1. Introduction
In this paper, we investigate the existence of global and decay of solutions
for the p-biharmonic parabolic equation with logarithmic nonlinearity
\begin{equation}
\begin{cases}
u_{t}+\Delta \left( \left\vert \Delta u\right\vert ^{p-2}\Delta u\right)
-\Delta u_{t}=u\left\vert u\right\vert ^{q-2}\ln \left\vert u\right\vert , & x\in \Omega ,\text{ }t>0, \\
u(x,t)=\Delta \left( x,t\right) =0, & x\in \partial \Omega ,\text{ }t>0, \\
u(x,0)=u_{0}(x), & x\in \Omega ,
\end{cases}
\label{10}
\end{equation}
(1)
where \(\Omega \) is bounded domain \(
\mathbb{R}
^{n}\) with smooth boundary \(\partial \Omega ,\) \(p,\) \(q\) are positive
constants, \(2< p< q< p\left( 1+\frac{4}{n}\right) ,\) and \(u_{0}\in \left(
W_{0}^{1,p}\left( \Omega \right) \cap W^{2,p}\left( \Omega \right) \right)
\backslash \left\{ 0\right\} .\) The term \(\Delta \left( \left\vert \Delta
u\right\vert ^{p-2}\Delta u\right) \) is called a \(p\)-biharmonic operator.
Studies of logarithmic nonlinearity have a long history in physics as it
occurs naturally in different areas of physics such as supersymmetric field
theories, inflationary cosmology, nuclear physics, optics and quantum
mechanics [1,2]. Peng and Zhou [3] studied the
following heat equation with logarithmic nonlinearity
\begin{equation*}
u_{t}-\Delta u_{t}=\left\vert u\right\vert ^{p-2}u\ln \left\vert
u\right\vert .
\end{equation*}
They obtained the global existence and blow-up of solutions. Also, they
discussed the upper bound of blow-up time under suitable conditions. Nhan
and Truong [
4] studied the following nonlinear pseudo-parabolic
equation
\begin{equation*}
u_{t}-\Delta u_{t}-div\left( \left\vert \nabla u\right\vert
^{p-2}\nabla u\right) =\left\vert u\right\vert ^{p-2}u\ln \left\vert
u\right\vert .
\end{equation*}
They obtained results as regard the existence or non-existence of global
solutions. Also, He
et al., [
5] proved the decay and the finite time
blow-up for weak solutions of the equation. Cao and Liu [
6] studied
the following nonlinear evolution equation with logarithmic source
\begin{equation*}
u_{t}-\Delta u_{t}-div\left( \left\vert \nabla u\right\vert
^{p-2}\nabla u\right) -k\Delta u_{t}=\left\vert u\right\vert ^{p-2}u\ln
\left\vert u\right\vert .
\end{equation*}
They established the existence of global weak solutions. Moreover, they
considered global boundedness and blowing-up at \(\infty \).
Wang and Liu [7] considered the following p-biharmonic parabolic equation with
the logarithmic nonlinearity
\begin{equation*}
u_{t}+\Delta \left( \left\vert \Delta u\right\vert ^{p-2}\Delta u\right)
=\left\vert u\right\vert ^{q-2}u\ln \left\vert u\right\vert
\end{equation*}
They studied existence of weak solutions by potential well method, blow up
at finite time by concative method.
Recently some authors studied the hyperbolic and parabolic equation with logarithmic source term (see [8,9,10,11,12,13,14,15,16,17,18,19,20]). This paper is organized as follows: In the §2, we introduce some lemma which will be needed later. In §3, under some conditions, we obtain the unique global weak solution of the
problem (1). Meanwhile, we find that the solution is decay polynomially.
It is necessary to note that prence of the logarithmic nonlinearity causes
some difficulties in deploying the potantial well method. In order to handle
this situation we need the following logarithmic Sobolev inequality which
was introduced by ([4,21,22]).
Proposition 1.
Let \(u\) be any function in \(H^{1}\left(
\mathbb{R}
^{n}\right) \) and \(\mu >0\) be any number. Then
\begin{equation}
p\int\nolimits_{
\mathbb{R}
^{n}}\left\vert u(x)\right\vert ^{p}\ln \left( \frac{\left\vert
u(x)\right\vert }{\left\Vert u(x)\right\Vert _{L^{p}(
\mathbb{R}
^{n})}}\right) dx+\frac{n}{p}\ln \left( \frac{p\mu e}{n\mathcal{L} _{p}}
\right) \int\nolimits_{
\mathbb{R}
^{n}}\left\vert u(x)\right\vert ^{p}dx\leq \mu \int\nolimits_{
\mathbb{R}
^{n}}\left\vert \nabla u(x)\right\vert ^{p}dx. \label{75}
\end{equation}
(2)
where
\begin{equation*}
\mathcal{L} _{p}=\frac{p}{n}\left( \frac{p-1}{e}\right) ^{p-1}\pi ^{-\frac{p
}{2}}\left[ \frac{\Gamma \left( \frac{\pi }{2}+1\right) }{\Gamma \left( n
\frac{p-1}{p}+1\right) }\right] ^{\frac{p}{n}}.
\end{equation*}
2. Preliminaries
For simplicity, we denote
\begin{equation*}
\text{ }\left\Vert u\right\Vert _{s}=\left\Vert u\right\Vert _{L^{s}(\Omega
)},\text{ }\left\Vert u\right\Vert _{W_{0}^{2,p}\left( \Omega \right)
}=\left\Vert u\right\Vert _{2,s}=\left( \left\Vert \Delta u\right\Vert
_{s}^{s}+\left\Vert \nabla u\right\Vert _{s}^{s}+\left\Vert u\right\Vert
_{s}^{s}\right) ^{\frac{1}{s}},
\end{equation*}
for \(1< s< \infty \) (see [
23,
24], for details). We also use notation \(X_{0}\) to denote \(\left(
W_{0}^{1,p}\left( \Omega \right) \cap W^{2,p}\left( \Omega \right) \right)
\backslash \left\{ 0\right\} \) and \(W^{-2,p^{\prime }}\left( \Omega \right) \)
to denote the dual space of \(W^{2,s}\left( \Omega \right) \), where \(
s^{\prime }\) is Hölder conjugate functional of \(s>1.\)
Let us introduce the energy functional \(J\) and Nehari functional \(I\) defined
on \(X_{0}\) as follow
\begin{equation}
J(u)=\frac{1}{p}\left\Vert \Delta u\right\Vert _{p}^{p}-\frac{1}{q}
\int\nolimits_{\Omega }\left\vert u\right\vert ^{q}\ln \left\vert
u\right\vert dx+\frac{1}{q^{2}}\left\Vert u\right\Vert _{q}^{q}, \label{30}
\end{equation}
(3)
and
\begin{equation}
I(u)=\left\Vert \Delta u\right\Vert _{p}^{p}-\int\nolimits_{\Omega
}\left\vert u\right\vert ^{q}\ln \left\vert u\right\vert dx. \label{40}
\end{equation}
(4)
By (3) and (4), we get
\begin{equation}
J(u)=\frac{1}{q}I(u)+\left( \frac{1}{p}-\frac{1}{q}\right) \left\Vert \Delta
u\right\Vert _{p}^{p}+\frac{1}{q^{2}}\left\Vert u\right\Vert _{q}^{q}.
\label{50}
\end{equation}
(5)
Let
\begin{equation*}
\mathcal{N}=\{u\in X_{0}:I(u)=0\},
\end{equation*}
be the Nehari manifold. Thus, we may define
\begin{equation}
d=\underset{u\in \mathcal{N}}{\inf }J(u). \label{70}
\end{equation}
(6)
\(d\) is positive and is obtained by some \(u\in \mathcal{N}.\) Then it is
obvious that
\begin{equation*}
M=\frac{1}{p^{2}}\left( \frac{p^{2}e}{n\mathcal{L} _{p}}\right) ^{\frac{n}{p}
}.
\end{equation*}
From [
4], we know \(d\geq M.\)
The local existence of the weak solutions can be obtained via the standard
parabolic theory. It is easy to obtain the following equality
\begin{equation}
\int\nolimits_{0}^{t}\left\Vert u_{s}(s)\right\Vert _{H_{0}^{1}(\Omega
)}^{2}ds+J(u(t))\leq J(u_{0}),\text{ }0\leq t\leq T. \label{73}
\end{equation}
(7)
Lemma 1.
Let \(u\in X_{0}\). Then we possess
- (i) \(\lim_{\lambda \to 0^{+}}j(\lambda )=0\) and \(\lim_{\lambda
\to +\infty }j(\lambda )=-\infty ;\)
- (ii) there is a unique \(\lambda ^{\ast }>0\) such that \(j^{\prime }(\lambda
^{\ast })=0;\)
- (iii) \(j(\lambda )\) is increasing on \((0,\lambda ^{\ast }),\) decreasing on \(
(\lambda ^{\ast },+\infty )\) and attains the maximum at \(\lambda ^{\ast };\)
- (iv) \(I(\lambda u)>0\) for \(0< \lambda < \lambda ^{\ast },\) \(I(\lambda u)< 0\)
for \(\lambda ^{\ast }< \lambda < +\infty \) and \(I(\lambda ^{\ast }u)=0.\)
Proof.
For \(u\in X_{0},\) by the definition of \(j,\) we get
\begin{eqnarray}
j(\lambda ) &=&\frac{1}{p}\left\Vert \Delta \left( \lambda u\right)
\right\Vert _{p}^{p}-\frac{1}{q}\int\nolimits_{\Omega }\left\vert \lambda
u\right\vert ^{q}\ln \left\vert \lambda u\right\vert dx+\frac{1}{q^{2}}
\left\Vert \lambda u\right\Vert _{q}^{q} \notag \\
&=&\frac{\lambda ^{p}}{p}\left\Vert \Delta u\right\Vert _{p}^{p}-\frac{
\lambda ^{q}}{q}\int\nolimits_{\Omega }\left\vert u\right\vert ^{q}\ln
\left\vert u\right\vert dx+\frac{\lambda ^{q}}{q}\ln \lambda \left\Vert
u\right\Vert _{q}^{q}+\frac{\lambda ^{q}}{q^{2}}\left\Vert u\right\Vert
_{q}^{q}. \label{101}
\end{eqnarray}
(8)
It is clear that (i) holds due to \(\int\nolimits_{\Omega
}\left\vert u\right\vert ^{q}dx\neq 0.\) We have
\begin{eqnarray*}
\frac{d}{d\lambda }j(\lambda ) &=&\lambda ^{p-1}\left\Vert \Delta
u\right\Vert _{p}^{p}-\lambda ^{q-1}\int\nolimits_{\Omega }\left\vert
u\right\vert ^{q}\ln \left\vert u\right\vert dx-\lambda ^{q-1}\ln \lambda
\left\Vert u\right\Vert _{q}^{q}, \\
&=&\lambda ^{p-1}\left( \left\Vert \Delta u\right\Vert _{p}^{p}-\lambda
^{q-p}\int\nolimits_{\Omega }\left\vert u\right\vert ^{q}\ln \left\vert
u\right\vert dx-\lambda ^{q-p}\ln \lambda \left\Vert u\right\Vert
_{q}^{q}\right) .
\end{eqnarray*}
Since \(\lambda >0,\) let \(\varphi \left( \lambda \right) =\lambda
^{1-p}j^{\prime }(\lambda ),\) through direct calculation, we get
\begin{equation*}
\varphi ^{\prime }(\lambda )=-\lambda ^{q-p-1}\left( \left( q-p\right)
\int\nolimits_{\Omega }\left\vert u\right\vert ^{q}\ln \left\vert
u\right\vert dx+\left( q-p\right) \ln \lambda \left\Vert u\right\Vert
_{q}^{q}+\left\Vert u\right\Vert _{q}^{q}\right) .
\end{equation*}
Hence, there exists a
\begin{equation*}
\lambda ^{\ast }=\exp \left( \frac{\left( p-q\right) \int\nolimits_{\Omega
}\left\vert u\right\vert ^{q}\ln \left\vert u\right\vert dx-\left\Vert
u\right\Vert _{q}^{q}}{\left( q-p\right) \left\Vert u\right\Vert _{q}^{q}}
\right) >0,
\end{equation*}
such that \(\varphi ^{\prime }(\lambda )>0\) on \((0,\lambda ^{\ast }),\) \(
\varphi ^{\prime }(\lambda )0,\) \(\lim_{\lambda \to +\infty }\) \(\varphi
(\lambda )=-\infty ,\) there exists a unique \(\lambda ^{\ast }>0\) such that \(
\varphi (\lambda ^{\ast })=0,\) i.e., \(j^{\prime }(\lambda ^{\ast })=0.\) So
(ii) holds. Then, \(j^{\prime }(\lambda )=\lambda \varphi (\lambda )\)
is positive on \((0,\lambda ^{\ast }),\) negative on \((\lambda ^{\ast
},+\infty ).\) Thus, \(j(\lambda )\) is increasing on \((0,\lambda ^{\ast }),\)
decreasing on \((\lambda ^{\ast },+\infty )\) and attains the maximum at \(
\lambda ^{\ast }.\) So (iii) holds. The last property, (iv)
, is only a simple corallary of the fact that
\begin{eqnarray*}
I(\lambda u) &=&\left\Vert \Delta \left( \lambda u\right) \right\Vert
_{p}^{p}-\int\nolimits_{\Omega }\left\vert \lambda u\right\vert ^{q}\ln
\left\vert \lambda u\right\vert dx \\
&=&\lambda ^{p}\left\Vert \Delta u\right\Vert _{p}^{p}-\lambda
^{q}\int\nolimits_{\Omega }\left\vert u\right\vert ^{q}\ln \left\vert
u\right\vert dx-\lambda ^{q}\ln \lambda \left\Vert u\right\Vert _{q}^{q} \\
&=&\lambda j^{\prime }(\lambda ).
\end{eqnarray*}
Thus, \(I(\lambda u)>0\) for \(0< \lambda < \lambda ^{\ast },\ I(\lambda u)< 0\)
for \(\lambda ^{\ast }< \lambda < +\infty \) and \(I(\lambda ^{\ast }u)=0.\) So
(iv) holds. The proof is complete.
Next we denote
\begin{equation*}
R:=\left( \frac{p^{2}e}{n\mathcal{L} _{p}}\right) ^{n/p^{2}}.
\end{equation*}
Lemma 2.
- (i) if \(I(u)>0\) then \(0< \left\Vert u\right\Vert _{p}< R,\)
- (ii) if \(I(u)R,\)
- (iii) if \(I(u)=0\) then \(\left\Vert u\right\Vert _{p}\geq R.\)
Proof.
By the definition of \(I(u)\), we get
\begin{eqnarray*}
I(u) &=&\left\Vert \Delta u\right\Vert _{p}^{p}-\int\nolimits_{\Omega
}\left\vert u\right\vert ^{q}\ln \left\vert u\right\vert dx \\
&\geq &\left\Vert \Delta u\right\Vert _{p}^{p}-\int\nolimits_{\Omega
}\left\vert u\right\vert ^{p}\left( \ln \frac{\left\vert u\right\vert }{
\left\Vert u\right\Vert _{p}}+\ln \left\Vert u\right\Vert _{p}\right) dx \\
&\geq &\left( 1-\frac{\mu }{p}\right) \left\Vert \Delta u\right\Vert
_{p}^{p}+\left( \frac{n}{p^{2}}\ln \left( \frac{p\mu e}{n\mathcal{L} _{p}}
\right) -\ln \left\Vert u\right\Vert _{p}\right) \left\Vert u\right\Vert
_{p}^{p}.
\end{eqnarray*}
Choosing \(\mu =p,\) we have
\begin{equation*}
I(u)\geq \left( \frac{n}{p^{2}}\ln \left( \frac{p^{2}e}{n\mathcal{L} _{p}}
\right) -\ln \left\Vert u\right\Vert _{p}\right) \left\Vert u\right\Vert
_{p}^{p}.
\end{equation*}
(i) if \(I(u)>0,\) then
\begin{equation*}
\ln \left\Vert u\right\Vert _{p}< \ln \left( \frac{p^{2}e}{n\mathcal{L} _{p}}
\right) ^{\frac{n}{p^{2}}},
\end{equation*}
that's mean
\begin{equation*}
\left\Vert u\right\Vert _{p}< \left( \frac{p^{2}e}{n\mathcal{L} _{p}}\right)
^{\frac{n}{p^{2}}}=R,
\end{equation*}
and (ii) if \(I(u)\left( \frac{p^{2}e}{n\mathcal{L} _{p}}\right)
^{\frac{n}{p^{2}}}=R,
\end{equation*}
property (iii) we can argue similarly the proof of (ii).
The proof of lemma is complete.
Lemma 3.
[25] For any \(u\in W_{0}^{1,p}(\Omega )\), \(p\geq 1\), \(r\geq 1\) and \(p_{\ast } =\frac{np}{n-p}\), the
inequality
\begin{equation*}
\left\Vert u\right\Vert _{q}\leq C\left\Vert \nabla u\right\Vert
_{p}^{\theta }\left\Vert u\right\Vert _{r}^{1-\theta },
\end{equation*}
is valid, where
\begin{equation*}
\theta =\left( \frac{1}{r}-\frac{1}{q}\right) \left( \frac{1}{n}-\frac{1}{p}
+ \frac{1}{r}\right) ^{-1},
\end{equation*}
and for \(p\geq n=1,\) \(r\leq q\leq \infty ;\) for \(n>1\) and \(p< n,\) \(q\in
\lbrack r,p_{\ast }]\) if \(r
1,\) \(r\leq q\leq \infty ;\) for \(p>n>1,\) \(r\leq
q\leq \infty .\)
Here, the constant \(C\) depends on \(n,p,q\) and \(r.\)
Lemma 4.
[26] Let \(f:R^{+}\to R^{+}\) be a nonincreasing function
and \(\sigma \) is a nonnegative constant such that
\begin{equation*}
\int\nolimits_{t}^{+\infty }f^{1+\sigma }(s)ds\leq \frac{1}{\omega }
f^{\sigma }(0)f(t),\text{ }\forall t\geq 0.
\end{equation*}
Hence
- (a) \(f(t)\leq f(0)e^{1-\omega t},\) for all \(t\geq 0,\) whenever \(\sigma =0,\)
- (b) \(f(t)\leq f(0)\left( \frac{1+\sigma }{1+\omega \sigma t}\right) ^{\frac{1
}{\sigma }},\) for all \(t\geq 0,\) whenever \(\sigma >0.\)
3. Main results
Now as in ([
4]), we introduce the follows sets:
\begin{eqnarray*}
\mathcal{W}_{1} &=&\{u\in X_{0}:J(u)0\},\text{ }\mathcal{W}
_{2}^{+}=\{u\in \mathcal{W}_{2}:I(u)>0\},\text{ }\mathcal{W}^{+}=\mathcal{W
}_{1}^{+}\cup \mathcal{W}_{2}^{+}, \\
\mathcal{W}_{1}^{-} &=&\{u\in \mathcal{W}_{1}:I(u)< 0\},\text{ }\mathcal{W}
_{2}^{-}=\{u\in \mathcal{W}_{2}:I(u)< 0\},\text{ }\mathcal{W}^{-}=\mathcal{W
}_{1}^{-}\cup \mathcal{W}_{2}^{-}.
\end{eqnarray*}
Definition 1.
(Maximal Existence Time). Assume that \(u\) be weak solutions of problem (1). We define the maximal existence time \(T_{\max }\) as follows
\begin{equation*}
T_{\max }=\sup \{T>0:u(t)\text{ exists on }[0,T]\}.
\end{equation*}
Then
- (i) If \(T_{\max }< \infty ,\) we say that \(u\) blows up in finite time and \(
T_{\max }\) is the blow-up time;
- (ii) If \(T_{\max }=\infty ,\) we say that \(u\) is global.
Definition 2.
(Weak solution). We define a function \(u\in L^{\infty }(0,T;X_{0})\) with \(
u_{t}\in L^{2}(0,T;H_{0}^{1}(\Omega ))\) to be a weak solution of problem (1) over \([0,T],\) if it satisfies the initial condition \(
u(0)=u_{0}\in X_{0},\) and
\begin{equation*}
\left\langle u_{t},w\right\rangle +\left\langle \left\vert \Delta
u\right\vert ^{p-1},\Delta w\right\rangle +\left\langle \nabla u_{t},\nabla
w\right\rangle=\int\nolimits_{\Omega }u\left\vert u\right\vert ^{q-2}\ln \left(
\left\vert u\right\vert \right) wdx,\text{ }
\end{equation*}
for all \(w\in X_{0},\) and for a.e. \(t\in \lbrack 0,T].\)
Theorem 1.
(Global Existence). Let \(u_{0}\in \) \(\mathcal{W}^{+},\) \(0< J(u_{0})0.\) Then there is a unique global weak solution \(u\) of (1)
satisfying \(u(0)=u_{0}.\) We have \(u(t)\in \mathcal{W}^{+}\)holds for all \(
0\leq t< +\infty ,\) and the energy estimate
\begin{equation*}
\int\nolimits_{0}^{t}\left\Vert u_{s}(s)\right\Vert _{H_{0}^{1}(\Omega
)}^{2}ds+J(u(t))\leq J(u_{0}),\text{ }0\leq t\leq +\infty .
\end{equation*}
Also, the solution decay polynomially provided \(u_{0}\in \mathcal{W}
_{1}^{+}. \)
Proof.
The Faedo-Galerkin’s methods is used. In the space \(W_{0}^{1,p}\left( \Omega
\right) \cap W^{2,p}\left( \Omega \right) ,\) we take a bases \(
\{w_{j}\}_{j=1}^{\infty }\) and define the finite orthogonal space
\begin{equation*}
V_{m}=span\{w_{1},w_{2},…,w_{m}\}.
\end{equation*}
Let \(u_{0m}\) be an element of \(V_{m}\) such that
\begin{equation}
u_{0m}=\sum\limits_{j=1}^{m}a_{mj}w_{j}\to u_{0}\text{ strongly
in }W_{0}^{1,p}\left( \Omega \right) \cap W^{2,p}\left( \Omega \right) ,
\label{320}
\end{equation}
(9)
as \(m\to \infty .\) We construct the following approximate solution \(
u_{m}(x,t)\) of the problem (1)
\begin{equation}
u_{m}(x,t)=\sum\limits_{j=1}^{m}a_{mj}(t)w_{j}(x), \label{340}
\end{equation}
(10)
where the coefficients \(a_{mj}\) \((1\leq j\leq m)\) satisfy the ordinary
differential equations
\begin{equation}
\int\nolimits_{\Omega }u_{mt}w_{i}dx+\int\nolimits_{\Omega }\left\vert
\Delta u_{m}\right\vert ^{p-1}\Delta w_{i}dx+\int\nolimits_{\Omega }\nabla
u_{mt}\nabla w_{i}dx=\int\nolimits_{\Omega }u\left\vert u_{m}\right\vert
^{q-2}\ln \left( \left\vert u_{m}\right\vert \right) w_{i}dx,\text{ }
\label{361}
\end{equation}
(11)
for \(i\in \{1,2,…,m\},\) with the initial condition
\begin{equation}
a_{mj}(0)=a_{mj},\text{ }j\in \{1,2,…,m\}. \label{380}
\end{equation}
(12)
We multiply both sides of (11) by \(a_{mi}^{\prime },\) sum for \(
i=1,…,m\) and integrating with respect to time variable on \([0,t],\) we get
\begin{equation}
\int\nolimits_{0}^{t}\left\Vert u_{ms}(s)\right\Vert _{H_{0}^{1}(\Omega
)}^{2}ds+J(u_{m}(t))\leq J(u_{0m}),\text{ }0\leq t\leq T_{\max },
\label{540}
\end{equation}
(13)
where \(T_{\max }\) is the maximal existence time of solution \(u_{m}(t).\) We
shall prove that \(T_{\max }=+\infty .\) From (9), (13) and the
continuity of \(J\), we obtain
\begin{equation}
J(u_{m}(0)) \to J(u_{0m}),\text{ as }m\to \infty ,
\label{900}
\end{equation}
(14)
Thanks to \(J(u_{0})< d\) and the continuity of functional \(J,\) it follows from
(14) that
\begin{equation*}
J(u_{0m})< d,\text{ for sufficiently large }m.
\end{equation*}
And therefore, from (13), we obtain
\begin{equation}
\int\nolimits_{0}^{t}\left\Vert u_{ms}(s)\right\Vert _{H_{0}^{1}(\Omega
)}^{2}ds+J(u_{m}(t))< d,\text{ }0\leq t\leq T_{\max }, \label{915}
\end{equation}
(15)
for sufficiently large \(m.\) Next, we will study
\begin{equation}
u_{m}(t)\in \mathcal{W}_{1}^{+},\text{ }t\in \lbrack 0,T_{\max }),
\label{920}
\end{equation}
(16)
for sufficiently large \(m.\) We assume that (16) does not process and
think that there exists a sufficiently small time \(t_{0}\) such that \(
u_{m}(t_{0})\notin \mathcal{W}_{1}^{+}.\) Then, by continuity of \(
u_{m}(t_{0})\in \partial \mathcal{W}_{1}^{+}.\) So, we get
\begin{equation}
J(u_{m}(t_{0}))=d, \label{930}
\end{equation}
(17)
and
\begin{equation}
I(u_{m}(t_{0}))=0. \label{940}
\end{equation}
(18)
Nevertheless, by definition of \(d,\) we see that (17) could not
consist by (15) while if (18) holds then, we get
\begin{equation*}
J(u_{m}(t_{0}))\geq \underset{u\in \mathcal{N}}{\inf }J(u)=d,
\end{equation*}
which also contradicts with (15). Moreover, we have (16),
i.e., \(J(u_{m}(t))0,\) for any \(t\in \lbrack 0,T_{\max
}),\) for sufficiently large \(m.\) Then, from (5), we obtain
\begin{eqnarray*}
d &>&J(u_{m}(t)) \\
&=&\frac{1}{q}I(u_{m})+\left( \frac{1}{p}-\frac{1}{q}\right) \left\Vert
\Delta u_{m}\right\Vert _{p}^{p}+\frac{1}{q^{2}}\left\Vert u_{m}\right\Vert
_{q}^{q} \\
&\geq &\left( \frac{1}{p}-\frac{1}{q}\right) \left\Vert \Delta
u_{m}\right\Vert _{p}^{p}+\frac{1}{q^{2}}\left\Vert u_{m}\right\Vert _{q}^{q},
\end{eqnarray*}
which gives
\begin{equation}
\left\Vert u_{m}\left( t\right) \right\Vert _{q}^{q}< q^{2}d, \label{945}
\end{equation}
(19)
and
\begin{equation}
\left\Vert \Delta u_{m}\right\Vert _{p}^{p}< \frac{pq}{q-p}d. \label{947}
\end{equation}
(20)
Since \(u_{m}(x,t)\in \mathcal{W}_{1}^{+}\) for \(m\) large enough, it follows
from (5) that \(J(u_{m})\geq 0\) for \(s\) large enough. So, by (15
) it follows for \(m\) large enough
\begin{equation}
\int\nolimits_{0}^{t}\left\Vert u_{ms}(s)\right\Vert _{H_{0}^{1}(\Omega
)}^{2}ds< d. \label{950}
\end{equation}
(21)
By (20), we know that
\begin{equation*}
T_{\max }=+\infty .
\end{equation*}
It follows from (19) and (21) that there exist a function \(
X_{0}\) and a subsequence of \(\{u_{m}\}_{j=1}^{\infty }\) is indicated by \(
\{u_{m}\}_{j=1}^{\infty }\) such that
\begin{equation}
u_{m}\to u\text{ weakly* in }L^{\infty }(0,\infty
;W_{0}^{1,p}\left( \Omega \right) \cap W^{2,p}\left( \Omega \right) ),
\label{620}
\end{equation}
(22)
\begin{equation}
u_{mt}\to u_{t}\text{ weakly in }L^{2}(0,\infty ;H_{0}^{1}(\Omega
)), \label{640}
\end{equation}
(23)
\begin{equation*}
\left\vert \Delta u\right\vert ^{p-2}\Delta u\to \chi \text{ weakly
in }L^{\infty }\left( 0,\infty ;W^{-2,p^{\prime }}\left( \Omega \right)
\right).
\end{equation*}
By (22), (23) and Aubin-Lions compactness theorem, we obtain
\begin{equation*}
u_{m}\to u\text{ strongly in }C([0,+\infty ];L^{2}(\Omega )).
\end{equation*}
This yields that
\begin{equation}
u_{m}\left\vert u_{m}\right\vert ^{q-2}\ln \left\vert u_{m}\right\vert
\to u\left\vert u\right\vert ^{q-2}\ln \left\vert u\right\vert \text{
a.e. }(x,t)\in \Omega \times (0,+\infty ). \label{660}
\end{equation}
(24)
Moreover, since
\begin{equation*}
\alpha ^{r-1}\ln \alpha =-(e\left( r-1\right) )^{-1}\text{ for }\alpha
>1,
\end{equation*}
and
\begin{equation*}
\ln \alpha =2\ln \left( \alpha ^{\frac{1}{2}}\right) \leq 2\alpha ^{\frac{1}{
2}}\text{ for }\alpha >0.
\end{equation*}
By (19), we have
\begin{eqnarray}
\int\nolimits_{\Omega }\left( \left\vert u_{m}(t)\right\vert ^{q-1}\ln
\left\vert u_{m}(t)\right\vert \right) ^{\frac{2q}{2q-1}}dx
&=&\int\nolimits_{\Omega _{1}}\left( \left\vert u_{m}(t)\right\vert
^{q-1}\ln \left\vert u_{m}(t)\right\vert \right) ^{\frac{2q}{2q-1}}dx +\int\nolimits_{\Omega _{2}}\left( \left\vert u_{m}(t)\right\vert
^{q-1}\ln \left\vert u_{m}(t)\right\vert \right) ^{\frac{2q}{2q-1}}dx \notag
\\
&\leq &\left[ e\left( r-1\right) \right] ^{-\frac{2q}{2q-1}}\left\vert
\Omega \right\vert +2^{\frac{2q}{2q-1}}\int\nolimits_{\Omega _{2}}\left\vert
u_{m}(t)\right\vert ^{\frac{2q\left( q-1+\frac{1}{2}\right) }{2q-1}}dx
\notag \\
&=&\left[ e\left( r-1\right) \right] ^{-\frac{2q}{2q-1}}\left\vert \Omega
\right\vert +2^{\frac{2q}{2q-1}}\int\nolimits_{\Omega _{2}}\left\vert
u_{m}(t)\right\vert ^{q}dx \notag \\
&\leq &C_{d}:=\left[ e\left( r-1\right) \right] ^{-\frac{2q}{2q-1}
}\left\vert \Omega \right\vert +2^{\frac{2q}{2q-1}}q^{2}d, \label{960}
\end{eqnarray}
(25)
where
\begin{equation*}
\Omega _{1}=\{x\in \Omega :\left\vert u_{m}(t)\right\vert \leq 1\},\text{
and }\Omega _{2}=\{x\in \Omega :\left\vert u_{m}(t)\right\vert \geq 1\}.
\end{equation*}
Hence, it follows from (24) and (25) that
\begin{equation*}
u_{m}\left\vert u_{m}\right\vert ^{q-2}\ln \left\vert u_{m}\right\vert
\to u\left\vert u\right\vert ^{q-2}\ln \left\vert u\right\vert \text{
weakly* in }L^{\infty }(0,+\infty ;L^{\frac{2q}{2q-1}}(\Omega ))\text{ }.
\end{equation*}
Then integrating (11) respect to \(t\) for \(0\leq t< \infty ,\) we obtain
\begin{equation*}
\left\langle u_{t},w\right\rangle +\left\langle \chi (t),\Delta
w\right\rangle +\left\langle \nabla u_{t},\nabla w\right\rangle
=\int\nolimits_{\Omega }u\left\vert u\right\vert ^{p-2}\ln \left( \left\vert
u\right\vert \right) wdx,
\end{equation*}
for all \(w\in W_{0}^{2,p}\left( \Omega \right) \) and for almost every \(t\in
\left[ 0,\infty \right] .\) Finally, well known arguments of the theory of
monotone operators implied
\begin{equation*}
\chi =\left\vert \Delta u\right\vert ^{p-2}\Delta u,
\end{equation*}
which yields
\begin{equation*}
\left\langle u_{t},w\right\rangle +\left\langle \left\vert \Delta
u\right\vert ^{p-1},\Delta w\right\rangle +\left\langle \nabla u_{t},\nabla
w\right\rangle =\int\nolimits_{\Omega }u\left\vert u\right\vert ^{p-2}\ln
\left\vert u\right\vert wdx.
\end{equation*}
for all \(w\in W_{0}^{2,p}\left( \Omega \right) \) and for a.e. \(t\in \left[
0,\infty \right] .\)
Finally, we discuss the decay results.
Thanks to \(u(t)\in \mathcal{W}_{1}^{+},\) we deduce from (13) that
\begin{equation*}
\left( \frac{1}{p}-\frac{1}{q}\right) \left\Vert \Delta u\right\Vert
_{p}^{p}+\frac{1}{q^{2}}\left\Vert u\right\Vert _{q}^{q}\leq J(u(t))\leq
J(u_{0}),\text{ }t\in \lbrack 0,T].
\end{equation*}
By using (5) and Proposition 1, we put \(p\left( \frac{J(u_{0})}{M}
\right) ^{\frac{p}{n}}< \mu < p,\) we know
\begin{eqnarray*}
I(u(t)) &\geq &\left( 1-\frac{\mu }{p}\right) \left\Vert \Delta u\right\Vert
_{p}^{p}+\left( \frac{n}{p^{2}}\ln \left( \frac{p\mu e}{n\mathcal{L} _{p}}
\right) -\ln \left\Vert u(t)\right\Vert _{p}\right) \left\Vert
u(t)\right\Vert _{p}^{p} \\
&\geq &\left( 1-\frac{\mu }{p}\right) \left\Vert \Delta u\right\Vert
_{p}^{p}+\frac{1}{p}\ln \left( \frac{M}{J(u_{0})}\left( \frac{\mu }{p}
\right) ^{\frac{n}{p}}\right) \left\Vert u(t)\right\Vert _{p}^{p} \\
&=&C_{1}\left\Vert u(t)\right\Vert _{W^{2,p}\left( \Omega \right) }^{p}.
\end{eqnarray*}
Integrating the \(I(u(s))\) with respect to \(s\) over \((t,T)\), we obtain
\begin{eqnarray}
\int\nolimits_{t}^{T}I(u(s))ds
&=&-\int\nolimits_{t}^{T}\int\nolimits_{\Omega
}u_{s}(s)u(s)dxds-\int\nolimits_{t}^{T}\int\nolimits_{\Omega }\nabla
u_{s}(s)\nabla u(s)dxds \notag \\
&=&\frac{1}{2}\left\Vert u(t)\right\Vert _{H_{0}^{1}\left( \Omega \right)
}^{2}-\frac{1}{2}\left\Vert u(T)\right\Vert _{H_{0}^{1}\left( \Omega \right)
}^{2} \notag \\
&\leq &C_{2}\left\Vert u(t)\right\Vert _{W^{2,p}\left( \Omega \right) }^{2}.
\label{1040}
\end{eqnarray}
(26)
where \(C_{2}\) stand by the best constant in the embedding \(W^{2,p}\left(
\Omega \right) \hookrightarrow \to H_{0}^{1}\left( \Omega \right) \) From (26), we have
\begin{equation}
\int\nolimits_{t}^{T}\left\Vert u(t)\right\Vert _{W^{2,p}\left( \Omega
\right) }^{p}ds\leq \frac{1}{\omega }\left\Vert u(t)\right\Vert
_{W^{2,p}\left( \Omega \right) }^{2}\text{ for all }t\in \lbrack 0,T].
\label{1060}
\end{equation}
(27)
Let \(T\to +\infty \) in (27), we can get
\begin{equation*}
\int\nolimits_{t}^{\infty }\left\Vert u(t)\right\Vert _{W^{2,p}\left(
\Omega \right) }^{p}ds\leq \frac{1}{\omega }\left\Vert u(t)\right\Vert
_{W^{2,p}\left( \Omega \right) }^{2}.
\end{equation*}
From Lemma 5, we have \(f(t)=\left\Vert u(t)\right\Vert _{W^{2,p}\left(
\Omega \right) }^{2},\) \(\sigma =\frac{p}{2}-1,\) \(f(0)=1\)
\begin{equation*}
\left\Vert u(t)\right\Vert _{W^{2,p}\left( \Omega \right) }\leq \left\Vert
u_{0}\right\Vert _{W^{2,p}\left( \Omega \right) }\left( \frac{p}{2+\omega
\left\Vert u_{0}\right\Vert _{W_{0}^{2,p}\left( \Omega \right) }^{p-2}\left(
p-2\right) t}\right) ^{\frac{1}{p-2}},\text{ }t\geq 0.
\end{equation*}
The above inequality implies that the solution \(u\) decays polynomially.
Acknowledgments :
The author would like to thank Prof. Charles N. Moore of Washington State University, USA for his valuable suggestions on this article.
Author Contributions:
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Conflicts of Interest:
”The authors declares no conflict of interest.”
Data Availability:
All data required for this research is included within this paper.
Funding Information:
No funding is available for this research.
