the potential well method.
Moreover, the decay rate of the energy functional was obtained for global solutions
and upper and lower bounds for the blow-up time were also derived for blow-up solutions.
However, the case that the initial energy is larger than the depth of the potential well was not considered in \cite{Di},
and we do not know whether or not problem \eqref{sec-2} admits finite time blow-up solutions for this case.
In addition, the lower bound for the blow-up time was obtained when $p$ is subcritical,
i.e., $p<\frac{2n-2}{n-2}$. When $p\in[\frac{2n-2}{n-2},\frac{2n}{n-2})$ for $n\geq 3$,
whether a lower bound for the blow-up time can be obtained is still open.

Motivated mainly by \cite{Di,Guo-Li,Wu}, we will consider problem \eqref{1.1} and investigate how the damping terms and logarithmic
nonlinearity determine the blow-up conditions and blow-up time of the solutions.
More precisely, we shall present some sufficient conditions for the solutions to problem \eqref{1.1} to blow up in
finite time with both lower and high initial energy and derive an upper bound for the blow-up time for each case.
Moreover, we also estimate a lower bound for the blow-up time, which, thanks to the strong damping term, also includes some supercritical case.
For simplicity, we only consider \eqref{1.1} for the case $\omega=1$ and $\alpha(t)\equiv 1$.
The main results can be extended to the general case with little difficulty.

The organization of this paper is as follows. In Section 2, as preliminaries,
some notations, definitions and lemmas that will be used in the sequel are introduced.
Finite time blow-up of solutions and upper bound for the blow-up time with lower and
high initial energy will be considered in Section 3 and Section 4, respectively.
In Section 5 we derive a lower bound for the blow-up time.

\section{Preliminaries}
\setcounter{equation}{0}

In this section, we introduce some notations and lemmas which will be used in the sequel.
In what follows, we denote by $\|\cdot\|_r$ the $L^r(\Omega)$-norm ($1\leq r\leq\infty$),
by $(\cdot ,\cdot )$ the $L^2(\Omega)$-inner product
and by $\lambda_1>0$ the first eigenvalue of $-\Delta$ in $\Omega$ under homogeneous Dirichlet boundary condition.
 Set
$$H=\{u\in H^2(\Omega)\cap H_0^1(\Omega):\ u=\Delta u=0\  \text{on}\  \partial\Omega\},$$
and equip it with the norm
$$\|u\|_H=\sqrt{\|\Delta u\|_2^2+\|\nabla u\|_2^2}.$$
For simplicity, we also denote the $H^1(\Omega)$-norm by
$$\|u\|=\sqrt{\|u\|_2^2+\|\nabla u\|_2^2}.$$
Obviously, for any $u\in H$, we have
\begin{equation}\label{lam}
 \lambda_1\|u\|^2\leq \|u\|_H^2.
\end{equation}
For any $u\in H$, define
\begin{equation}\label{j}
J(u)=\dfrac{1}{2}\|u\|_
  H^2-\dfrac{1}{p}\int_\Omega|u|^p\ln|u|{\rm d}x+\dfrac{1}{p^2} \|u\|_p^p,
\end{equation}
\begin{equation}\label{i}
I(u)=\|u\|_H^2-\int_\Omega|u|^p\ln|u|{\rm d}x,
\end{equation}
\begin{equation}\label{N}
\mathcal{N}=\{u\in H\setminus\{0\}: I(u)=0\},
\end{equation}
\begin{equation}\label{d}
d=\inf\limits_{u\in H\setminus\{0\}}\sup\limits_{\lambda>0}J(\lambda u)=\inf\limits_{u\in \mathcal{N}}J(u),
\end{equation}
where $\mathcal{N}$ is called the Nehari manifold and $d$ is the depth of the potential well (also called mountain pass level). In what follows, we shall show that $\mathcal{N}$ is non-empty and $d$ is positive.

The following lemma gives some properties of the so-called fibering map $J(\lambda u)$.
Since the proof is more or less standard (see \cite{Di} for example), we omit it here.

\begin{lemma}\label{fibering}
Let $p$ satisfy (A). Then for any $u\in H\setminus\{0\}$, we have
\begin{eqnarray*}
\mathrm{(i)} \ \lim_{\lambda\rightarrow0^+}J(\lambda u)=0, \ \lim_{\lambda\rightarrow+\infty}J(\lambda u)=-\infty. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{eqnarray*}

$\mathrm{(ii)}$ \ there exists a unique $\lambda^*=\lambda^*(u)>0$ such that $\frac{d}{d\lambda}J(\lambda u)|_{\lambda=\lambda^*}=0$.
$J(\lambda u)$ is increasing on
$0<\lambda<\lambda^*$, decreasing on $\lambda^*<\lambda<+\infty$ and takes its maximum at $\lambda=\lambda^*$.

$\mathrm{(iii)}$ $I(\lambda u)>0$ on $0<\lambda<\lambda^*$, $I(\lambda u)<0$ on $\lambda^*<\lambda<+\infty$ and $I(\lambda^* u)=0$.
\end{lemma}

Let $\sigma$ be any positive number such that $p+\sigma<2_*$.
Then it is well known that the embedding from $H$ to $L^{p+\sigma}(\Omega)$ is compact and
there is a positive constant $B_\sigma$ such that
\begin{equation}\label{embedding}
  \|u\|_{p+\sigma}\leq B_\sigma\|u\|_H,\qquad \forall\ u\in H.
\end{equation}

\begin{lemma}\label{c0}
There is a positive constant $C_*$ such that $\|u\|_H\geq C_*$ for any $u\in \mathcal{N}$.
\end{lemma}

\begin{proof}
First, it follows from Lemma \ref{fibering} (iii) that $\mathcal{N}$ is non-empty.
For any $u\in \mathcal{N}$, using \eqref{embedding} and the basic inequality $\ln s\leq\dfrac{1}{e\sigma}s^\sigma$
for $s\geq 1$ and $\sigma>0$, we have
  \begin{equation}\label{2.6}
   \begin{split}
     \|u\|_H^2=&\int_\Omega|u|^p\ln|u|{\rm d}x=\int_{\Omega_1}|u|^p\ln|u|{\rm d}x+\int_{\Omega_2}|u|^p\ln|u|{\rm d}x\\
     \leq&\int_{\Omega_2}|u|^p\ln|u|{\rm d}x\leq \dfrac{1}{e\sigma}\int_{\Omega_2}|u|^{p+\sigma}{\rm d}x\\
     \leq&\dfrac{1}{e\sigma}\|u\|_{p+\sigma}^{p+\sigma}\leq \dfrac{B^{p+\sigma}_\sigma}{e\sigma}\|u\|_{H}^{p+\sigma},
   \end{split}
  \end{equation}
where $\Omega_1=\{x\in\Omega: |u(x)|<1\}$ and $\Omega_2=\{x\in\Omega: |u(x)|\geq1\}$.
Recalling that $p>2$, we obtain from \eqref{2.6} that $\|u\|_H\geq\Big(
\dfrac{e\sigma}{B^{p+\sigma}_\sigma}\Big)^{1/(p+\sigma-2)}\triangleq C_*$.
The proof is complete.
\end{proof}

\begin{lemma}\label{depth}
  The depth $d$ of the potential well is positive and
  there is a nonnegative function $v_0\in \mathcal{N}$ such that $J(v_0)=d$.
\end{lemma}

\begin{proof}
  By \eqref{j} and \eqref{i} we have
  \begin{equation}\label{i-j}
    J(u)=\dfrac{p-2}{2p}\|u\|^2_H+\dfrac{1}{p}I(u)+\dfrac{1}{p^2}\|u\|_p^p, \quad\ u\in H.
  \end{equation}
Therefore, for any $u\in \mathcal{N}$, by combining Lemma \ref{N} with \eqref{i-j} we obtain
 \begin{equation}\label{d0}
 J(u)\geq \dfrac{p-2}{2p}\|u\|^2_H\geq \dfrac{p-2}{2p}C_*^2 \triangleq d_0>0.
\end{equation}
By the definition of $d$ one sees that $d\geq d_0$, i.e., $d$ is positive.

To show that $d$ can be attained,
let $\{v_k\}_{k=1}^\infty\subset\mathcal{N}$ be a minimizing sequence of $J$.
It is easy to check that $\{|v_k|\}_{k=1}^\infty\subset\mathcal{N}$ is also a minimizing sequence of $J$.
Therefore, without loss of generality, we may assume that $v_k\geq 0$ a.e. in $\Omega$ for all $k\in \mathbb{N}$.
Then $J(v_k)$ is bounded, which, together with \eqref{i-j}, implies
that $\{v_k\}_{k=1}^\infty\subset\mathcal{N}$ is bounded in $H$.
Noticing that the embedding from $H$ to $L^{p+\sigma}(\Omega)$ is compact,
we see that there is a subsequence of $\{v_k\}_{k=1}^\infty\subset\mathcal{N}$,
which we still denote by $\{v_k\}_{k=1}^\infty\subset\mathcal{N}$, and a $v_0\in H$ such that
\begin{equation*}
\begin{split}
v_k\rightharpoonup v_0 \ & weakly \ in \ H \ as\ k \rightarrow\infty,\\
v_k\rightarrow v_0 \ & \ strongly \ in \ L^{p+\sigma}(\Omega) \ as\ k \rightarrow\infty,\\
v_k\rightarrow v_0 \ & \ a.e. \ in \ \Omega \ as\ k \rightarrow\infty.
\end{split}
\end{equation*}
Hence, $v_0\geq 0$ a.e. in $\Omega$. Furthermore, using the dominated convergence theorem, we obtain
\begin{equation}\label{ln}
\int_\Omega |v_0|^p\ln|v_0|{\rm d}x=\lim\limits_{k\rightarrow \infty}\int_\Omega |v_k|^p\ln|v_k|{\rm d}x,
\end{equation}
\begin{equation}\label{vp}
\int_\Omega |v_0|^p{\rm d}x=\lim\limits_{k\rightarrow \infty}\int_\Omega |v_k|^p{\rm d}x.
\end{equation}
Moreover, by the weak lower semicontinuity of $\|\cdot\|_H$, we have
\begin{equation}\label{h}
\|v_0\|_H\leq\liminf\limits_{k\rightarrow\infty}\|v_k\|_H.
\end{equation}
Therefore, it follows from \eqref{ln}-\eqref{h} that
\begin{equation}\label{jv0}
J(v_0)\leq\liminf\limits_{k\rightarrow\infty}J(v_k)=d,
\end{equation}
and
\begin{equation}\label{iv0}
I(v_0)\leq\liminf\limits_{k\rightarrow\infty}I(v_k)=0.
\end{equation}

It remains to show that $v_0\not\equiv 0$ and $I(v_0)=0$ to complete the proof.
By \eqref{ln} and Lemma \ref{c0} we know
\begin{equation*}
 \begin{split}
    \int_\Omega |v_0|^p\ln|v_0|{\rm d}x=&\lim\limits_{k\rightarrow \infty}\int_\Omega |v_k|^p\ln|v_k|{\rm d}x\\
      =&\lim\limits_{k\rightarrow \infty}\|v_k\|_H^2\geq C_*^2,
 \end{split}
\end{equation*}
which implies that $v_0\not\equiv 0$.

If $I(v_0)<0$, then by Lemma \ref{fibering} (iii) we know that there exists a $\lambda^*\in(0,1)$ such that
$I(\lambda^*v_0)=0$, i.e., $\lambda^*v_0\in\mathcal{N}$. By the definition of $d$, we see
\begin{equation*}
  \begin{split}
     d\leq&J(\lambda^*v_0)=\dfrac{(p-2)\lambda^{*2}}{2p}\|v_0\|_H^2+\dfrac{\lambda^{*p}}{p^2}\|v_0\|_p^p \\
       =&\lambda^{*2}\Big[\dfrac{p-2}{2p}\|v_0\|_H^2+\dfrac{\lambda^{*p-2}}{p^2}\|v_0\|_p^p\Big]\\
       <&\lambda^{*2}\Big[\dfrac{p-2}{2p}\|v_0\|_H^2+\dfrac{1}{p^2}\|v_0\|_p^p\Big]\\
       \leq&\lambda^{*2}\liminf\limits_{k\rightarrow\infty}\Big[\dfrac{p-2}{2p}\|v_k\|_H^2+\dfrac{1}{p^2}\|v_k\|_p^p\Big]\\
       =&\lambda^{*2}\liminf\limits_{k\rightarrow\infty}J(v_k)=\lambda^{*2}d,
  \end{split}
\end{equation*}
a contradiction. Therefore, $I(v_0)=0$ and $v_0\not\equiv 0$, which means that $v_0\in\mathcal{N}$.
Recalling \eqref{jv0} and the definition of $d$ again one sees that $J(v_0)=d$.
The proof is complete.
\end{proof}

In this paper, we consider weak solutions to problem \eqref{1.1}.
For completeness, we state, without proof, the local existence theorem
which can be established by slightly modifying the argument in \cite{Liu-Xu}.
Sometimes $u(x,t)$ will be simply written as $u(t)$ if no confusion arises.

\begin{theorem}\label{local-solution}
(\cite{Guo-Li,Wu}) Let $u_0\in H$ and $u_1\in L^2(\Omega)$. Then the problem \eqref{1.1} admits a unique weak solution
$u\in L^\infty(0,T_0;H)$, $u_t\in L^\infty(0,T_0;L^2(\Omega))\cap L^2(0,T_0;H_0^1(\Omega))$, for $T_0>0$ suitably small.
Moreover, the energy functional satisfies
\begin{equation}\label{ei}
 E'(t)=-\int_\Omega (u_t^2+|\nabla u_t|^2){\rm d}x\leq 0,
\end{equation}
where
\begin{equation}\label{e}
 E(t)=E(u(t))=\dfrac{1}{2}\|u_t\|_2^2+J(u(t)).
\end{equation}
\end{theorem}

At the end of this section, we present the well-known concavity lemma
which will play essential role in proving the blow-up result.

\begin{lemma}\label{concave}(See \cite{Han2018,Levine})
Suppose that a positive, twice-differentiable function $\psi(t)$ satisfies the inequality
$$\psi''(t)\psi(t)-(1+\theta)(\psi'(t))^2\geq0,$$
where $\theta>0$. If $\psi(0)>0$, $\psi'(0)>0$, then $\psi(t)\rightarrow\infty$ as $t\rightarrow t_*\leq t^*=\frac{\psi(0)}{\theta\psi'(0)}$.
\end{lemma}

\section{Blow-up for lower initial energy}
\setcounter{equation}{0}

In this section, we will investigate the blow-up phenomena of solutions to problem \eqref{1.1}
with lower initial energy. We first show that the unstable set $\mathcal{U}$ is invariant under the
flow of problem \eqref{1.1}, where
\begin{equation}\label{u}
\mathcal{U}=\{u\in H: I(u)<0,\ J(u)<d\},
\end{equation}
and $d$ is the depth of the potential well defined in \eqref{d}.

\begin{lemma}\label{inv}
  Let $u_0\in \mathcal{U}$ and $u_1\in L^2(\Omega)$ such that $E(0)<d$.
  Then $u(t)\in \mathcal{U}$ for all $t\in[0,T)$ and
  \begin{equation}\label{lower-bound-in-u}
   \dfrac{p-2}{2p}\|u(t)\|_H^2+\dfrac{1}{p^2}\|u(t)\|_p^p>d,\quad\forall\ t\in[0,T).
  \end{equation}
\end{lemma}

\begin{proof}
First, it follows from \eqref{j}, \eqref{e} and \eqref{ei} that
$$J(u(t))\leq E(t)\leq E(0)<d,\quad\forall\  t\in[0,T).$$
Therefore, in order to prove $u(t)\in \mathcal{U}$ for all $t\in[0,T)$,
it suffices to show that $I(u(t))<0$ for all $t\in[0,T)$.
Assume by contradiction that there exists a $t_1\in(0,T)$ such that $u(t_1)\in \mathcal{N}$.
Then by the variational definition of $d$, we obtain
$$d\leq J(u(t_1))\leq E(t_1)\leq E(0)<d,$$
a contradiction.

For any $t\in[0,T)$, since $I(u(t))<0$, it follows from Lemma \ref{fibering} (iii) that there exists a  $\lambda(t)\in(0,1)$ such that $I(\lambda(t)u(t))=0$, i.e., $\lambda(t)u(t)\in \mathcal{N}$.
By the definition of $d$ and \eqref{i-j}, we have
\begin{equation*}
  \begin{split}
  \dfrac{p-2}{2p}\|u(t)\|_H^2+\dfrac{1}{p^2}\|u(t)\|_p^p\geq& \dfrac{(p-2)\lambda^2(t)}{2p}\|u(t)\|_H^2+\dfrac{\lambda^p(t)}{p^2}\|u(t)\|_p^p\\
  =&J(\lambda^2(t)u(t))\geq d.
  \end{split}
\end{equation*}
The proof is complete.
\end{proof}

With the preliminaries given above, we can show the first blow-up results for problem \eqref{1.1}
with lower initial energy.

\begin{theorem}\label{lower}
Let $p$ satisfy (A), $u_0\in \mathcal{U}$ and $u_1\in L^2(\Omega)$ such that $E(0)<d$.
Then the solution $u(x,t)$ to problem \eqref{1.1} blows up at a finite time $T$ in the sense that
\begin{equation}\label{blow-up}
\lim\limits_{t\rightarrow T^{-}}\Big(\|u(t)\|_2^2+\int_0^t \|u(s)\|^2{\rm d}s\Big)=\infty.
\end{equation}
Moreover, the blow-up time $T$ can be estimated from above as follows
\begin{equation}\label{upper bound}
 T\leq \dfrac{4\Big[\big(a^2+(p-2)^2b\|u_0\|^2_2\big)^{1/2}+a\Big]}{(p-2)^2b},
\end{equation}
where $a,b$ are constants that will be fixed in the proof.
\end{theorem}

\begin{proof}
Assume by contradiction that the solution $u$ exists globally. As was done in \cite{Wu},
fix $T^*>0$ and define the functional
\begin{equation}\label{g}
G(t)=\|u(t)\|_2^2+\int_0^t \|u(s)\|^2{\rm d}s+(T^*-t)\|u_0\|^2+b(t+\tau)^2,\quad t\in[0,T^*],
\end{equation}
where $T^*$, $b$ and $\tau$ are positive constants to be fixed later.
Taking derivative we have
\begin{equation}\label{g1}
G'(t)=2(u,u_t)+2\int_0^t\int_\Omega (uu_s+\nabla u\cdot\nabla u_s){\rm d}x{\rm d}s+2b(t+\tau).
\end{equation}
Taking derivative again and using \eqref{ei}, \eqref{e} and Lemma \ref{inv},
we obtain
\begin{equation}\label{g2}
\begin{split}
G''(t)=&2\|u_t\|_2^2+2(u,u_{tt})+2\int_\Omega (uu_t+\nabla u\cdot\nabla u_t){\rm d}x+2b\\
      =&(p+2)\|u_t\|_2^2+(p-2)\|u\|_H^2-2pE(t)+2b+\frac{2}{p}\|u\|_p^p\\
   \geq&(p+2)\|u_t\|_2^2+2pd-2pE(t)+2b\\
      =&(p+2)\|u_t\|_2^2+2p(d-E(0))+2p\int_0^t\|u_s(s)\|^2{\rm d}s+2b.
\end{split}
\end{equation}
Choosing $b=d-E(0)>0$ and noticing $p>2$ we get
\begin{equation}\label{g3}
G''(t)\geq (p+2)[\|u_t\|_2^2+\int_0^t\|u_s(s)\|^2{\rm d}s+b]>0.
\end{equation}
Combining \eqref{g}, \eqref{g1} with \eqref{g3} we know, for any $t\in[0,T^*]$, that
\begin{equation*}\
\begin{split}
&G(t)G''(t)-\dfrac{p+2}{4}(G'(t))^2\\
\geq&(p+2)\Big[\|u(t)\|_2^2+\int_0^t \|u(s)\|^2{\rm d}s+b(t+\tau)^2\Big]\cdot\Big[\|u_t\|_2^2+\int_0^t\|u_s(s)\|^2{\rm d}s+b\Big]\\
&-(p+2)\Big[(u,u_t)+\int_0^t\int_\Omega (uu_s+\nabla u\cdot\nabla u_s){\rm d}x{\rm d}s+b(t+\tau)\Big]^2.
\end{split}
\end{equation*}
Since
\begin{equation}\label{cauchy-1}
(u,u_t)\leq \|u(t)\|_2\|u_t\|_2,
\end{equation}
\begin{equation}\label{cauchy-2}
\int_0^t\int_\Omega uu_s{\rm d}x{\rm d}s\leq \Big(\int_0^t\int_\Omega u^2{\rm d}x{\rm d}s\Big)^{1/2}\Big(\int_0^t\int_\Omega u^2_s{\rm d}x{\rm d}s\Big)^{1/2},
\end{equation}
\begin{equation}\label{cauchy-3}
\int_0^t\int_\Omega \nabla u\cdot\nabla u_s{\rm d}x{\rm d}s\leq \Big(\int_0^t\int_\Omega |\nabla u|^2{\rm d}x{\rm d}s\Big)^{1/2}\Big(\int_0^t\int_\Omega |\nabla u_s|^2{\rm d}x{\rm d}s\Big)^{1/2},
\end{equation}
it can be directly verified by using Cauchy-Schwarz inequality that
$$G(t)G''(t)-\dfrac{p+2}{4}(G'(t))^2\geq 0,\quad t\in[0,T^*].$$
Take
\begin{equation}\label{tau}
\tau=\max\Big\{0,\frac{2\|u_0\|^2-(p-2)(u_0,u_1)}{(p-2)b}\Big\},
\end{equation}
then
$$G(0)=\|u_0\|_2^2+T^*\|u_0\|^2+b\tau^2>0,$$
$$G'(0)=2(u_0,u_1)+2b\tau>0,$$
and
\begin{equation}\label{T*}
\frac{4G(0)}{(p-2)G'(0)}=\frac{2[\|u_0\|_2^2+T^*\|u_0\|^2+b\tau^2]}{(p-2)[(u_0,u_1)+b\tau]}\leq T^*,
\end{equation}
for suitably large $T^*$.
According to Lemma \ref{concave}, there exists a $T_*>0$ satisfying
\begin{equation}\label{T*}
T_*\leq \frac{4G(0)}{(p-2)G'(0)}
\end{equation}
such that
$$G(t)\rightarrow\infty\ \text{as}\ t\rightarrow T_*^-.$$
This contradicts with the assumption that $G(t)$ is well defined on the closed $[0,T^*]$ for any $T^*>0$.

To derive an upper bound for the blow-up time, we proceed as follows:
Let $T$ be the maximal existence time of $u(x,t)$ (which is finite by the above argument)
and let $G(t)$ be given in \eqref{g}, with the exception that $T^*$ is replaced by $T$ and $t\in[0,\overline{T}]$,
where $\overline{T}\in(0,T)$. Similarly to the foregoing arguments, one can show that
$$\overline{T}\leq \frac{2[\|u_0\|_2^2+T\|u_0\|^2+b\tau^2]}{(p-2)[(u_0,u_1)+b\tau]},$$
where we require that $\tau$, which is independent of $\overline{T}$, still satisfies \eqref{tau}.
It then follows from the arbitrariness of $\overline{T}<T$ that
\begin{equation}\label{t-1}
T\leq \frac{2[\|u_0\|_2^2+T\|u_0\|^2+b\tau^2]}{(p-2)[(u_0,u_1)+b\tau]},
\end{equation}
which guarantees
\begin{equation}\label{t-2}
 T\leq T(\tau)\triangleq\frac{2(\|u_0\|^2_2+b\tau^2)}{(p-2)[(u_0,u_1)+b\tau]-2\|u_0\|^2}.
\end{equation}
Set $a=2\|u_0\|^2-(p-2)(u_0,u_1)$ and $\tau_0=\dfrac{[a^2+(p-2)^2b\|u_0\|_2^2]^{1/2}+a}{(p-2)b}$.
Then it is an easy matter to verify that $\tau_0$ satisfies \eqref{tau}, $T(\tau)$ attains its minimum at $\tau_0$
and
$$T(\tau_0)=\dfrac{4\Big[\big(a^2+(p-2)^2b\|u_0\|^2_2\big)^{1/2}+a\Big]}{(p-2)^2b}.$$
Therefore,
$$T\leq \dfrac{4\Big[\big(a^2+(p-2)^2b\|u_0\|^2_2\big)^{1/2}+a\Big]}{(p-2)^2b}.$$
The proof is complete.
\end{proof}

\begin{remark}\label{rem3.1}
  By \eqref{e} and \eqref{i-j} and recalling that $p>2$ one sees $E(0)<0$ implies $I(u_0)<0$.
  Therefore, Theorem \ref{lower} implies that the solution $u(x,t)$ to problem \eqref{1.1} blows up in finite time
  for negative initial energy.
\end{remark}

\section{Blow-up for high initial energy}
\setcounter{equation}{0}

In this section we shall build a blow-up criterion for problem \eqref{1.1} at high initial energy level.
Some ideas used in this section are borrowed from \cite{Guo-Li} and \cite{Li-Liu}.
As a preliminary, we first establish a lemma that will play a fundamental role.

\begin{lemma}\label{lem1}
Let $p$ satisfy $(A)$. Assume that $u_0\in H$ and $u_1\in L^2(\Omega)$
such that
\begin{equation}\label{initial}
0<E(0)<\dfrac{C_0}{p}(u_0,u_1).
\end{equation}
Then the solution $u(x,t)$ to problem \eqref{1.1} satisfies
\begin{equation}\label{growth}
(u,u_t)-\frac{p}{C_0}E(t)\geq \Big[(u_0,u_1)-\dfrac{p}{C_0}E(0)\Big]e^{C_0t},\quad t\in[0,T).
\end{equation}
Here
\begin{equation}\label{c0}
C_0=\min\Big\{p+2,p(p-2)\lambda_1,\frac{(p-2)(\lambda_1+\lambda^2_1)}{2}\Big\}>0.
\end{equation}
\end{lemma}

\begin{proof}
Set $F(t)=(u,u_t)$. By direct calculations and recalling \eqref{e} we have
\begin{equation}\label{4.3}
\begin{split}
   F'(t)=&\|u_t\|_2^2+(u,u_{tt}) \\
     =&\|u_t\|_2^2+(u,-\Delta^2u+\Delta u+\Delta u_t-u_t+|u|^{p-2}u\ln|u|)\\
     =&\|u_t\|_2^2-\|u\|_H^2-\int_\Omega \nabla u\cdot\nabla u_t{\rm d}x-(u,u_t)+\int_\Omega |u|^p\ln|u|{\rm d}x\\
     =&\dfrac{p+2}{2}\|u_t\|_2^2+\dfrac{p-2}{2}\|u\|_H^2-\int_\Omega \nabla u\cdot\nabla u_t{\rm d}x-(u,u_t)-pE(t)+\dfrac{1}{p}\|u\|_p^p\\
     \geq&\dfrac{p+2}{2}\|u_t\|_2^2+\dfrac{p-2}{2}\|u\|_H^2-\int_\Omega \nabla u\cdot\nabla u_t{\rm d}x-(u,u_t)-pE(t).
\end{split}
\end{equation}
By using Cauchy inequality, we can estimate the third and fourth terms in the last inequality as follows
\begin{equation}\label{4.4}
 |\int_\Omega \nabla u\cdot\nabla u_t{\rm d}x|\leq\dfrac{C_0}{4p}\|\nabla u\|_2^2+\dfrac{p}{C_0}\|\nabla u_t\|_2^2,
\end{equation}
\begin{equation}\label{4.5}
 |(u,u_t)|\leq\dfrac{C_0}{4p}\|u\|_2^2+\dfrac{p}{C_0}\|u_t\|_2^2.
\end{equation}
Substituting \eqref{4.4} and \eqref{4.5} into \eqref{4.3} we arrive at
\begin{equation}\label{4.6}
 F'(t)\geq \dfrac{p+2}{2}\|u_t\|_2^2+\dfrac{p-2}{2}\|u\|_H^2-\dfrac{C_0}{4p}\|u\|^2-\dfrac{p}{C_0}\| u_t\|^2-pE(t).
\end{equation}

Set $H(t)=F(t)-\dfrac{p}{C_0}E(t)$.
Then in view of \eqref{lam}, \eqref{ei}, \eqref{c0} and \eqref{4.6} we obtain
\begin{equation}\label{4.7}
\begin{split}
   H'(t)=&F'(t)-\dfrac{p}{C_0}E'(t)=F'(t)+\dfrac{p}{C_0}\|u_t\|^2\\
     \geq&\dfrac{p+2}{2}\|u_t\|_2^2+\dfrac{p-2}{2}\|u\|_H^2-\dfrac{C_0}{4p}\|u\|^2-pE(t)\\
     \geq&\dfrac{p+2}{2}\|u_t\|_2^2+\Big[\dfrac{(p-2)\lambda_1}{2}-\dfrac{p(p-2)\lambda_1}{4p}\Big]\|u\|^2-pE(t)\\
     =&\dfrac{p+2}{2}\|u_t\|_2^2+\dfrac{(p-2)\lambda_1}{4}\|u\|^2-pE(t)\\
     \geq&\dfrac{p+2}{2}\|u_t\|_2^2+\dfrac{(p-2)(\lambda_1+\lambda^2_1)}{4}\|u\|_2^2-pE(t)\\
     \geq&C_0\Big[\dfrac{1}{2}\|u_t\|_2^2+\dfrac{1}{2}\|u\|_2^2-\dfrac{p}{C_0}E(t)\Big]\geq C_0H(t).
\end{split}
\end{equation}
Since $H(0)>0$ by \eqref{initial}, \eqref{growth} follows after an application of Gronwall's inequality to $H(t)$.
The proof is complete.
\end{proof}

With Lemma \ref{lem1} at hand, we are now in the position to
prove high initial energy blow-up and estimate an upper bound
for the blow-up time for problem \eqref{1.1}.

\begin{theorem}\label{high}
Let all the assumptions in Lemma \ref{lem1} hold.
Then the solution $u(x,t)$ to problem \eqref{1.1} blows up at some finite time $T$ in the sense of \eqref{blow-up}.
Moreover, if
\begin{equation}\label{c2}
  E(0)<\frac{C_0}{p}\|u_0\|_2^2,
\end{equation}
then
\begin{equation}\label{t}
  T\leq \dfrac{2(\|u_0\|_2^2+\beta t_0^2)}{(p-2)[(u_0,u_1)+\beta t_0]-2\|u_0\|^2}.
\end{equation}
Here $C_0$ is the positive constant given in \eqref{c0},
\begin{equation}\label{beta}
\beta=2\Big[\frac{C_0}{p}\|u_0\|_2^2-E(0)\Big]>0,
\end{equation}
$t_0$ is suitably large such that
\begin{equation}\label{t0}
(p-2)\Big[(u_0,u_1)+\beta t_0\Big]>2\|u_0\|^2.
\end{equation}
\end{theorem}

\begin{proof}
We divide the proof into two steps.

{\bf Step I: Finite time blow-up.}
Suppose by contradiction that \eqref{blow-up} will not happen for any finite $T$.
Then $\|u(\cdot,t)\|_2$ is well-defined for all $t\geq0$.
Without loss of generality, we may assume that $E(t)\geq0$ for all $t\geq0$.
Otherwise by Remark \ref{rem3.1} we know that $u(x,t)$ blows up in finite time.

On one hand, it follows from Lemma \ref{lem1} that
\begin{equation}\label{4.13}
\dfrac{d}{dt}\|u(t)\|_2^2=2(u,u_t)\geq 2H(0)e^{C_0t}+\dfrac{2p}{C_0}E(t)\geq 2H(0)e^{C_0t}.
\end{equation}
Integration of \eqref{4.13} over $[0,t]$ yields
\begin{equation}\label{4.14}
\begin{split}
\|u(t)\|_2^2=&\|u_0\|_2^2+2\int_0^t\int_\Omega uu_\tau {\rm d}x{\rm d}\tau\geq\|u_0\|_2^2+2\int_0^t H(0)e^{C_0\tau}{\rm d}\tau\\
=&\|u_0\|_2^2+\dfrac{2H(0)}{C_0}(e^{C_0t}-1).
\end{split}
\end{equation}

On the other hand, by virtue of Minkowski inequality,
H\"{o}lder inequality, \eqref{ei}, the definition of $\lambda_1$ and the fact $E(t)\geq0$ one gets
\begin{equation*}
\begin{split}
\|u(t)\|_2\leq&\|u_0\|_2+\|u(t)-u_0\|_2=\|u_0\|_2+\|\int_0^t u_\tau{\rm d}\tau\|_2\\
\leq&\|u_0\|_2+\int_0^t \|u_\tau\|_2{\rm d}\tau\leq \|u_0\|_2
+\dfrac{1}{\sqrt{1+\lambda_1}}\int_0^t \|u_\tau\|{\rm d}\tau\\
\leq&\|u_0\|_2
+\dfrac{\sqrt{t}}{\sqrt{1+\lambda_1}}\Big(\int_0^t \|u_\tau\|^2{\rm d}\tau\Big)^{1/2}=\|u_0\|_2+\dfrac{\sqrt{t}}{\sqrt{1+\lambda_1}}(E(0)-E(t))^{1/2}\\
\leq&\|u_0\|_2+\sqrt{\dfrac{E(0)}{1+\lambda_1}}t^{1/2},
\end{split}
\end{equation*}
which contradicts \eqref{4.14} when $t$ is sufficiently large.
Therefore, $u(x,t)$ blows up in finite time.

{\bf Step II: Upper bound for the blow-up time.} From now on, we assume that $T>0$ is the blow-up time of $u(x,t)$,
which is finite by Step I.
According to Lemma \ref{lem1} and the assumption that $E(t)\geq 0$, we see that
\begin{equation*}
\dfrac{d}{dt}\|u(t)\|_2^2=2(u,u_t)\geq 2H(0)e^{C_0t}+\dfrac{2p}{C_0}E(t)> 0,\quad t\in[0,T),
\end{equation*}
which implies $\|u(t)\|_2^2$ is increasing with respect to $t$.
To estimate $T$ from above, as was done in the proof of Theorem \ref{lower}, we define
\begin{equation}\label{g}
K(t)=\|u(t)\|_2^2+\int_0^t \|u(s)\|^2{\rm d}s+(T-t)\|u_0\|^2+\beta(t+t_0)^2,\quad \quad t\in[0,T),
\end{equation}
where $\beta$ and $t_0$ are given in \eqref{beta} and \eqref{t0}, respectively.
By applying similar argument to that in the proof of the first part of Theorem \ref{lower}, we obtain
\begin{equation}\label{concave-inequality}
\begin{split}
&K(t)K''(t)-\dfrac{p+2}{4}(K'(t))^2\\
=&2K(t)\Big[\|u_t\|_2^2-\|u\|_H^2+\int_\Omega |u|^p\ln|u|{\rm d}x+\beta\Big]\\
&-(p+2)\Big[(u,u_t)+\int_0^t\int_\Omega (uu_s+\nabla u\cdot\nabla u_s){\rm d}x{\rm d}s+\beta(t+t_0)\Big]^2\\
=&2K(t)\Big[\|u_t\|_2^2-\|u\|_H^2+\int_\Omega |u|^p\ln|u|{\rm d}x+\beta\Big]\\
&+(p-2)\Big[\eta(t)-\big(K(t)-(T-t)\|u_0\|^2\big)\big(\|u_t\|_2^2+\int_0^t \|u_s\|^2{\rm d}s+\beta\big)\Big],
\end{split}
\end{equation}
where
\begin{equation}\label{concave-inequality-1}
\begin{split}
\eta(t)=&\Big[\|u(t)\|_2^2+\int_0^t \|u(s)\|^2{\rm d}s+\beta(t+t_0)^2\Big]\Big[\|u_t\|_2^2+\int_0^t \|u_s\|^2{\rm d}s+\beta\Big]\\
&-\Big[(u,u_t)+\int_0^t\int_\Omega (uu_s+\nabla u\cdot\nabla u_s){\rm d}x{\rm d}s+\beta(t+t_0)\Big]^2.
\end{split}
\end{equation}
Using \eqref{cauchy-1}-\eqref{cauchy-3} and Cauchy-Schwarz inequality we can show that $\eta(t)\geq 0$ on $[0,T)$.
Therefore, recalling \eqref{ei}, \eqref{e}, \eqref{beta} and the monotonicity of $\|u(t)\|_2^2$, we have
\begin{equation}\label{concave-inequality-2}
\begin{split}
&K(t)K''(t)-\dfrac{p+2}{4}(K'(t))^2\\
\geq &2K(t)\Big[\|u_t\|_2^2-\|u\|_H^2+\int_\Omega |u|^p\ln|u|{\rm d}x+\beta\Big]-(p+2)K(t)\Big[\|u_t\|_2^2+\int_0^t \|u_s\|^2{\rm d}s+\beta\Big]\\

=&K(t)\Big[(p-2)\|u\|_H^2-2pE(0)+(p-2)\int_0^t \|u_s\|^2{\rm d}s+\frac{2}{p}\|u\|_p^p-p\beta\Big]\\
\geq&K(t)\Big[(p-2)(\lambda_1+\lambda^2_1)\|u\|_2^2-2pE(0)-p\beta\Big]\\
\geq&K(t)\Big[(p-2)(\lambda_1+\lambda^2_1)\|u_0\|_2^2-2pE(0)-p\beta\Big]\\
=&2pK(t)\Big[\frac{C_0}{p}\|u_0\|_2^2-E(0)-\beta/2\Big]\geq0,\quad t\in[0,T).
\end{split}
\end{equation}
Besides, $K(0)=\|u_0\|_2^2+T\|u_0\|^2+\beta t_0^2>0$ and $K'(0)=2(u_0,u_1)+2\beta t_0>0$ by \eqref{t0}.
Applying Lemma \ref{concave} to $K(t)$ yields
\begin{equation*}
T\leq\dfrac{4K(0)}{(p-2)K'(0)}=\dfrac{2(\|u_0\|_2^2+T\|u_0\|^2+\beta t_0^2)}{(p-2)[(u_0,u_1)+\beta t_0]}.
\end{equation*}
Since $\dfrac{2\|u_0\|^2}{(p-2)[(u_0,u_1)+\beta t_0]}<1$ by \eqref{t0}, we further obtain
\begin{equation*}
T\leq\dfrac{2(\|u_0\|_2^2+\beta t_0^2)}{(p-2)[(u_0,u_1)+\beta t_0]-2\|u_0\|^2}.
\end{equation*}
The proof of Theorem \ref{high} is complete.
\end{proof}

\begin{remark}
As was done in deriving \eqref{upper bound}, one can also minimize the right-hand side term of \eqref{t} for $t_0$ satisfying \eqref{t0}
to obtain a more accurate upper bound for $T$. Interested reader may check it.
\end{remark}

\section{Lower bound for the blow-up time}
\setcounter{equation}{0}

Since the lower bound for the blow-up time provides a safe time interval for the system under consideration,
it is more important in practice to estimate $T$ from below. In this section,
our aim is to determine a lower bound for the blow-up time of problem \eqref{1.1} by constructing a new auxiliary functional.
Throughout this section we shall use $C,C_1,C_2,\cdots,$ to denote generic positive constants
which may depend on $\Omega,p,n$, but are independent of the solution $u(x,t)$.

\begin{theorem}\label{lower-bound}
Assume that $p$ satisfies $\dfrac{2n}{n+2}<\dfrac{2n(p-1)}{n+2}<2_*$, i.e.,
\begin{equation}\label{p}
 p\in(2,\infty)\quad if\quad  n\leq 4;\quad p\in\big(2,\frac{2n-2}{n-4}\big)\quad  if \quad n\geq 5.
\end{equation}
Let $u(x,t)$ be a weak solution to problem \eqref{1.1} that blows up at $T$ in the sense of \eqref{blow-up}.
Then $$T\geq \int_{N(0)}^\infty\dfrac{{\rm d}s}{C_4+C_5s^{p-1+\mu}},$$
where $N(0)=\|u_1\|_2^2+\|u_0\|_H^2$.
\end{theorem}

\begin{proof}
For simplicity, we only prove this theorem for $n\geq 3$. The case for $n=1,2$ is similar (and simpler).
We aim to determine a time interval $(0,T_0)$ on which the quantity $\|u(t)\|_H^2$ is bounded.
Clearly $T_0$ is a lower bound for $T$ since both $\|u(t)\|_2^2$ and $\|u(t)\|^2$ can be bounded by $\|u(t)\|_H^2$.

Define
\begin{equation}\label{n1}
N(t)=\|u_t(t)\|_2^2+\|u(t)\|_H^2,\quad t\in[0,T_0).
\end{equation}
Then
\begin{equation}\label{n2}
\lim_{t\rightarrow T_0^{-}}N(t)=+\infty.
\end{equation}
Differentiating \eqref{n1} and making use of Green's second identity,
we obtain
\begin{align}\label{n3}
N'(t)&=2[(u_t,u_{tt})+(\Delta u,\Delta u_t)+(\nabla u,\nabla u_t)]\nonumber\\
&=2(u_t,u_{tt}+\Delta^2 u-\Delta u)\nonumber\\
&=2(u_t,\Delta u_t-u_t+|u|^{p-2}\ln|u|)\nonumber\\
&=-2\|u_t\|^2+2\int_\Omega u_t|u|^{p-2}u\ln|u|{\rm d}x.
\end{align}

Set
$\Omega_1=\Omega_1(t)=\{x\in\Omega: |u(x,t)|<1\}$ and $\Omega_2=\Omega_2(t)=\{x\in\Omega: |u(x,t)|\geq1\}$.
Since $p$ satisfies \eqref{p}, we can choose $\mu>0$ suitably small such that $\frac{2n(p-1+\mu)}{n+2}<2_*$,
which implies that $H$ can be embedded into $L^{\frac{2n(p-1+\mu)}{n+2}}(\Omega)$ continuously.
We use $B_\mu$ to denote the embedding constant from $H$ to $L^{\frac{2n(p-1+\mu)}{n+2}}(\Omega)$,
i.e.,
\begin{equation}\label{n4}
\|v\|_{\frac{2n(p-1+\mu)}{n+2}}\leq B_\mu\|v\|_H,\qquad\forall\ v\in H.
\end{equation}
Using H\"{o}lder's inequality, Cauchy inequality, \eqref{n4} and the basic inequalities $|s^{p-2}\ln s|\leq (e(p-1))^{-1}$ for $0<s<1$ and $\ln s\leq\dfrac{1}{e\mu}s^\mu$
for $s\geq 1$, we can estimate the second term on the right-hand side of \eqref{n3} as follows
\begin{align}\label{n5}
&\int_\Omega u_t|u|^{p-2}u\ln|u|{\rm d}x\nonumber=\int_{\Omega_1} u_t|u|^{p-2}u\ln|u|{\rm d}x+\int_{\Omega_2} u_t|u|^{p-2}u\ln|u|{\rm d}x\nonumber\\
\leq&\Big(\int_{\Omega_1} |u_t|^{\frac{2n}{n-2}}{\rm d}x\Big)^{\frac{n-2}{2n}}\Big(\int_{\Omega_1} \big||u|^{p-2}u\ln|u|\big|^{\frac{2n}{n+2}}{\rm d}x\Big)^{\frac{n+2}{2n}}\nonumber\\
&+\Big(\int_{\Omega_2} |u_t|^{\frac{2n}{n-2}}{\rm d}x\Big)^{\frac{n-2}{2n}}\Big(\int_{\Omega_2} \big||u|^{p-2}u\ln|u|\big|^{\frac{2n}{n+2}}{\rm d}x\Big)^{\frac{n+2}{2n}}\nonumber\\
\leq&\|u_t\|_{\frac{2n}{n-2}}\Big[(e(p-1))^{-1}|\Omega_1|^{\frac{n+2}{2n}}+(e\mu)^{-1}\Big(\int_{\Omega} |u|^{\frac{2n(p-1+\mu)}{n+2}}{\rm d}x\Big)^{\frac{n+2}{2n}}\Big]\nonumber\\
\leq&C\|u_t\|\Big[(e(p-1))^{-1}|\Omega_1|^{\frac{n+2}{2n}}+(e\mu)^{-1}B^{p-1+\mu}_\mu\|u\|^{p-1+\mu}_H\Big]\nonumber\\
\leq&\varepsilon \|u_t\|^2+C(\varepsilon)\Big[C_1+C_2\|u\|^{2(p-1+\mu)}_H\Big]\nonumber\\
\leq&\varepsilon \|u_t\|^2+C(\varepsilon)\Big[C_1+C_3N^{p-1+\mu}(t)\Big].
\end{align}
Therefore, it follows by taking $\varepsilon\leq 1$ and substituting \eqref{n5} into \eqref{n3} that
\begin{equation}\label{n6}
N'(t)\leq C_4+C_5N^{p-1+\mu}(t).
\end{equation}
Integrating \eqref{n6} over $[0,t]$, we have
\begin{equation}\label{n7}
\int_0^t\dfrac{N'(\tau)}{C_4+C_5N^{p-1+\mu}(\tau)}{\rm d}\tau\leq t.
\end{equation}
Letting $t\rightarrow T_0^{-}$ and recalling \eqref{n2}, we obtain
\begin{equation}\label{n8}
\int_{N(0)}^\infty\dfrac{{\rm d}s}{C_4+C_5s^{p-1+\mu}}\leq T_0\leq T.
\end{equation}
Recalling that $p-1+\mu>1$, the left-hand side term in \eqref{n8} is finite.
The proof is complete.
\end{proof}

\begin{remark}
  By making full use of the damping term, we obtain the lower bound for the blow-up time not only for subcritical exponent $p$, but also for some supercritical ones. We point out that this observation can also be applied to
  problem \eqref{sec-2} considered in \cite{Di}.
\end{remark}

{\bf Acknowledgement}\\
The authors would like to express their sincere gratitude to Professor Wenjie Gao for his enthusiastic
guidance and constant encouragement.


text:
\section{Conclusion}
\label{sec:conc}
We have studied the variation diminishing property of the Toeplitz and the Hankel operator of finite-dimensional causal LTI systems. Each class defines different refinements of the system property of  external positivity. While the theory of totally positive operators is classical in mathematics \cite{karlin1968total}, the state-space realization of finite-dimensional LTI systems sheds new light onto the properties of $k$-totally positive systems. In particular, we have provided a bridge between external positivity and total positivity. A key insight of the present paper is that the $k$-positivity of a LTI system can be studied via the classical positivity property of associated compound systems.

The present work has focused on external open-loop k-positivity. Future work should investigate the relationship to the internal property as studied in the recent work \cite{margaliot2018revisiting,weiss2019generalization} as well as feedback interconnections. Implications of $k$`-positivity for model reduction are studied in \cite{grussler2020balanced}.


TITLE: Similar quadrilaterals in rectangle
QUESTION [1 upvotes]: While working on a problem on finding the shaded region in a rectangle, I realized that if a line cuts a rectangle into two quadrilaterals, then these two quadrilaterals are similar as the opposite sides of a rectangle are parallel. (angle $AFE = $ angle $CEF$). It can be shown that quadrilateral $AFED$ is similar to quadrilateral $CEFB$. If the quadrilaterals are similar, the sides should be in proportion, $$\frac{AF}{CE}=\frac{AD}{CB}$$ and we know that $ad=cb$ from the properties of a rectangle. This should mean that $AF=CE$ but from drawing many rectangles, I understand that this is not the case. Please will someone tell me what I am doing wrong, why $AF$ is not equal to $CE$.
Diagram:

REPLY [1 votes]: The line segment $\overline{EF}$ may be shifted to one side, so that the angles of the two small quadrilaterals are equal but the sides are not proportional.

For triangles, equal angles guarantee similarity (the AA or AAA postulate), but that is not true for polygons with more than three sides. For those you also need to somehow state that the sides are proportional.


TITLE: Count the number of ways n different-sided dice can add up to a given number
QUESTION [1 upvotes]: I am trying to find a way to count the number of ways n different-sided dice can add up to a given number.
For example, 2 dice, 4- and 6-sided, can add up to 8 in 3 different ways: ($(2,6),(3,5),(4,4)$).
I've found a similar question but I cannot find a way to generalize it to my problem.

REPLY [0 votes]: I can't provide a nice formula as in your referenced question, but I can show you at least half of the way.
Let's start with your

Example: There are two dice with 4 and 6 sides. How many ways are there, so that rolling both dice once result in 8 pips?

We encode the dies with polynomials and use the exponents to label the pips

\begin{align*}
\text{4-sided die}&\qquad x^1+x^2+x^3+x^4\\
\text{6-sided die}&\qquad x^1+x^2+x^3+x^4+x^5+x^6
\end{align*}

If we denote with $[x^k]$ the coefficient $a_k$ of $x^k$ of a polynomial $\sum_{j=0}^{n}a_j x^j$, we observe
\begin{align*}
[x^8]&(x^1+x^2+x^3+x^4)(x^1+x^2+x^3+x^4+x^5+x^6)\\
&=[x^8]x\frac{1-x^4}{1-x}x\frac{1-x^6}{1-x}\\
&=[x^6](1-x^4)(1-x^6)\frac{1}{(1-x)^2}\\
&=[x^6](1-x^4-x^6+x^{10})\sum_{j\geq 0}\binom{-2}{j}(-x)^j\\
&=[x^6](1-x^4-x^6+x^{10})\sum_{j\geq 0}\binom{j+1}{j}x^j\tag{1}\\
&=([x^6]-[x^2]-[x^0])\sum_{j\geq 0}(j+1)x^j\\
&=7-3-1\\
&=3
\end{align*}
So, the result is $3$ in accordance with your example. Now the same generalized:

Generalisation: Let's assume there are $N$ dice with different sides. How many ways are there rolling $N$ dies once result in $K$ pips? Since we do not require pairwise different sides, we my assume $j_1$ dice with $k_1$ sides, $j_2$ dice with $k_2$ and finally $j_l$ dice with $k_l$ sides together with  $j_1+\dots+j_l=N$.

We can now calculate similarly to the special case above:
\begin{align*}
[x^K]&(x^1+\cdots+x^{k_1})^{j_1}\cdot...\cdot(x^1+\cdots+x^{k_l})^{j_l}\\
&=[x^K]x^{j_1}\left(\frac{1-x^{k_1}}{1-x}\right)^{j_1}\cdot...\cdot x^{j_l}\left(\frac{1-x^{k_l}}{1-x}\right)^{j_l}\\
&=[x^{K-j_1-...-j_l}](1-x^{k_1})^{j_1}\cdot...\cdot(1-x^{k_l})^{j_l}\frac{1}{(1-x)^N}\\
&=[x^{K-j_1-...-j_l}]\prod_{m=1}^{l}\left(1-x^{k_m}\right)^{j_m}\sum_{n\geq 0}\binom{-N}{n}(-x)^n\\
&=[x^{K-j_1-...-j_l}]\prod_{m=1}^{l}\left(1-x^{k_m}\right)^{j_m}\sum_{n\geq 0}\binom{N+n-1}{n}x^n\tag{2}\\
\end{align*}
Observe, that formula $(2)$ corresponds to formula $(1)$ in the special case above. So, for specific values we can continue at least in principle as we did from $(1)$.


TITLE: Suficient condition for tensor product of vector spaces..
QUESTION [1 upvotes]: Can anyone help me showing the following:
Let $E$, $F$, $G$ and $H$ vector spaces and $\varphi:E\times F\rightarrow G$ a bilinear map. If for every $\psi:E\times F\rightarrow H$ bilinear there is an unique linear map $f:G\rightarrow H$ such that $\psi=f\circ \varphi$ then $(G, \varphi)$ is a tensor product of $E$ and $F$.
The definition I have for tensor product of vector spaces is:
Definition: Let $E, F$ and $G$ be vector spaces and $\varphi:E\times F\rightarrow G$ a bilinear map. We say the pair $(G, \varphi)$ is a tensor product of $E$ and $F$ if:
(i) $\textrm{im}(\varphi)=G$.
(ii) For every $\psi:E\times F\rightarrow H$ bilinear, where $H$ is an arbitrary vector space, there is linear map $f:G\rightarrow H$ such that $\psi=f\circ \varphi$.
So using this definition it suffices showing the map $\varphi:E\times F\rightarrow G$ is surjective for solving my problem...
Any help will be welcome..Thanks

REPLY [0 votes]: Your confusion stems from a lack of understanding of the unconventional notation used by your book (Greub's Multilinear Algebra, 2nd ed).
In p. 2, the book states: "We shall denote $\text{Im}\,\varphi$ the subspace of $G$ generated by $S$", where $S$ is the usual image of $\varphi$ and $G$ is the codomain of $\varphi$.
The author does indeed formulate the way you've mentioned. However, the first thing in the book (p. 1) is to show that the image of a bilinear map is not in general a subspace. The definition you cite is in p. 4; so just read the previous pages, which are worthwhile.


TITLE: The length of a gap between the rationals
QUESTION [2 upvotes]: In Terence Tao's book Analysis I, he says

there are still an infinite number of “gaps” or “holes” between the
  rationals, although this denseness property does ensure that these
  holes are in some sense infinitely small.

I think a gap between the rationals should have  zero length.
Supposing $A_{1}=\{a\in {\mathbb  {Q}}:a^{2}<2{\text{ or }}a<0\},
{\displaystyle A_{2}=\{a\in \mathbb {Q} :a^{2}>2{\text{ and }}a>0\}} $,
I define the "length" of the gap between $A_{1}$ and $A_{2}$ to be the greatest lower bound of $A =\left\{ a \middle| a = a_{2} - a_{1},  {\ a}_{1} \in A_{1}{,a}_{2} \in A_{2} \right\}$,  so how to prove the greatest lower bound is $0$ ? especially using the density property of rational numbers to prove it?

Maybe I have a lack of understanding in the density property of rational numbers , so I am unable to give a proof to my question here.

REPLY [0 votes]: It is possible to prove  the greatest lower bound of $A =\left\{ a \middle| a = a_{2} - a_{1},  {\ a}_{1} \in A_{1}{,a}_{2} \in A_{2} \right\}$ is 0 in the rational number system， since if any other positive number $b$ is the greatest lower bound, we could always find a positive number $c=a_{2} - a_{1}<b$ .
I think what is important here is that  the question won't make much sense unless we prove  the greatest lower bound of $A$ is 0 in a continuous number system ($\mathbb R$). We may use Archimedean property of $\mathbb R$ to prove the conclusion .


TITLE: Relative horizontal velocity and Origin question
QUESTION [1 upvotes]: Question:
  A police boat is chasing a boat with criminals along a straight river
  by moving against the stream. The speed of the river stream is 3 miles
  per hour, the speed of the boat with criminals relative to the river
  is 30 miles per hour, and the police boat is 4 miles per hour faster
  than the boat with criminals.
Currently the criminals are ahead of the police, and horizontally
  throw a stone at the police boat at a speed 16 miles per hour relative
  to their boat (i.e. relative to the boat of criminals).
What is the horizontal velocity of the stone relative to the police
  boat and to the river bank? You need to state what the origin and the
  positive direction of motion are.

What should the origin be?

REPLY [1 votes]: There's not "right/wrong" answer for where to put the origin (or which direction you choose to be the positive direction), but your answer will be relative to those choices and that's why they are asking you to declare them.
I'd select the origin to be at the front of the criminals' boat and the positive direction to be pointing back toward the police boat.