1. Introduction
In this work, we investigate the
existence of global, decay and finite time blow up of solutions for the
parabolic type Kirchhoff equation with logarithmic source term
\begin{equation}
\left\{
\begin{array}{l}
u_{t}-M(\left\Vert \nabla u\right\Vert ^{2})\Delta u-\Delta u_{t}=u^{k-1}\ln
\left\vert u\right\vert -\oint\nolimits_{\Omega }u^{k-1}\ln \left\vert
u\right\vert dx, \  \  x\in \Omega ,\  t>0, \\
u(x,t)=0,\  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  x\in \partial \Omega ,\  t>0, \\
u(x,0)=u_{0}(x),\  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  x\in \Omega ,
\end{array}
\right. \label{10}
\end{equation}
(1)
where \(\Omega \) is a bound domain in \( \mathbb{R}^{n}(n\geq 1)\) with smooth boundary \(\partial \Omega \). Also, \(
M(s)=1+s^{\gamma },\) \(\left( \gamma >0\right) ,\) \(\oint\nolimits_{\Omega
}u_{0}dx=\frac{1}{\left\vert \Omega \right\vert }\int\nolimits_{\Omega
}u_{0}dx=0\) and
\[
\begin{cases}
2\gamma +2\leq k\leq +\infty ,& n=1,2, \\
2\gamma +2\leq k\leq \frac{2n}{n-2},& n\geq 3.
\end{cases}
\]
Many other authors studied the problem (1) in a more general form
\[
\begin{cases}
u_{t}-\Delta u=f(u)-\oint\nolimits_{\Omega }f(u)dx,& x\in \Omega,\ \ t>0, \\
\frac{\partial u(x,t)}{\partial \eta }=0,& x\in \partial \Omega ,\ \ t>0, \\
u(x,0)=u_{0}(x),& x\in \Omega ,
\end{cases}
\]
where \(\Omega \) is a bound domain in \(\mathbb{R}^{n}\) \((n\geq 1)\) with \(\left\vert \Omega \right\vert \) denoting its
Lebesgue measure, the function \(f(u)\) is usually taken to be a power of \(u\),
and \(n\) is the outer normal vector of \(\partial \Omega \). Studies of
logarithmic nonlinearity have a long history in physics as it occurs
naturally in different areas of physics such as supersymmetric field
theories, optics, quantum mechanics and inflationary cosmology [
1,
2]. When \(M\equiv 1\) and \(k=2,\) (1) become semilinear
pseudo-parabolic equation as follow
\begin{equation}
u_{t}-\Delta u-\Delta u_{t}=u\log \left\vert u\right\vert . \label{15}
\end{equation}
(2)
Chen and Tian [
3] obtained the global existence, behavior
of vacuum isolation and blow-up of solutions at \(+\infty \) of the Equation (2). Without \(\Delta u,\) (2) become the following semilinear
parabolic equation
\begin{equation}
u_{t}-\Delta u_{t}=u\log \left\vert u\right\vert . \label{18}
\end{equation}
(3)
Chen
et al., [
4] studied the global existence, decay and blow-up at \(
+\infty \) of solutions of the Equation (3). Yan and Yang [
5]
studied nonlocal parabolic equation with logarithmic nonlinearity
\[
u_{t}-\Delta u_{t}=u\log \left\vert u\right\vert -\oint\nolimits_{\Omega
}u\log \left\vert u\right\vert dx.
\]
Recently, they obtained the results under appropriate conditions on blow-up
and existion of the solutions.
Toualbia
et al., [
6] studied the initial boundary value problem
of a nonlocal parabolic equation with logarithmic nonlinearity
\[
u_{t}-div(\left\vert \nabla u\right\vert ^{p-2}\nabla u)=\left\vert
u\right\vert ^{p-2}u\log \left\vert u\right\vert -\oint\nolimits_{\Omega
}\left\vert u\right\vert ^{p-2}u\log \left\vert u\right\vert dx.
\]
They obtained the decay, blow up and nonextinction of solutions under
appropriate condition. Also, recently some authors studied the parabolic and
hyperbolic type equation with logarithmic source term (see [
7,
8,
9,
10,
11,
12,
13,
14,
15]).
The organization of the remaining part of this paper is as follows: In the
next Section 2, we introduce some lemmas which will be needed later. In Section
3, under some conditions, we get the unique global weak solution of the
problem (1). Moreover, we find that the decay of solutions. In the
lastly, we get the blow up theorem.
2. Preliminaries
Throughout this work, we adopt the following abbreviations
\[
\text{ }\left\Vert u\right\Vert _{s}=\left\Vert u\right\Vert _{L^{s}(\Omega
)},\text{ }\left\Vert u\right\Vert _{H_{0}^{1}(\Omega )}=\left( \left\Vert
\nabla u\right\Vert ^{2}+\left\Vert u\right\Vert ^{2}\right) ^{\frac{1}{2}},
\]
for \(1< s< \infty .\)
The energy functional \(J\) and Nehari functional \(I\) defined on \(
H_{0}^{1}(\Omega )\backslash \{0\}\) as follow
\begin{equation}
J(u)=\frac{1}{2}\left\Vert \nabla u\right\Vert ^{2}+\frac{1}{2\left( \gamma
+1\right) }\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right) }-\frac{
1}{k}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert
u\right\vert dx+\frac{1}{k^{2}}\int\nolimits_{\Omega }\left\vert
u\right\vert ^{k}dx, \label{30}
\end{equation}
(4)
and
\begin{equation}
I(u)=\left\Vert \nabla u\right\Vert ^{2}+\left\Vert \nabla u\right\Vert
^{2\left( \gamma +1\right) }-\int\nolimits_{\Omega }\left\vert u\right\vert
^{k}\ln \left\vert u\right\vert dx. \label{40}
\end{equation}
(5)
By (4) and (5), we obtain
\begin{equation}
J(u)=\frac{1}{k}I(u)+\frac{k-2}{2k}\left\Vert \nabla u\right\Vert ^{2}+\frac{
k-2\gamma -2}{2k\left( \gamma +1\right) }\left\Vert \nabla u\right\Vert
^{2\left( \gamma +1\right) }+\frac{1}{k^{2}}\int\nolimits_{\Omega
}\left\vert u\right\vert ^{k}dx. \label{50}
\end{equation}
(6)
Let
\[
\mathcal{N}=\{u\in H_{0}^{1}(\Omega )\backslash \{0\}:I(u)=0\},
\]
be the Nehari manifold. Thus, we may define
\begin{equation}
d=\underset{u\in \mathcal{N}}{\inf }J(u), \label{70}
\end{equation}
(7)
\(d\) is positive and is obtained by some \(u\in \mathcal{N}.\)
Lemma 1.
Let \(u\in H_{0}^{1}(\Omega )\backslash \{0\},\) we consider the function \(
j:\lambda \to J(\lambda u)\) for \(\lambda >0\). 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 strictly increasing on \((0,\lambda ^{\ast }),\)
strictly decreasing on \((\lambda ^{\ast },+\infty )\) and takes the maximum
at \(\lambda ^{\ast };\) \(I(\lambda u)=\lambda j^{\prime }(\lambda )\) and
\[
I(\lambda u)\left\{
\begin{array}{l}
>0,\text{ }0< \lambda < \lambda ^{\ast }, \\
=0,\text{ }\lambda =\lambda ^{\ast }, \\
< 0,\text{ }\lambda ^{\ast }< \lambda < +\infty .
\end{array}
\right.
\]
Proof.
For \(u\in H_{0}^{1}(\Omega )\backslash \{0\},\) by the definition of \(j,\) we
get
\begin{align}
j(\lambda ) &=\frac{1}{2}\left\Vert \nabla \left( \lambda u\right)
\right\Vert ^{2}+\frac{1}{2\left( \gamma +1\right) }\left\Vert \nabla \left(
\lambda u\right) \right\Vert ^{2\left( \gamma +1\right) }-\frac{1}{k}
\int\nolimits_{\Omega }\left\vert \lambda u\right\vert ^{k}\ln \left\vert
\lambda u\right\vert dx+\frac{1}{k^{2}}\int\nolimits_{\Omega }\left\vert
\lambda u\right\vert ^{k}dx \notag \\
&=\frac{\lambda ^{2}}{2}\left\Vert \nabla u\right\Vert ^{2}+\frac{\lambda
^{2\left( \gamma +1\right) }}{2\left( \gamma +1\right) }\left\Vert \nabla
u\right\Vert ^{2\left( \gamma +1\right) }-\frac{\lambda ^{k}}{k}
\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert
u\right\vert dx-\frac{\lambda ^{k}}{k}\int\nolimits_{\Omega }\left\vert u\right\vert
^{k}\ln \lambda dx+\frac{\lambda ^{k}}{k^{2}}\int\nolimits_{\Omega
}\left\vert u\right\vert ^{k}dx. \label{100}
\end{align}
(8)
It is clear that
(i) holds due to \(\int\nolimits_{\Omega
}\left\vert u\right\vert ^{k}dx\neq 0.\) We get
\begin{align*}
\frac{d}{d\lambda }j(\lambda ) =&\lambda \left\Vert \nabla u\right\Vert
^{2}+\lambda ^{2\gamma +1}\left\Vert \nabla u\right\Vert ^{2\left( \gamma
+1\right) }-\lambda ^{k-1}\int\nolimits_{\Omega }\left\vert u\right\vert
^{k}\ln \left\vert u\right\vert dx \\
&-\lambda ^{k-1}\ln \lambda \int\nolimits_{\Omega }\left\vert u\right\vert
^{k}dx-\frac{\lambda ^{k-1}}{k}\int\nolimits_{\Omega }\left\vert
u\right\vert ^{k}dx+\frac{\lambda ^{k-1}}{k}\int\nolimits_{\Omega
}\left\vert u\right\vert ^{k}dx \\
=&\lambda \left\Vert \nabla u\right\Vert ^{2}+\lambda ^{2\gamma
+1}\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right) }-\lambda
^{k-1}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert
u\right\vert dx-\lambda ^{k-1}\ln \lambda \int\nolimits_{\Omega }\left\vert
u\right\vert ^{k}dx \\
=&\lambda \left( \left\Vert \nabla u\right\Vert ^{2}+\lambda ^{2\gamma
}\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right) }-\lambda
^{k-2}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert
u\right\vert dx-\lambda ^{k-2}\ln \lambda \int\nolimits_{\Omega }\left\vert
u\right\vert ^{k}dx\right) .
\end{align*}
Let
\begin{equation*}
\varphi \left( \lambda \right) :=\lambda ^{2\gamma }\left\Vert \nabla
u\right\Vert ^{2\left( \gamma +1\right) }-\lambda
^{k-2}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert
u\right\vert dx-\lambda ^{k-2}\ln \lambda \int\nolimits_{\Omega }\left\vert
u\right\vert ^{k}dx.
\end{equation*}
Then from \(2\gamma +2\leq k\) we can conclude that \(\lim_{\lambda
\to \infty }\varphi \left( \lambda \right) =-\infty ,\) \(\varphi
\left( \lambda \right) \) is monotone decreasing when \(\lambda >\lambda
^{\ast }\) and there exists a unique \(\lambda ^{\ast }\) such that \(\varphi
\left( \lambda \right) \mid _{\lambda =\lambda ^{\ast }=0}\). Hence, we
obtain there is a \(\lambda ^{\ast }>0\) such that \(\left\Vert \nabla
u\right\Vert ^{2}+\varphi \left( \lambda \right) =0\), which means \(\frac{d}{
d\lambda }J\left( \lambda u\right) \mid _{\lambda =\lambda ^{\ast }=0}\). The
conclusion
(iii) directly follows from the proof of the conclusion
(ii) and
\begin{align*}
I(\lambda u) =&\left\Vert \nabla (\lambda u)\right\Vert ^{2}+\left\Vert
\nabla (\lambda u)\right\Vert ^{2\left( \gamma +1\right)
}-\int\nolimits_{\Omega }\left\vert \lambda u\right\vert ^{k}\ln \left\vert
\lambda u\right\vert dx \\
=&\lambda ^{2}\left\Vert \nabla u\right\Vert ^{2}+\lambda ^{2\left( \gamma
+1\right) }\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right)
}-\lambda ^{k}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln
\left\vert u\right\vert dx-\lambda ^{k}\int\nolimits_{\Omega }\left\vert
u\right\vert ^{k}\ln \lambda dx \\
=&\lambda ^{2}\left\Vert \nabla u\right\Vert ^{2}+\lambda ^{2\left( \gamma
+1\right) }\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right)
}-\lambda ^{k}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln
\left\vert u\right\vert dx-\lambda ^{k}\ln \lambda \int\nolimits_{\Omega
}\left\vert u\right\vert ^{k}dx \\
=&\lambda \left( \lambda \left\Vert \nabla u\right\Vert ^{2}+\lambda
^{2\gamma +1}\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right)
}-\lambda ^{k-1}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln
\left\vert u\right\vert dx-\lambda ^{k-1}\ln \lambda \int\nolimits_{\Omega
}\left\vert u\right\vert ^{k}dx\right) \\
=&\lambda j^{\prime }(\lambda ).
\end{align*}
Thus, \(I(\lambda u)>0\) for \(0< \lambda < \lambda ^{\ast },\ I(\lambda u)< 0\)
for \(\lambda ^{\ast }< \lambda < +\infty \) and \(I(\lambda ^{\ast }u)=0.\) The
proof is complete.
Lemma 2. [16]
- a)   For any function \(u\in W_{0}^{1,p}(\Omega )\) such that
\(
\left\Vert u\right\Vert _{q}\leq B_{q,p}\left\Vert \nabla u\right\Vert _{p,}
\)
for all \(1\leq q\leq p^{\ast }\) where \(p^{\ast }=\frac{np}{n-p}\) if \(n>p\)
and \(p^{\ast }=\infty \) if \(n\leq p.\) The best constant \(B_{q,p}\) depends
only on \(\Omega ,\) \(n,\) \(p\) and \(q.\) We will denote the constant \(B_{p,p}\)
by \(B_{p}.\)
- b)   For any \(u\in W_{0}^{1,p}(\Omega ),\) \(p\geq 1,\) and \(r\geq 1,\) the
inequality
\[
\left\Vert u\right\Vert _{q}\leq C\left\Vert \nabla u\right\Vert
_{p}^{\theta }\left\Vert u\right\Vert _{r}^{1-\theta },
\]
is valid, where
\(
\theta =\left( \frac{1}{r}-\frac{1}{q}\right) \left( \frac{1}{n}-\frac{1}{p}
+ \frac{1}{r}\right) ^{-1}
\)
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 3. [17] Assume that \(f:R^{+} \to R^{+}\) be a nonincreasing
function and \(\sigma \) is a nonnegative constant such that
\[
\int\nolimits_{t}^{+\infty }f^{1+\sigma }(s)ds\leq \frac{1}{\omega }
f^{\sigma }(0)f(t), \  \  \forall t\geq 0.
\]
Hence
- (a)   \(f(t)\leq f(0)e^{1-\omega t},\) \(\forall t\geq 0,\) whenever \(\sigma =0;\)
- (b)   \(f(t)\leq f(0)\left( \frac{1+\sigma }{1+\omega \sigma t}\right) ^{\frac{
1 }{\sigma }},\) \(\forall t\geq 0,\) whenever \(\sigma >0.\)
3. Main results
Now as in [
18], we introduce the follows sets:
\begin{align*}
\mathcal{W}_{1} &=\{u\in H_{0}^{1}(\Omega )\backslash \{0\}:J(u)0\},\\
\mathcal{W}
_{2}^{+}&=\{u\in \mathcal{W}_{2}:I(u)>0\},\\
\mathcal{W}^{+}&=\mathcal{
W }_{1}^{+}\cup \mathcal{W}_{2}^{+}, \\
\mathcal{W}_{1}^{-} &=\{u\in \mathcal{W}_{1}:I(u)< 0\},\\
\mathcal{W}
_{2}^{-}&=\{u\in \mathcal{W}_{2}:I(u)< 0\},\\
\mathcal{W}^{-}&=\mathcal{
W }_{1}^{-}\cup \mathcal{W}_{2}^{-}.
\end{align*}
Definition 4.
(Maximal existence time). Assume that \(u\) be weak solutions of problem (1). We define the maximal existence time \(T_{\max }\) as follows
\[
T_{\max }=\sup \{T>0:u(t)\text{ exists on }[0,T]\}.
\]
Then
- (i) If \(T_{\max }< \infty ,\) we see that \(u\) blows up in finite time and \(
T_{\max }\) is the blow-up time;
- (ii) If \(T_{\max }=\infty ,\) we see that \(u\) is global.
Definition 5.
(Weak solution). We define a function \(u\in L^{\infty }(0,T;H_{0}^{1}(\Omega
))\) 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 \(
u(0)=u_{0}\in H_{0}^{1}(\Omega )\backslash \{0\},\) and
\begin{align*}
& + ++ \\
& =\int\nolimits_{\Omega }u^{k-1}\ln \left\vert u\right\vert
wdx-\int\nolimits_{\Omega }\oint\nolimits_{\Omega }u^{k-1}\ln \left\vert
u\right\vert wdsdx,\text{ }
\end{align*}
for all \(w\in H_{0}^{1}(\Omega ),\) and for a.e. \(t\in \lbrack 0,T].\)
Theorem 6.
(Global existence). Let \(u_{0}\in \) \(\mathcal{W}^{+},\) \(0< J(u_{0})0.\) Hence, there is a unique global weak solution \(u\) of (1)
satisfying \(u(0)=u_{0}.\) We obtain \(u(t)\in \mathcal{W}^{+}\) holds for all \(
0\leq t< +\infty ,\) and the energy estimate
\[
\int\nolimits_{0}^{t}\left\Vert u_{s}(s)\right\Vert _{H_{0}^{1}(\Omega
)}^{2}ds+J(u(t))\leq J(u_{0}),\text{ }0\leq t\leq +\infty .
\]
Also, the solution decay exponentially provided \(u_{0}\in \mathcal{W}
_{1}^{+}.\)
Proof. (Global existence) In the space \(H_{0}^{1}(\Omega ),\) we take a
Galerkin bases \(\{w_{j}\}_{j=1}^{\infty }\) and define the finite dimensional
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 }H_{0}^{1}(\Omega ), \label{320}
\end{equation}
(9)
as \(m\to \infty .\) We can find the approximate solution \(u_{m}(x,t)\)
of the problem (1) in the form
\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{align}
&\int\nolimits_{\Omega }u_{mt}w_{i}dx+\int\nolimits_{\Omega }\nabla
u_{m}\nabla w_{i}dx +\int\nolimits_{\Omega }\left\Vert \nabla u_{m}\right\Vert ^{2\gamma
}\nabla u_{m}\nabla w_{i}dx\text{ }+\int\nolimits_{\Omega }\nabla
u_{mt}\nabla w_{i}dx \notag \\
&=\int\nolimits_{\Omega }\left\vert u_{m}\right\vert ^{k-1}\ln \left\vert
u_{m}\right\vert )w_{i}dx-\int\nolimits_{\Omega }\oint\nolimits_{\Omega
}\left\vert u_{m}\right\vert ^{k-1}\ln \left\vert u_{m}\right\vert
)w_{i}dsdx,\text{ } \label{360}
\end{align}
(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)
Now, multiplying (11) by \(a_{mi}^{\prime },\) summing over \(i\) from \(1\)
to \(m\) and integrating with related to time variable on \([0,t],\) 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))\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 \(u_{m}(t).\) By (9), (13) and the continuity of \(J\) that
\begin{equation}
J(u_{m}(0))\to J(u_{0m}),\text{ as }m\to \infty ,
\label{900}
\end{equation}
(14)
with \(J(u_{0})< d\) and the continuity of functional \(J,\) by (14), we
have
\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 show that
\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 suppose that (16) does not hold and
think that there exists a smallest 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 obtain
\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)
It is clear that (17) could not occur by (15) while if (18) holds then, by definition of \(d,\) we get
\begin{equation*}
J(u_{m}(t_{0}))\geq \underset{u\in \mathcal{N}}{\inf }J(u)=d,
\end{equation*}
which contradicts with (15). Thus, we get (16), i.e., \(
J(u_{m}(t))0,\) for any \(t\in \lbrack 0,T_{\max }),\)
for sufficiently large \(m.\) Hence, by (6), we get
\begin{align*}
d &>J(u_{m}(t)) \\
&=\frac{1}{k}I(u_{m}(t))+\frac{k-2}{2k}\left\Vert \nabla
u_{m}(t)\right\Vert ^{2}+\frac{k-2\gamma -2}{2k(\gamma +1)}\left\Vert \nabla
u_{m}(t)\right\Vert ^{2\left( \gamma +1\right) }+\frac{1}{k^{2}}
\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx \\
&\geq \frac{k-2}{2k}\left\Vert \nabla u_{m}(t)\right\Vert ^{2}+\frac{\gamma
}{2(\gamma +1)}\left\Vert \nabla u_{m}(t)\right\Vert ^{2\left( \gamma
+1\right) }+\frac{1}{k^{2}}\int\nolimits_{\Omega }\left\vert u\right\vert
^{k}dx \geq \frac{k-2}{2k}\left\Vert \nabla u_{m}(t)\right\Vert ^{2}.
\end{align*}
And therefore, we deduce from (15) that
\begin{equation*}
\int\nolimits_{0}^{t}\left\Vert u_{ms}(s)\right\Vert _{H_{0}^{1}(\Omega
)}^{2}ds+\frac{k-2}{2k}\left\Vert \nabla u_{m}(t)\right\Vert ^{2}< d.
\end{equation*}
This inequality implies \(T_{\max }=+\infty .\) Further, by the logarithmic
inequality, we get
\begin{align*}
\left\Vert \nabla u_{m}(t)\right\Vert ^{2} =&2J(u_{m}(t))+\frac{2}{k}
\int\nolimits_{\Omega }\left\vert u_{m}(t)\right\vert ^{k}\ln \left\vert
u_{m}(t)\right\vert dx -\frac{2}{k^{2}}\left\Vert u_{m}(t)\right\Vert ^{2}-\frac{1}{\left( \gamma
+1\right) }\left\Vert \nabla u_{m}(t)\right\Vert ^{2\left( \gamma +1\right) }
\\
\leq &2J(u_{m}(0))+\frac{2}{k}\int\nolimits_{\Omega }\left\vert
u_{m}(t)\right\vert ^{k}\ln \left\vert u_{m}(t)\right\vert dx.
\end{align*}
This implies that
\begin{equation*}
\left\Vert \nabla u_{m}(t)\right\Vert ^{2}\leq 2J(u_{m}(0))+\frac{2}{k}
\int\nolimits_{\Omega }\left\vert u_{m}(t)\right\vert ^{k}\ln \left\vert
u_{m}(t)\right\vert dx.
\end{equation*}
We deduce that
\begin{equation*}
\left\Vert \nabla u_{m}(t)\right\Vert ^{2}\leq C_{d},\text{ }\forall t\in
\lbrack 0,T_{\max }).
\end{equation*}
(Decay estimates) Thanks to \(u_{0}\in \mathcal{W}_{1}^{+}\), we
deduce from (6) that
\begin{align}
J(u_{0}) &>J(u(t)) \notag \\
&=\frac{1}{k}I(u(t))+\frac{k-2}{2k}\left\Vert \nabla u(t)\right\Vert ^{2}+
\frac{k-2\gamma -2}{2k\left( \gamma +1\right) }\left\Vert \nabla
u(t)\right\Vert ^{2\left( \gamma +1\right) }+\frac{1}{k^{2}}
\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx, \notag \\
&\geq \frac{k-2}{2k}\left\Vert \nabla u(t)\right\Vert ^{2}+\frac{k-2\gamma
-2}{2k\left( \gamma +1\right) }\left\Vert \nabla u(t)\right\Vert ^{2\left(
\gamma +1\right) }+\frac{1}{k^{2}}\int\nolimits_{\Omega }\left\vert
u\right\vert ^{k}dx. \label{970}
\end{align}
(19)
By \(I(u(t))>0,\) (7) and Lemma 1, there exists a \(\lambda ^{\ast }>1\)
such that \(I(\lambda ^{\ast }u(t))=0.\) We have
\begin{align}
d &\leq J(\lambda ^{\ast }u(t)) \notag \\
&=\frac{1}{k}I(\lambda ^{\ast }u(t))+\frac{k-2}{2k}\left\Vert \nabla \left(
\lambda ^{\ast }u(t)\right) \right\Vert ^{2}+\frac{k-2\gamma -2}{2k\left(
\gamma +1\right) }\left\Vert \nabla \left( \lambda ^{\ast }u(t)\right)
\right\Vert ^{2\left( \gamma +1\right) }+\frac{1}{k^{2}}\int\nolimits_{
\Omega }\left\vert \lambda ^{\ast }u\left( t\right) \right\vert ^{k}dx
\notag \\
&=\frac{k-2}{2k}\left\Vert \nabla \left( \lambda ^{\ast }u(t)\right)
\right\Vert ^{2}+\frac{k-2\gamma -2}{2k\left( \gamma +1\right) }\left\Vert
\nabla \left( \lambda ^{\ast }u(t)\right) \right\Vert ^{2\left( \gamma
+1\right) }+\frac{1}{k^{2}}\int\nolimits_{\Omega }\left\vert \lambda ^{\ast
}u\left( t\right) \right\vert ^{k}dx \notag \\
&=(\lambda ^{\ast })^{2}\left( \frac{k-2}{2k}\right) \left\Vert \nabla
u(t)\right\Vert ^{2}+(\lambda ^{\ast })^{2\left( \gamma +1\right) }\left(
\frac{k-2\gamma -2}{2k\left( \gamma +1\right) }\right) \left\Vert \nabla
u(t)\right\Vert ^{2\left( \gamma +1\right) }
+(\lambda ^{\ast })^{k}\left( \frac{1}{k^{2}}\right) \int\nolimits_{\Omega
}\left\vert u\left( t\right) \right\vert ^{k}dx \notag \\
&\leq (\lambda ^{\ast })^{k}\left( \frac{k-2}{2k}\left\Vert \nabla
u(t)\right\Vert ^{2}+\frac{k-2\gamma -2}{2k\left( \gamma +1\right) }
\left\Vert \nabla u(t)\right\Vert ^{2\left( \gamma +1\right) }+\frac{1}{k^{2}
}\int\nolimits_{\Omega }\left\vert u\left( t\right) \right\vert
^{k}dx\right) . \label{980}
\end{align}
(20)
Using (19) and (20), we have
\( d\leq (\lambda ^{\ast })^{k}J(u_{0}),
\)
which implies that
\begin{equation}
\lambda ^{\ast }\geq \left( \frac{d}{J(u_{0})}\right) ^{\frac{1}{k}}.
\label{985}
\end{equation}
(21)
By (5), we get
\begin{align}
0 =&I(\lambda ^{\ast }u(t))=\left\Vert \nabla (\lambda ^{\ast }u(t))\right\Vert ^{2}+\left\Vert
\nabla (\lambda ^{\ast }u(t))\right\Vert ^{2\left( \gamma +1\right)
}-\int\nolimits_{\Omega }\left\vert \lambda ^{\ast }u(t)\right\vert ^{k}\ln
\left\vert \lambda ^{\ast }u(t)\right\vert dx \notag \\
=&(\lambda ^{\ast })^{2}\left\Vert \nabla u(t)\right\Vert ^{2}+(\lambda
^{\ast })^{2\left( \gamma +1\right) }\left\Vert \nabla u(t)\right\Vert
^{2\left( \gamma +1\right) }
-(\lambda ^{\ast })^{k}\int\nolimits_{\Omega }\left\vert u(t)\right\vert
^{k}\ln \left\vert u(t)\right\vert dx-(\lambda ^{\ast })^{k}\ln (\lambda
^{\ast })\int\nolimits_{\Omega }\left\vert u\left( t\right) \right\vert
^{k}dx \notag \\
=&(\lambda ^{\ast })^{k}I(u(t))+(\lambda ^{\ast })^{2}\left\Vert \nabla
u(t)\right\Vert ^{2}+(\lambda ^{\ast })^{2\left( \gamma +1\right)
}\left\Vert \nabla u(t)\right\Vert ^{2\left( \gamma +1\right) } \notag \\
&-(\lambda ^{\ast })^{k}\left\Vert \nabla u(t)\right\Vert ^{2}-(\lambda
^{\ast })^{k}\left\Vert \nabla u(t)\right\Vert ^{2\left( \gamma +1\right)
}-(\lambda ^{\ast })^{k}\ln (\lambda ^{\ast })\int\nolimits_{\Omega
}\left\vert u\left( t\right) \right\vert ^{k}dx \notag \\
=&(\lambda ^{\ast })^{k}I(u(t))+\left[ (\lambda ^{\ast })^{2}-(\lambda
^{\ast })^{k}\right] \left\Vert \nabla u(t)\right\Vert ^{2} \notag \\
&+\left[ (\lambda ^{\ast })^{2\left( \gamma +1\right) }-(\lambda ^{\ast
})^{k}\right] \left\Vert \nabla u(t)\right\Vert ^{2\left( \gamma +1\right)
}-(\lambda ^{\ast })^{k}\ln (\lambda ^{\ast })\int\nolimits_{\Omega
}\left\vert u\left( t\right) \right\vert ^{k}dx. \label{990}
\end{align}
(22)
Using (21) and (22), we have
\begin{align*}
(\lambda ^{\ast })^{k}I(u(t)) =&\left[ (\lambda ^{\ast })^{k}-(\lambda
^{\ast })^{2}\right] \left\Vert \nabla u(t)\right\Vert ^{2}+\left[ (\lambda
^{\ast })^{k}-(\lambda ^{\ast })^{2\left( \gamma +1\right) }\right]
\left\Vert \nabla u(t)\right\Vert ^{2\left( \gamma +1\right) }
+(\lambda ^{\ast })^{k}\ln (\lambda ^{\ast })\int\nolimits_{\Omega
}\left\vert u\left( t\right) \right\vert ^{k}dx \\
\geq &\left[ (\lambda ^{\ast })^{k}-(\lambda ^{\ast })^{2}\right]
\int\nolimits_{\Omega }\left\vert u\left( t\right) \right\vert ^{k}dx,
\end{align*}
which implies that
\begin{equation}
I(u(t))\geq \left[ 1-(\lambda ^{\ast })^{2-k}\right] \left\Vert \nabla
u(t)\right\Vert ^{2}. \label{995}
\end{equation}
(23)
It follows from (21) and (23) that
\begin{align}
I(u(t)) \geq &\left[ 1-(\lambda ^{\ast })^{2-k}\right] \left\Vert \nabla
u(t)\right\Vert ^{2} \notag \\
\geq &\left[ 1-\left( \frac{d}{J(u_{0})}\right) ^{\frac{2-k}{k}}\right]
\left\Vert \nabla u(t)\right\Vert ^{2} \notag \\
\geq &C\left[ 1-\left( \frac{d}{J(u_{0})}\right) ^{\frac{2-k}{k}}\right]
\left\Vert u(t)\right\Vert ^{2}, \label{1000}
\end{align}
(24)
where \(C\) is constant. Hence, by (24), we get
\begin{align}
I(u(t)) \geq &\frac{1}{2}\left[ 1-\left( \frac{d}{J(u_{0})}\right) ^{\frac{
2-k}{k}}\right] \left\Vert \nabla u(t)\right\Vert ^{2}+\frac{C}{2}\left[
1-\left( \frac{d}{J(u_{0})}\right) ^{\frac{2-k}{k}}\right] \left\Vert
u(t)\right\Vert ^{2} \notag \\
\geq &C_{1}\left( \left\Vert \nabla u(t)\right\Vert ^{2}+\left\Vert
u(t)\right\Vert ^{2}\right) \notag \\
=&C_{1}\left\Vert u(t)\right\Vert _{H_{0}^{1}\left( \Omega \right) }^{2},
\label{1020}
\end{align}
(25)
where
\begin{equation*}
C_{1}=\max \left\{ \frac{1}{2}\left[ 1-\left( \frac{d}{J(u_{0})}\right) ^{
\frac{2-k}{k}}\right] ,\frac{C}{2}\left[ 1-\left( \frac{d}{J(u_{0})}\right)
^{\frac{2-k}{k}}\right] \right\} .
\end{equation*}
Integrating the \(I(u(s))\) with respect to \(s\) over \((t,T)\), we get
\begin{align}
\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 &\frac{1}{2}\left\Vert u(t)\right\Vert _{H_{0}^{1}\left( \Omega
\right) }^{2}. \label{1040}
\end{align}
(26)
From (25) and (26), we have
\begin{equation}
\int\nolimits_{t}^{T}C_{1}\left\Vert u(t)\right\Vert _{H_{0}^{1}\left(
\Omega \right) }^{2}ds\leq \frac{1}{2}\left\Vert u(t)\right\Vert
_{H_{0}^{1}\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 have
\begin{equation*}
\int\nolimits_{t}^{\infty }\left\Vert u(t)\right\Vert _{H_{0}^{1}\left(
\Omega \right) }^{2}ds\leq \frac{1}{2C_{1}}\left\Vert u(t)\right\Vert
_{H_{0}^{1}\left( \Omega \right) }^{2}.
\end{equation*}
From Lemma 3, we get
\begin{equation*}
\left\Vert u(t)\right\Vert _{H_{0}^{1}\left( \Omega \right) }^{2}\leq
\left\Vert u(0)\right\Vert _{H_{0}^{1}\left( \Omega \right)
}^{2}e^{1-2C_{1}t}.
\end{equation*}
The above inequality satisfies that the solution \(u\) decays exponentially.
Theorem 7.
(Blow up). Let \(u_{0}\in W_{1}^{-}\) and assume that \(u(t)\) be a unique weak
solution to the problem (1). Then \(u(t)\) blows up in finite time,
that is, there exists \(T_{\ast }>0\) such that
\[
\underset{t\to T_{\ast }}{\lim }\left\Vert u(t)\right\Vert
_{H_{0}^{1}(\Omega )}^{2}=\infty .
\]
Proof.
We show that \(u(t)\) blows up at a finite time. Using contradiction, we
suppose that \(u(t)\) is global. We contract a function \(\phi :[0,\infty
)\to\mathbb{R}^{+},\) and
\begin{equation}
\phi (t)=\int\nolimits_{0}^{t}\left\Vert u(s)\right\Vert _{H_{0}^{1}(\Omega
)}^{2}ds. \label{1475}
\end{equation}
(28)
Then, thorough direct calculation, we have
\begin{align}
\phi ^{\prime }(t) =&\left\Vert u(s)\right\Vert _{H_{0}^{1}(\Omega )}^{2}
=2\int\nolimits_{0}^{t}\int\nolimits_{\Omega }\left( u_{s}u+\nabla
u_{s}\nabla u\right) dxds. \label{1500}
\end{align}
(29)
By (5) and (29), we have
\begin{align}
\phi ^{\prime \prime }(t) &=2\int\nolimits_{\Omega }\left( u_{s}u+\nabla
u_{s}\nabla u\right) dx \notag \\
&=2\int\nolimits_{\Omega }u\left( u_{s}-\Delta u_{s}\right) dx \notag \\
&=2\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert
u\right\vert dx+2\int\nolimits_{\Omega }M(\left\Vert \nabla u\right\Vert
^{2})u\Delta udx \notag \\
&=2\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert
u\right\vert dx-2\int\nolimits_{\Omega }(1+\left\Vert \nabla u\right\Vert
^{2\gamma })(\nabla u)^{2}dx \notag \\
&=-2I(u). \label{1525}
\end{align}
(30)
By (30) and \(I(u)0\), hence
\begin{equation}
\phi ^{\prime }(t)>\phi ^{\prime }(0)=\left\Vert u_{0}\right\Vert
_{H_{0}^{1}(\Omega )}^{2}>0. \label{1530}
\end{equation}
(31)
From the Hölder inequality and combining (30), we get
\begin{align}
\frac{1}{4}\left( \phi ^{\prime }(t)-\phi ^{\prime }(0)\right) ^{2} &=\frac{
1}{4}\left( \int\nolimits_{0}^{t}\phi ^{\prime \prime }(s)ds\right) ^{2}
\notag \\
&=\left( \int\nolimits_{0}^{t}\int\nolimits_{\Omega }\left( u_{s}u+\nabla
u_{s}\nabla u\right) dxds\right) ^{2} \notag \\
&\leq \int\nolimits_{0}^{t}\left\Vert u(s)\right\Vert _{H_{0}^{1}(\Omega
)}^{2}ds\int\nolimits_{0}^{t}\left\Vert u_{s}\right\Vert _{H_{0}^{1}(\Omega
)}^{2}ds. \label{1540}
\end{align}
(32)
It follows from (6) and (30) that
\begin{align}
\phi ^{\prime \prime }(t) &=-2I(u)
=-2kJ(u)+\left( k-2\right) \left\Vert \nabla u\right\Vert ^{2}+\frac{
k-2\gamma -2}{\gamma +1}\left\Vert \nabla u\right\Vert ^{2\left( \gamma
+1\right) }+\frac{2}{k}\left\Vert u\right\Vert _{k}^{k} \notag \\
&\geq -2kJ(u_{0})+2k\int\nolimits_{0}^{t}\left\Vert u_{s}(s)\right\Vert
_{H_{0}^{1}(\Omega )}^{2}ds+\left( k-2\right) \left\Vert \nabla u\right\Vert
^{2}+\frac{k-2\gamma -2}{\gamma +1}\left\Vert \nabla u\right\Vert ^{2\left(
\gamma +1\right) }+\frac{2}{k}\left\Vert u\right\Vert _{k}^{k} \notag \\
&\geq 2k\left( d-J(u_{0})\right) +2k\int\nolimits_{0}^{t}\left\Vert
u_{s}(s)\right\Vert _{H_{0}^{1}(\Omega )}^{2}ds. \label{1550}
\end{align}
(33)
Using (28), (32) and (33), we get
\begin{align}
\phi (t)\phi ^{\prime \prime }(t) &\geq 2k\int\nolimits_{0}^{t}\left\Vert
u(s)\right\Vert _{H_{0}^{1}(\Omega )}^{2}ds\int\nolimits_{0}^{t}\left\Vert
u_{s}(s)\right\Vert _{H_{0}^{1}(\Omega )}^{2}ds+2k\left( d-J(u_{0})\right)
\phi (t) \notag \\
&\geq 2k\left( d-J(u_{0})\right) \phi (t)+\frac{k}{2}\left( \phi ^{\prime
}(t)-\phi ^{\prime }(0)\right) ^{2}. \label{1625}
\end{align}
(34)
By (34), we get
\begin{equation}
\phi (t)\phi ^{\prime \prime }(t)-\frac{k}{2}\left( \phi ^{\prime }(t)-\phi
^{\prime }(0)\right) ^{2}\geq 2k(d-J(u_{0}))\left\Vert u_{0}\right\Vert
_{H_{0}^{1}(\Omega )}^{2}t_{0}>0. \label{1675}
\end{equation}
(35)
Choose \(T>t_{0}\) sufficiently large and let
\begin{equation*}
\psi (t)=\phi (t)+(T-t)\left\Vert u_{0}\right\Vert _{H_{0}^{1}(\Omega
)}^{2}t_{0},\text{ }\forall t\in \lbrack 0,T].
\end{equation*}
Hence, \(\mu (t)\geq \phi (t)>0,\) \(\mu ^{\prime }(t)=\phi ^{\prime }(t)-\phi
^{\prime }(0)\) and \(\mu ^{\prime \prime }(t)=\phi ^{\prime \prime }(t)>0,\)
so (35) implies
\begin{equation}
\mu (t)\mu ^{\prime \prime }(t)-\frac{k}{2}\mu ^{\prime }(t)^{2}\geq
2k\left( d-J(u_{0})\right) \left\Vert u_{0}\right\Vert _{H_{0}^{1}(\Omega
)}^{2}t_{0}>0,\text{ for all }t\in \lbrack t_{0},T]. \label{1700}
\end{equation}
(36)
Let \(\psi (t)=\mu (t)^{-\frac{k-2}{2}}.\) Thus,
\begin{equation}
\psi ^{\prime }(t)=-\frac{k-2}{2}\mu (t)^{-\frac{k}{2}}\mu ^{\prime }(t).
\label{1725}
\end{equation}
(37)
From (36) and (37), we get
\begin{equation*}
\psi ^{\prime \prime }(t)=\frac{k-2}{2}\mu (t)^{-\frac{k+2}{2}}\left[ \frac{k
}{2}\mu ^{\prime }(t)^{2}-\mu (t)\mu ^{\prime \prime }(t)\right] t_{0},\) \(\psi (t)\) is a concave
function in \([t_{0},T].\) Since \(\psi (t_{0})>0\) and \(\psi ^{\prime
}(t_{0})< 0,\) there exists a finite time \(T_{\ast }\) such that
\begin{equation*}
\underset{t\to T_{\ast }^{-}}{\lim }\psi (t)=0.
\end{equation*}
Consequently,
\begin{equation*}
\underset{t\to T_{\ast }^{-}}{\lim }\mu (t)=\infty ,
\end{equation*}
which satisfies
\begin{equation*}
\underset{t\to T_{\ast }^{-}}{\lim }\int\nolimits_{0}^{t}\left\Vert
u(s)\right\Vert _{H_{0}^{1}(\Omega )}^{2}ds=\infty ,
\end{equation*}
therefore, we have
\begin{equation*}
\underset{t\to T_{\ast }^{-}}{\lim }\left\Vert u(s)\right\Vert
_{H_{0}^{1}(\Omega )}^{2}=\infty .
\end{equation*}
This contradicts with \(u(t)\) being a global solution.
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 declare no conflict of interest.
