1 Introduction
In this paper, we study the following problem:
where Ω is a bound domain in \(\mathbb {R}^{n}\) with smooth boundary, \(2< q<2+\frac{4}{n}\), \(u_{0}(x)\in H_{0}^{2}(\varOmega)\setminus\{0\}\).
$$\begin{aligned} \textstyle\begin{cases} u_{t}+\Delta^{2}u= \vert u \vert ^{q-2}u\log \vert u \vert ,\quad x\in\varOmega, t>0, \\ u(x,t)=\Delta u(x,t)=0,\quad x\in\partial\varOmega, t>0, \\ u(x,0)=u_{0}(x),\quad x\in\varOmega, \end{cases}\displaystyle \end{aligned}$$
(1.1)
Many papers have been devoted to the fourth-order parabolic equation. Qu and Zhou [1] studied the following fourth-order equation:
Using the method of potential wells, they established a threshold result for the global existence and blow-up for the sign-changing weak solutions. Zhou [2] proved new blow-up conditions and the maximum of the blow-up time for Eq. (1.2). Li, Gao and Han [3] considered
They obtained the existence, uniqueness and blow-up of solutions. Liu and Liu [4] considered the following equation:
They combine the potential well method, the classical Galerkin method and the energy method to give a threshold result for the global existence and non-existence of sign-changing weak solutions to the problem. The relevant equations have also been studied in [5, 6].
$$ u_{t}+D^{4}u= \vert u \vert ^{p-1}u- \frac{1}{ \vert \varOmega \vert } \int_{\varOmega } \vert u \vert ^{p-1}u\,dx. $$
(1.2)
$$\begin{aligned} \textstyle\begin{cases} u_{t}+D^{4}u- ( \vert u_{x} \vert ^{p-2}u_{x} )_{x}= \vert u \vert ^{p-1}u- \frac{1}{ \vert \varOmega \vert } \int _{\varOmega} \vert u \vert ^{p-1}u\,dx,\quad x\in\varOmega, \\ Du=D^{3}u=0,\qquad (x,t)\in\partial\varOmega\times(0,T), \\ u(x,0)=u_{0}(x),\quad x\in\varOmega. \end{cases}\displaystyle \end{aligned}$$
$$\begin{aligned} &u_{t}-D^{6}u+D \biggl(a(Du)D^{3}u+ \frac{a'(Du)}{2} \bigl(D^{2}u \bigr)^{2} \biggr) \\ &\quad = \vert u \vert ^{p-1}u-\frac{1}{ \vert \varOmega \vert } \int_{\varOmega} \vert u \vert ^{p-1}u\,dx, \quad (x,t) \in\varOmega \times(0,T). \end{aligned}$$
Anzeige
In this paper, we study a fourth-order parabolic equation with the logarithmic nonlinearity. The second-order parabolic equation with the logarithmic nonlinearity is diffusely studied. Chen, Luo and Liu [7] studied the heat equation with the logarithmic nonlinearity. Ji, Yin and Cao [8] established the existence of positive periodic solutions and discussed the instability of such solutions for the semilinear pseudo-parabolic equation with the logarithmic source. Nahn and Truong [9] studied the following nonlinear equation:
They obtained results as regards the existence or non-existence of global weak solutions. He, Gao and Wang [10] considered the following equation:
where \(2< p< q< p(1+\frac{2}{n})\), they proved the decay and the finite time blow-up for weak solutions.
$$ u_{t}-\Delta u_{t}-\operatorname{div} \bigl( \vert \nabla u \vert ^{p-2}\nabla u \bigr)= \vert u \vert ^{p-2}u\log\bigl( \vert u \vert \bigr). $$
(1.3)
$$ u_{t}-\Delta u_{t}-\operatorname{div} \bigl( \vert \nabla u \vert ^{p-2}\nabla u \bigr)= \vert u \vert ^{q-2}u\log\bigl( \vert u \vert \bigr), $$
(1.4)
In this paper, we prove the existence of the unique global weak solution of the problem (1.1) based on the potential well method. In addition, we also obtain some properties of the solutions. This paper is organized as follows: in Sect. 2, we introduce some lemmas. In Sect. 3, we mainly introduce the existence of the unique local weak solution of the problem (1.1). In Sect. 4, under some conditions, we obtain the existence of the unique global weak solution of the problem (1.1). Meanwhile, we find that the solution is decaying. In the last section, we prove the blow-up theorem.
2 Some lemmas
We first consider the energy functional J and Nehari functional I defined on \(H_{0}^{2}(\varOmega)\setminus\{0\}\) as follows:
We can see that J and I are continuous from the Gagliardo–Nirenberg multiplicative embedding inequality (see [11]).
$$\begin{aligned} &J(u)=\frac{1}{2} \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx- \frac{1}{q} \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx+ \frac{1}{q^{2}} \int_{\varOmega} \vert u \vert ^{q}\,dx, \end{aligned}$$
(2.1)
$$\begin{aligned} &I(u)= \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx- \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx. \end{aligned}$$
(2.2)
By (2.1) and (2.2), we have
Let \(\mathscr{N}=\{u\in H_{0}^{2}(\varOmega)\setminus\{0\}:I(u)=0\}\). Lemma 2.1 indicates \(\mathscr{N}\) is not empty. Thus, we can define
It is obvious that \(d>0\) by (2.3), (2.4), \(2< q<2+\frac {4}{n}\) and the definition of \(\mathscr{N}\). For a fixed \(u\in H_{0}^{2}(\varOmega)\setminus\{0\}\), we consider the function \(j:\lambda\rightarrow J(\lambda u)\) for \(\lambda>0\).
$$\begin{aligned} &J(u)=\frac{1}{q}I(u)+ \biggl(\frac{1}{2}- \frac{1}{q} \biggr) \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx+ \frac{1}{q^{2}} \int_{\varOmega} \vert u \vert ^{q}\,dx. \end{aligned}$$
(2.3)
$$\begin{aligned} d=\inf_{u\in\mathscr{N}}J(u). \end{aligned}$$
(2.4)
Anzeige
Lemma 2.1
Let
\(u\in H_{0}^{2}(\varOmega)\setminus\{0\}\). Then the following results hold:
(1)
\(\lim_{\lambda\to0^{+}}j(\lambda)=0, \lim_{\lambda\to+\infty }j(\lambda)=0\);
(2)
there exists a unique
\(\overline{\lambda}>0\)
such that
\(j'(\overline {\lambda})=0\);
(3)
\(j(\lambda)\)
is increasing on
\((0,\overline{\lambda})\), decreasing on
\((\overline{\lambda},+\infty)\)
and attains the maximum at
λ̅;
(4)
\(I(\lambda u)>0\)
for
\(0<\lambda<\overline{\lambda}\), \(I(\lambda u)<0\)
for
\(\overline{\lambda}<\lambda<+\infty\)
and
\(I(\overline{\lambda}u)=0\).
Proof
For \(u\in H_{0}^{2}(\varOmega)\setminus\{0\}\), by the definition of j, we have
It is obvious that (1) holds due to \(2< q<2+\frac{4}{n}\) and \(\int _{\varOmega}|u|^{q}\,dx\neq0\). We have
We construct a function \(\varphi(\lambda)=\lambda^{-1}j'(\lambda)\), thus we obtain
Then
which implies that there exists a \(\lambda_{1}>0 \) such that \(\varphi (\lambda)\) is increasing on \((0,\lambda_{1})\), decreasing on \((\lambda _{1},+\infty)\). Using the Poincaré inequality and \(u\in H_{0}^{2}(\varOmega)\setminus\{ 0\}\), we have \(0<\int_{\varOmega}|u|^{2}\,dx\leq C_{1}\int_{\varOmega}|Du|^{2}\,dx\leq C_{1}C_{2}\int _{\varOmega}|\Delta u|^{2}\,dx\), where \(C_{1},C_{2}\) is the Poincaré constants. Since \(\lim_{\lambda\to0^{+}}\varphi(\lambda)=\int_{\varOmega}|\Delta u|^{2}\,dx>0\), \(\lim_{\lambda\to+\infty}\varphi(\lambda)=-\infty\), there exists a unique \(\overline{\lambda}>0\) such that \(\varphi(\overline {\lambda})=0\), i.e. \(j'(\overline{\lambda})=0\). So (2) holds. Thus \(j'(\lambda)=\lambda\varphi(\lambda)>0\) for \(0<\lambda<\overline{\lambda }\) and \(j'(\lambda)<0\) for \(\overline{\lambda}<\lambda<+\infty\), which indicates \(j(\lambda)\) is increasing on \((0,\overline{\lambda})\), decreasing on \((\overline {\lambda},+\infty)\) and attains the maximum at λ̅. So (3) holds. From (2.2), we have
Thus, \(I(\lambda u)>0\) for \(0<\lambda<\overline{\lambda}\), \(I(\lambda u)<0\) for \(\overline{\lambda}<\lambda<+\infty\) and \(I(\overline{\lambda}u)=0\). So (4) holds. □
$$\begin{aligned} j(\lambda)=J(\lambda u)={}&\frac{1}{2} \int_{\varOmega}\bigl\vert \Delta(\lambda u) \bigr\vert ^{2}\,dx- \frac{1}{q} \int_{\varOmega} \vert \lambda u \vert ^{q}\log \vert \lambda u \vert \,dx+ \frac{1}{q^{2}} \int_{\varOmega} \vert \lambda u \vert ^{q}\,dx \\ ={}&\frac{\lambda^{2}}{2} \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx- \frac{\lambda ^{q}}{q} \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx- \frac{\lambda^{q}}{q}\log \lambda \int_{\varOmega} \vert u \vert ^{q}\,dx \\ &{}+\frac{\lambda^{q}}{q^{2}} \int_{\varOmega} \vert u \vert ^{q}\,dx. \end{aligned}$$
$$\begin{aligned} j'({\lambda})={}&\lambda \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx- \lambda^{q-1} \int _{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx -\lambda^{q-1} \log\lambda \int_{\varOmega} \vert u \vert ^{q}\,dx \\ &{}-\frac{\lambda^{q-1}}{q} \int_{\varOmega} \vert u \vert ^{q}\,dx + \frac{\lambda^{q-1}}{q} \int_{\varOmega} \vert u \vert ^{q}\,dx \\ ={}&\lambda \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx- \lambda^{q-1} \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx -\lambda^{q-1} \log\lambda \int_{\varOmega} \vert u \vert ^{q}\,dx. \end{aligned}$$
$$\begin{aligned} \varphi(\lambda)={}&\lambda^{-1}j'(\lambda) \\ ={}&\lambda^{-1} \biggl(\lambda \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx-\lambda ^{q-1} \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx -\lambda^{q-1} \log\lambda \int_{\varOmega} \vert u \vert ^{q}\,dx \biggr) \\ ={}& \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx- \lambda^{q-2} \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx -\lambda^{q-2} \log\lambda \int_{\varOmega} \vert u \vert ^{q}\,dx. \end{aligned}$$
$$\begin{aligned} &\varphi'(\lambda)=-(q-2)\lambda^{q-3} \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx \\ &\qquad{}-(q-2)\lambda^{q-3}\log\lambda \int_{\varOmega} \vert u \vert ^{q}\,dx - \lambda^{q-3} \int_{\varOmega} \vert u \vert ^{q}\,dx \\ &\quad =-\lambda^{q-3} \biggl((q-2) \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx +(q-2)\log\lambda \int_{\varOmega} \vert u \vert ^{q}\,dx+ \int_{\varOmega} \vert u \vert ^{q}\,dx \biggr), \end{aligned}$$
$$\begin{aligned} I(\lambda u)={}& \int_{\varOmega}\bigl\vert \Delta(\lambda u) \bigr\vert ^{2}\,dx- \int_{\varOmega} \vert \lambda u \vert ^{q}\log \vert \lambda u \vert \,dx \\ ={}&\lambda^{2} \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx- \lambda^{q} \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx -\lambda^{q} \log\lambda \int_{\varOmega} \vert u \vert ^{q}\,dx \\ ={}&\lambda \biggl(\lambda \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx- \lambda^{q-1} \int _{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx -\lambda^{q-1} \log\lambda \int_{\varOmega} \vert u \vert ^{q}\,dx \biggr) \\ ={}&\lambda j'(\lambda). \end{aligned}$$
Lemma 2.2
There exists a
\(u>0\)
with
\(u\in\mathscr{N}\)
such that
\(J(u)=d\).
Proof
By (2.4), we suppose \(\{u_{k}\}_{k=1}^{\infty}\subset\mathscr{N}\) is a minimizing sequence of J. Since \(\{|u_{k}|\}_{k=1}^{\infty}\subset \mathscr{N}\) is also a minimizing sequence of J, we consider the case where \(u_{k}>0\) a.e. in Ω, \(k\in\mathbb {N}\) without loss of generality. Thus,
which implies that \(\{J(u_{k})\}_{k=1}^{\infty}\) is bounded, i.e. there exists a constant \(C_{3}>0\) such that \(|J(u_{k})|\leq C_{3}\). Using (2.3), \(I(u_{k})=0\) and \(|J(u_{k})|\leq C_{3}\), we have
Combining \(2< q<2+\frac{4}{n}\) with (2.6), we have
Using (2.7) and the Poincaré inequality, we have
where \(C_{4}, C_{5}\) are the Poincaré constants. The above inequality implies that \(\{u_{k}\}_{k=1}^{\infty}\) is bounded in \(H_{0}^{2}(\varOmega)\). Let \(\mu_{1}>0\) be sufficiently small such that \(q+\mu_{1}<\frac{2n}{n-2}\). Since \(H_{0}^{2}(\varOmega)\hookrightarrow L^{q+\mu_{1}}(\varOmega)\) is a compact embedding, there exist a function u and a subsequence of \(\{ u_{k}\}_{k=1}^{\infty}\), still denoted by \(\{u_{k}\}_{k=1}^{\infty}\), such that
Then we have \(u\geq0\) a.e. in Ω. First, we prove \(u\neq0\). Using the dominated convergence theorem, we obtain
It follows from the weak lower semicontinuity of the \(L^{2}\) norm that
Using (2.1), (2.5), (2.8), (2.9) and (2.10), we have
Using (2.2), (2.8), (2.10) and \(u_{k}\in\mathscr{N}\), we have
By \(u_{k}\in\mathscr{N}\) and using the Sobolev embedding inequality and the Poincaré inequality, we have
where \(C_{6}\) is the Sobolev embedding constant, \(C_{7}\) is the Poincaré constant.
$$\begin{aligned} \lim_{k \to\infty}J(u_{k})=d, \end{aligned}$$
(2.5)
$$\begin{aligned} \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega} \vert \Delta u_{k} \vert ^{2}\,dx+ \frac{1}{q^{2}} \int_{\varOmega} \vert u_{k} \vert ^{q}\,dx \leq C_{3}. \end{aligned}$$
(2.6)
$$\begin{aligned} \int_{\varOmega} \vert \Delta u_{k} \vert ^{2}\,dx\leq \biggl(\frac{1}{2}-\frac{1}{q} \biggr)^{-1}C_{3}. \end{aligned}$$
(2.7)
$$\begin{aligned} \int_{\varOmega} \vert u_{k} \vert ^{2}\,dx \leq C_{4} \int_{\varOmega} \vert Du_{k} \vert ^{2}\,dx \leq C_{4}C_{5} \int _{\varOmega} \vert \Delta u_{k} \vert ^{2}\,dx\leq \biggl(\frac{1}{2}-\frac{1}{q} \biggr)^{-1}C_{3}C_{4}C_{5}, \end{aligned}$$
$$\begin{aligned} &u_{k}\rightarrow u, \quad\text{weakly in } H_{0}^{2}( \varOmega), \\ &u_{k}\rightarrow u, \quad\text{strongly in } L^{q+\mu_{1}}( \varOmega), \\ &u_{k}\rightarrow u(x),\quad \text{a.e. in } \varOmega. \end{aligned}$$
$$\begin{aligned} & \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx=\lim_{k \to\infty} \int_{\varOmega} \vert u_{k} \vert ^{q} \log \vert u_{k} \vert \,dx, \end{aligned}$$
(2.8)
$$\begin{aligned} & \int_{\varOmega} \vert u \vert ^{q}\,dx=\lim _{k \to\infty} \int_{\varOmega} \vert u_{k} \vert ^{q} \,dx. \end{aligned}$$
(2.9)
$$\begin{aligned} \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx\leq \liminf_{k \to\infty} \int_{\varOmega} \vert \Delta u_{k} \vert ^{2}\,dx. \end{aligned}$$
(2.10)
$$\begin{aligned} J(u)={}&\frac{1}{2} \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx- \frac{1}{q} \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx+ \frac{1}{q^{2}} \int_{\varOmega} \vert u \vert ^{q}\,dx \\ \leq{}&\liminf_{k \to\infty}\frac{1}{2} \int_{\varOmega} \vert \Delta u_{k} \vert ^{2}\,dx -\lim_{k \to\infty}\frac{1}{q} \int_{\varOmega} \vert u_{k} \vert ^{q} \log \vert u_{k} \vert \,dx +\lim_{k \to\infty} \frac{1}{q^{2}} \int_{\varOmega} \vert u_{k} \vert ^{q}\,dx \\ ={}&\liminf_{k \to\infty} \biggl(\frac{1}{2} \int_{\varOmega} \vert \Delta u_{k} \vert ^{2}\,dx- \frac{1}{q} \int_{\varOmega} \vert u_{k} \vert ^{q} \log \vert u_{k} \vert \,dx +\frac{1}{q^{2}} \int_{\varOmega} \vert u_{k} \vert ^{q}\,dx \biggr) \\ ={}&\liminf_{k \to\infty}J(u_{k})=d. \end{aligned}$$
(2.11)
$$\begin{aligned} I(u)={}& \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx- \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx \\ \leq{}&\liminf_{k \to\infty} \int_{\varOmega} \vert \Delta u_{k} \vert ^{2}\,dx-\lim_{k \to \infty} \int_{\varOmega} \vert u_{k} \vert ^{q} \log \vert u_{k} \vert \,dx \\ ={}&\liminf_{k \to\infty} \biggl( \int_{\varOmega} \vert \Delta u_{k} \vert ^{2}\,dx- \int _{\varOmega} \vert u_{k} \vert ^{q} \log \vert u_{k} \vert \,dx \biggr) \\ ={}&\liminf_{k \to\infty}I(u_{k})=0. \end{aligned}$$
(2.12)
$$\begin{aligned} \int_{\varOmega} \vert \Delta u_{k} \vert ^{2}\,dx={}& \int_{\varOmega} \vert u_{k} \vert ^{q}\log \vert u_{k} \vert \,dx \\ \leq{}&\frac{e^{-1}}{\mu_{1}} \int_{\varOmega} \vert u_{k} \vert ^{q+\mu_{1}}\,dx \\ \leq{}&C_{6}^{q+\mu_{1}}\frac{e^{-1}}{ \mu_{1}} \biggl( \int_{\varOmega} \vert Du_{k} \vert ^{2}\,dx \biggr)^{\frac{q+\mu_{1}}{2}} \\ \leq{}&C_{6}^{q+\mu_{1}}C_{7}^{\frac{q+\mu_{1}}{2}} \frac{e^{-1}}{ \mu_{1}} \biggl( \int_{\varOmega} \vert \Delta u_{k} \vert ^{2}\,dx \biggr)^{\frac{q+\mu_{1}}{2}}, \end{aligned}$$
(2.13)
By (2.13), we have, for some positive constant \(C_{8}\),
Using (2.8) and (2.14), we have
which indicates \(u\neq0\). Next, we will study \(I(u)=0\). If \(I(u)<0\), by Lemma 2.1, there exists a \(\overline{\lambda}_{1}\) such that \(I(\overline{\lambda}_{1}u)=0\) and \(0<\overline{\lambda}_{1}<1\). Thus, \(\overline{\lambda}_{1}u\in\mathscr{N}\). By (2.3), (2.4), (2.9) and (2.10), we have
which indicates \(\overline{\lambda}_{1}\geq1\) by \(d>0\). It contradicts \(0<\overline{\lambda}_{1}<1\). Then, by (2.12), we have \(I(u)=0\). Therefore, \(u\in\mathscr{N}\). By (2.4), we have \(J(u)\geq d\). By (2.11), we have \(J(u)\leq d\). So, \(J(u)=d\). □
$$\begin{aligned} \int_{\varOmega} \vert u_{k} \vert ^{q} \log \vert u_{k} \vert \,dx= \int_{\varOmega} \vert \Delta u_{k} \vert ^{2}\,dx \geq C_{8}. \end{aligned}$$
(2.14)
$$\begin{aligned} \frac{e^{-1}}{\mu_{1}} \int_{\varOmega} \vert u \vert ^{q+\mu_{1}}\,dx\geq \int_{\varOmega} \vert u \vert ^{q}\log \vert u \vert \,dx=\lim_{k \to\infty} \int_{\varOmega} \vert u_{k} \vert ^{q}\log \vert u_{k} \vert \,dx\geq C_{8}, \end{aligned}$$
$$\begin{aligned} d\leq{}&J(\overline{\lambda}_{1}u) \\ ={}&\frac{1}{q}I(\overline{ \lambda}_{1}u) + \biggl( \frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta(\overline{\lambda}_{1}u) \bigr\vert ^{2} \,dx+\frac{1}{q^{2}} \int_{\varOmega} \vert \overline{\lambda}_{1}u \vert ^{q} \,dx \\ ={}& \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta(\overline{\lambda}_{1}u) \bigr\vert ^{2} \,dx+\frac{1}{q^{2}} \int_{\varOmega} \vert \overline{\lambda}_{1}u \vert ^{q} \,dx \\ ={}& \biggl(\frac{1}{2}-\frac{1}{q} \biggr)\overline{ \lambda}_{1}^{2} \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx + \frac{\overline{\lambda}_{1}^{q}}{q^{2}} \int_{\varOmega} \vert u \vert ^{q}\,dx \\ \leq{}& \biggl(\frac{1}{2}-\frac{1}{q} \biggr)\overline{ \lambda}_{1}^{2} \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx+ \frac{\overline{\lambda}_{1}^{2}}{q^{2}} \int_{\varOmega} \vert u \vert ^{q}\,dx \\ ={}&\overline{\lambda}_{1}^{2} \biggl[ \biggl( \frac{1}{2}- \frac{1}{q} \biggr) \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx+ \frac{1}{q^{2}} \int_{\varOmega} \vert u \vert ^{q}\,dx \biggr] \\ \leq{}&\overline{\lambda}_{1}^{2}\liminf _{k \to\infty} \biggl[ \biggl(\frac{1}{2}- \frac{1}{q} \biggr) \int_{\varOmega} \vert \Delta u_{k} \vert ^{2}\,dx+ \frac{1}{q^{2}} \int_{\varOmega} \vert u_{k} \vert ^{q}\,dx \biggr] \\ ={}&\overline{\lambda}_{1}^{2}\liminf_{k \to\infty}J(u_{k}) \\ ={}&\overline{\lambda}_{1}^{2}d, \end{aligned}$$
Lemma 2.3
([9])
For any
\(u\in W_{0}^{1,p}(\varOmega)\), \(p\geq1\), and
\(r\geq1\), the inequality
is valid, where
and for
\(p\geq n=1, r\leq q\leq\infty\); for
\(n>1\)
and
\(p< n\), \(q\in[r,p^{*}]\)
if
\(r\leq p^{*}\)
and
\(q\in[p^{*},r]\)
if
\(r\geq p^{*}\); for
\(p=n>1, r\leq q\leq\infty\); for
\(p>n>1, r\leq q\leq\infty\).
$$\begin{aligned} \Vert u \Vert _{q}\leq C \Vert Du \Vert _{p}^{\theta} \Vert u \Vert _{r}^{1-\theta}, \end{aligned}$$
$$\begin{aligned} \theta= \biggl(\frac{1}{r}-\frac{1}{q} \biggr) \biggl( \frac{1}{n}-\frac{1}{p} +\frac{1}{r} \biggr)^{-1}, \end{aligned}$$
Here, the constant
C
depends on
n, p, q
and
r.
Lemma 2.4
([12])
Let
\(h:\mathbb {R}^{+}\rightarrow R^{+}\)
be a nonincreasing function. Assume that there is a constant
\(A>0\)
such that
Then
\(h(t)\leq h(0)e^{1-\frac{t}{A}}\), for all
\(t>0\).
$$\begin{aligned} \int_{s}^{+\infty}h(t)\,dt\leq Ah(s),\quad 0\leq s< +\infty. \end{aligned}$$
3 Local existence and uniqueness
Definition 3.1
(Weak solution)
A function u is a solution of problem (1.1) over \([0,T]\) if \(u\in L^{\infty}(0,T;H_{0}^{2}(\varOmega))\) with \(u_{t}\in L^{2}(0,T;L^{2}(\varOmega))\), satisfies the initial condition \(u(0)=u_{0}(x)\in H_{0}^{2}(\varOmega)\setminus\{0\}\), and
for any \(w\in H_{0}^{2}(\varOmega)\), and for a.e. \(t\in[0,T]\).
$$\begin{aligned} \int_{\varOmega}u_{t}w\,dx+ \int_{\varOmega}\Delta u\Delta w\,dx= \int_{\varOmega} \vert u \vert ^{q-2}u\log \vert u \vert w\,dx, \end{aligned}$$
(3.1)
Theorem 3.1
(Local existence)
Let
\(u_{0}\in H_{0}^{2}(\varOmega)\setminus\{0\}\). Then there exists a positive constant
\(T_{0}\)
such that the problem (1.1) has a unique weak solution
\(u(x,t)\)
on
\(\varOmega\times(0,T_{0})\). Furthermore, \(u(x,t)\)
satisfies the energy inequality
$$\begin{aligned} \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds+J \bigl(u(t) \bigr)\leq J(u_{0}),\quad t\in[0,T_{0}]. \end{aligned}$$
(3.2)
Proof
In the space of \(H_{0}^{2}(\varOmega)\), we take a basis \(\{w_{j}\} _{j=1}^{\infty}\) and define the finite dimensional space
Let \(u_{0m}\) be an element of \(V_{m}\) such that
as \(m\rightarrow\infty\). We can find the approximate solution \(u_{m}(x,t)\) of the problem (1.1) in the form
where \(\alpha_{mj}\ (1\leq j\leq m)\) satisfy the ordinary differential equations
for \(i\in\{1,2,\ldots,m\}\), with
We find from Peano’s theorem that (3.5)–(3.6) has a local solution \(\alpha_{mj}\), and there exists a positive \(T_{m}>0\) such that \(\alpha_{mj}\in C^{1}([0,T_{m}])\), therefore \(u_{m}\in C^{1}([0,T_{m}];H_{0}^{2}(\varOmega))\). Multiplying the ith equation in (3.5) by \(\alpha_{mi}\), summing over i from 1 to m, we have
Integrating the above formula with respect to s over \((0,t)\), we have
where
Choose \(\mu_{2}\) such that \(0<\mu_{2}<2+\frac{4}{n}-q\). Using Lemma 2.3, the Poincaré inequality and the Young inequality, we have
where \(C_{9}\) is the constant of Lemma 2.3, \(C_{10}\) is the Poincaré constant, \(0<\varepsilon<1\), and \(\theta=n(\frac {1}{2}-\frac{1}{q+\mu_{2}})\). Let \(\gamma=\frac{(1-\theta)(q+\mu_{2})}{2-\theta(q+\mu_{2})}\) and \(C_{11}= (\frac{\varepsilon\mu_{2}}{e^{-1}C_{9}^{q+\mu_{2}}C_{10}^{\frac{\theta(q+\mu_{2})}{2}}} ) ^{-\frac{\theta(q+\mu_{2})}{2-\theta(q+\mu_{2})}}\), thus (3.10) becomes
It is easy to check \(\gamma>1\) according to \(2< q<2+\frac{4}{n}\). Using (3.3), (3.8), (3.9) and (3.11), we have
Using \(0<\varepsilon<1\) and (3.12),
Using the integral inequality of Gronwall–Bellman–Bihari type and combining with (3.13), there exists \(T_{0}\) such that
where \(C_{13}(T_{0})\) is a positive constant dependent on \(T_{0}\). Multiplying equation (3.5) by \(\alpha'_{mi}\), summing over i from 1 to m and integrating with respect to time variable on \([0,t]\), we have
We find from (3.3) and the continuity of the J that there exists a constant \(C_{14}>0\) such that
Using (2.1), (3.9), (3.11), (3.14), (3.15) and (3.16), we have
which implies that
By the Poincaré inequality and (3.18), we obtain
where \(C_{15},C_{16}\) are the Poincaré constants. We can easily obtain from the above inequality
where \(C_{17}(T_{0})\) is a positive constant dependent on \(T_{0}\). Using (3.15)–(3.17), we have
which implies that
where \(C_{18}(T_{0})\) is a positive constant dependent on \(T_{0}\). It follows from (3.20) and (3.22) that there exist a function u and a subsequence of \(\{u_{m}\}_{m=1}^{\infty}\) still denoted \(\{u_{m}\}_{m=1}^{\infty}\) such that
We obtain from the Aubin–Lions–Simon lemma (see [13]) together with (3.23) and (3.24)
So, \(u_{m}\rightarrow u\) a.e. \((x,t)\in\varOmega\times(0,T_{0})\). This implies that
According to \(2< q<2+\frac{4}{n}\), we can choose \(\mu_{3}\) such that \(1<\frac{q(q-1+\mu_{3})}{q-1}<\frac{2n}{n-2}\). Then, using the Sobolev embedding inequality and combining (3.19), we have
where \(C_{19}\) is the embedding constant. Using (3.26), (3.27) and Lion’s lemma (see [13]), we obtain
Passing to the limit in (3.5) and (3.6) as \(m\rightarrow \infty\), by (3.23), (3.24) and (3.28), we see that u satisfies the initial condition \(u(0)=u_{0}\) and
for all \(w\in H_{0}^{2}(\varOmega)\), and for a.e. \(t\in[0,T_{0}]\). So, u is a desired solution of problem (1.1).
$$\begin{aligned} V_{m}=\operatorname{span}\{w_{1},w_{2}, \ldots,w_{m}\}. \end{aligned}$$
$$\begin{aligned} u_{0m}=\sum_{j=1}^{m}a_{mj}w_{j} \rightarrow u_{0}\quad \mbox{strongly in }H_{0}^{2}( \varOmega), \end{aligned}$$
(3.3)
$$\begin{aligned} u_{m}(x,t)=\sum_{j=1}^{m} \alpha_{mj}(t)w_{j}(x), \end{aligned}$$
(3.4)
$$\begin{aligned} \int_{\varOmega}u_{mt}w_{i}\,dx+ \int_{\varOmega}\Delta u_{m}\Delta w_{i}\,dx= \int _{\varOmega} \vert u_{m} \vert ^{q-2}u_{m} \log \vert u_{m} \vert w_{i}\,dx, \end{aligned}$$
(3.5)
$$\begin{aligned} \alpha_{mj}(0)=a_{mj},\quad i\in\{1,2, \ldots,m\}. \end{aligned}$$
(3.6)
$$\begin{aligned} \int_{\varOmega}u_{mt}u_{m}\,dx+ \int_{\varOmega} \vert \Delta u_{m} \vert ^{2}\,dx= \int_{\varOmega} \vert u_{m} \vert ^{q} \log \vert u_{m} \vert \,dx. \end{aligned}$$
(3.7)
$$\begin{aligned} y_{m}(t)=y_{m}(0)+ \int_{0}^{t} \int_{\varOmega} \vert u_{m} \vert ^{q}\log \vert u_{m} \vert \,dx\,ds, \end{aligned}$$
(3.8)
$$\begin{aligned} y_{m}(t)=\frac{1}{2} \int_{\varOmega} \vert u_{m} \vert ^{2}\,dx+ \int_{0}^{t} \int_{\varOmega} \vert \Delta u_{m} \vert ^{2}\,dx\,ds. \end{aligned}$$
(3.9)
$$\begin{aligned} & \int_{\varOmega} \vert u_{m} \vert ^{q}\log \vert u_{m} \vert \,dx \\ &\quad \leq\frac{e^{-1}}{\mu_{2}} \int_{\varOmega} \vert u_{m} \vert ^{q+\mu_{2}}\,dx \\ &\quad \leq\frac{e^{-1}}{\mu_{2}}C_{9}^{q+\mu_{2}} \Vert Du_{m} \Vert _{2}^{\theta(q+\mu_{2})} \Vert u_{m} \Vert _{2}^{(1-\theta)(q+\mu_{2})} \\ &\quad \leq\frac{e^{-1}}{\mu_{2}}C_{9}^{q+\mu_{2}}C_{10}^{\frac{\theta(q+\mu _{2})}{2}} \Vert \Delta u_{m} \Vert _{2}^{\theta(q+\mu_{2})} \Vert u_{m} \Vert _{2}^{(1-\theta)(q+\mu_{2})} \\ &\quad \leq\varepsilon \Vert \Delta u_{m} \Vert _{2}^{2}+ \biggl( \frac{\varepsilon\mu_{2}}{e^{-1}C_{9}^{q+\mu_{2}}C_{10}^{\frac{\theta(q+\mu_{2})}{2}}} \biggr)^{-\frac{\theta(q+\mu _{2})}{2-\theta(q+\mu_{2})}} \Vert u_{m} \Vert _{2}^{\frac{2(1-\theta)(q+\mu_{2})}{2-\theta (q+\mu_{2})}}, \end{aligned}$$
(3.10)
$$\begin{aligned} \int_{\varOmega} \vert u_{m} \vert ^{q}\log \vert u_{m} \vert \,dx\leq&\varepsilon \int_{\varOmega} \vert \Delta u_{m} \vert ^{2}\,dx +C_{11} \biggl( \int_{\varOmega} \vert u_{m} \vert ^{2}\,dx \biggr)^{\gamma}. \end{aligned}$$
(3.11)
$$\begin{aligned} y_{m}(t)={}&y_{m}(0)+ \int_{0}^{t} \int_{\varOmega} \vert u_{m} \vert ^{q}\log \vert u_{m} \vert \,dx\,ds \\ \leq{}&y_{m}(0)+ \int_{0}^{t} \biggl[\varepsilon \int_{\varOmega} \vert \Delta u_{m} \vert ^{2} \,dx+C_{11} \biggl( \int_{\varOmega} \vert u_{m} \vert ^{2}\,dx \biggr) ^{\gamma} \biggr]\,ds \\ ={}&\frac{1}{2} \int_{\varOmega}\bigl\vert u_{m}(0) \bigr\vert ^{2}\,dx+ \int_{0}^{t} \int_{\varOmega}\bigl\vert \Delta u_{m}(0) \bigr\vert ^{2}\,dx\,ds \\ &{}+ \int_{0}^{t} \biggl[\varepsilon \int_{\varOmega} \vert \Delta u_{m} \vert ^{2}\,dx +C_{11} \biggl( \int_{\varOmega} \vert u_{m} \vert ^{2}\,dx \biggr) ^{\gamma} \biggr]\,ds \\ \leq{}&C_{12}+ \int_{0}^{t} \biggl[\varepsilon \int_{\varOmega} \vert \Delta u_{m} \vert ^{2}\,dx +C_{11} \biggl( \int_{\varOmega} \vert u_{m} \vert ^{2}\,dx \biggr) ^{\gamma} \biggr]\,ds \\ \leq{}&C_{12}+\varepsilon \int_{0}^{t} \int_{\varOmega} \vert \Delta u_{m} \vert ^{2}\,dx\,ds +C_{11} \int_{0}^{t} \biggl( \int_{\varOmega} \vert u_{m} \vert ^{2}\,dx \biggr) ^{\gamma}\,ds \\ \leq{}&C_{12}+\frac{\varepsilon}{2} \int_{\varOmega} \vert u_{m} \vert ^{2}\,dx+ \varepsilon \int _{0}^{t} \int_{\varOmega} \vert \Delta u_{m} \vert ^{2}\,dx\,ds \\ &{}+C_{11}2^{\gamma} \int_{0}^{t} \biggl( \int_{0}^{s} \int_{\varOmega} \vert \Delta u_{m} \vert ^{2}\,dx\,dy \biggr)^{\gamma}\,ds \\ &{}+C_{11}2^{\gamma} \int_{0}^{t} \biggl(\frac{1}{2} \int_{\varOmega} \vert u_{m} \vert ^{2}\,dx \biggr) ^{\gamma}\,ds \\ \leq{}&C_{12}+\varepsilon y_{m}(t)+C_{11}2^{\gamma} \int_{0}^{t}y_{m}(s)^{\gamma}\,ds. \end{aligned}$$
(3.12)
$$\begin{aligned} y_{m}(t)\leq\frac{C_{12}}{1-\varepsilon}+\frac{C_{11}2^{\gamma}}{ 1-\varepsilon} \int_{0}^{t}y_{m}(s)^{\gamma}\,ds. \end{aligned}$$
(3.13)
$$\begin{aligned} y_{m}(t)\leq C_{13}(T_{0}),\quad t \in[0,T_{0}], \end{aligned}$$
(3.14)
$$\begin{aligned} \int_{0}^{t} \int_{\varOmega}u_{ms}^{2}\,dx\,ds+J \bigl(u_{m}(t) \bigr)=J \bigl(u_{m}(0) \bigr),\quad \mbox{for all }t \in[0,T_{0}]. \end{aligned}$$
(3.15)
$$\begin{aligned} J \bigl(u_{m}(0) \bigr)\leq C_{14},\quad \mbox{for all } m. \end{aligned}$$
(3.16)
$$\begin{aligned} C_{14}\geq{}& J \bigl(u_{m}(t) \bigr) \\ ={}&\frac{1}{2} \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx- \frac{1}{q} \int_{\varOmega}\bigl\vert u_{m}(t) \bigr\vert ^{q} \log \bigl\vert u_{m}(t) \bigr\vert \,dx + \frac{1}{q^{2}} \int_{\varOmega}\bigl\vert u_{m}(t) \bigr\vert ^{q}\,dx \\ \geq{}&\frac{1}{2} \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx- \frac{1}{q} \int _{\varOmega}\bigl\vert u_{m}(t) \bigr\vert ^{q} \log \bigl\vert u_{m}(t) \bigr\vert \,dx \\ \geq{}&\frac{1}{2} \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx- \frac{\varepsilon }{q} \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx - \frac{C_{11}}{q} \biggl( \int_{\varOmega}\bigl\vert u_{m}(t) \bigr\vert ^{2}\,dx \biggr)^{\gamma} \\ ={}& \biggl(\frac{1}{2}-\frac{\varepsilon}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx - \frac{C_{11}}{q} \biggl( \int_{\varOmega}\bigl\vert u_{m}(t) \bigr\vert ^{2}\,dx \biggr)^{\gamma} \\ \geq{}& \biggl(\frac{1}{2}-\frac{\varepsilon}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx - \frac{C_{11}}{q} \bigl(2C_{13}(T_{0}) \bigr)^{\gamma}, \end{aligned}$$
(3.17)
$$\begin{aligned} \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx\leq \biggl(\frac{1}{2}-\frac {\varepsilon}{q} \biggr)^{-1} \biggl[C_{14}+\frac{C_{11}}{ q} \bigl(2C_{13}(T_{0}) \bigr)^{\gamma} \biggr]. \end{aligned}$$
(3.18)
$$\begin{aligned} \int_{\varOmega}\bigl\vert u_{m}(t) \bigr\vert ^{2}\,dx\leq{}& C_{15} \int_{\varOmega}\bigl\vert Du_{m}(t) \bigr\vert ^{2}\,dx\leq C_{15}C_{16} \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx \\ \leq{}& C_{15}C_{16} \biggl(\frac{1}{2}- \frac{\varepsilon}{q} \biggr)^{-1} \biggl[C_{14}+ \frac{C_{11}}{ q} \bigl(2C_{13}(T_{0}) \bigr)^{\gamma} \biggr], \end{aligned}$$
(3.19)
$$\begin{aligned} \Vert u_{m} \Vert _{L^{\infty}(0,T_{0};H_{0}^{2}(\varOmega))}\leq C_{17}(T_{0}), \end{aligned}$$
(3.20)
$$\begin{aligned} \biggl(\frac{1}{2}-\frac{\varepsilon}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx - \frac{C_{11}}{q} \bigl(2C_{13}(T_{0}) \bigr)^{\gamma}+ \int_{0}^{t} \int _{\varOmega}u_{ms}^{2}\,dx\,ds\leq C_{14}, \end{aligned}$$
(3.21)
$$\begin{aligned} \Vert u_{mt} \Vert _{L^{2}(0,T_{0};L^{2}(\varOmega))}\leq C_{18}(T_{0}), \end{aligned}$$
(3.22)
$$\begin{aligned} &u_{m}\rightarrow u\quad \text{weakly star in } {L^{\infty} \bigl(0,T_{0};H_{0}^{2}( \varOmega) \bigr)}, \end{aligned}$$
(3.23)
$$\begin{aligned} & u_{mt}\rightarrow u_{t} \quad\text{weakly in } {L^{2} \bigl(0,T_{0};L^{2}(\varOmega) \bigr)}. \end{aligned}$$
(3.24)
$$\begin{aligned} u_{m}\rightarrow u\quad \mbox{strongly in }C \bigl(0,T_{0};L^{2}( \varOmega) \bigr). \end{aligned}$$
(3.25)
$$\begin{aligned} \vert u_{m} \vert ^{q-2}u_{m} \log \vert u_{m} \vert \rightarrow \vert u \vert ^{q-2}u\log \vert u \vert \quad \mbox{a.e. }(x,t)\in \varOmega \times(0,T_{0}). \end{aligned}$$
(3.26)
$$\begin{aligned} & \int_{\varOmega}\bigl\vert \vert u_{m} \vert ^{q-2}u_{m} \log \vert u_{m} \vert \bigr\vert ^{\frac{q}{q-1}}\,dx \\ &\quad= \int_{\{x\in\varOmega: \vert u_{m} \vert \leq1\}} \bigl\vert \vert u_{m} \vert ^{q-2}u_{m} \log \vert u_{m} \vert \bigr\vert ^{\frac{q}{q-1}}\,dx \\ &\qquad{}+ \int_{\{x\in\varOmega: \vert u_{m} \vert \geq1\}} \bigl\vert \vert u_{m} \vert ^{q-2}u_{m} \log \vert u_{m} \vert \bigr\vert ^{\frac{q}{q-1}}\,dx \\ &\quad \leq \bigl(e(q-1) \bigr)^{-\frac{q}{q-1}} \vert \varOmega \vert + \biggl(\frac{e^{-1}}{\mu _{3}} \biggr) ^{-\frac{q}{q-1}} \int_{\varOmega} \vert u_{m} \vert ^{\frac{q(q-1+\mu_{3})}{q-1}}\,dx \\ &\quad \leq \bigl(e(q-1) \bigr)^{-\frac{q}{q-1}} \vert \varOmega \vert + \biggl(\frac{e^{-1}}{\mu _{3}} \biggr) ^{-\frac{q}{q-1}}C_{19}^{{\frac{q(q-1+\mu_{3})}{q-1}}} \biggl( \int_{\varOmega} \vert Du_{m} \vert ^{2}\,dx \biggr)^{{\frac{q(q-1+\mu_{3})}{2(q-1)}}} \\ &\quad \leq \bigl(e(q-1) \bigr)^{-\frac{q}{q-1}} \vert \varOmega \vert + \biggl(\frac{e^{-1}}{\mu _{3}} \biggr) ^{-\frac{q}{q-1}}C_{19}^{{\frac{q(q-1+\mu_{3})}{q-1}}} \\ &\qquad{}\times \biggl(C_{16} \biggl(\frac{1}{2}- \frac{\varepsilon}{q} \biggr)^{-1} \biggl[C_{14}+ \frac{C_{11}}{ q} \bigl(2C_{13}(T_{0}) \bigr)^{\gamma} \biggr] \biggr)^{{\frac{q(q-1+\mu_{3})}{2(q-1)}}}, \end{aligned}$$
(3.27)
$$\begin{aligned} \vert u_{m} \vert ^{q-2}u_{m} \log \vert u_{m} \vert \rightarrow \vert u \vert ^{q-2}u\log \vert u \vert \quad\text{weakly$^{*}$ in } {L^{\infty} \bigl(0,T_{0};L^{\frac{q}{q-1}}(\varOmega) \bigr)}. \end{aligned}$$
(3.28)
$$\begin{aligned} \int_{\varOmega}u_{t}(t)w\,dx+ \int_{\varOmega}\Delta u(t)\Delta w\,dx= \int _{\varOmega}\bigl\vert u(t) \bigr\vert ^{q-2}u(t) \log \bigl\vert u(t) \bigr\vert w\,dx, \end{aligned}$$
(3.29)
Next, we will study uniqueness of the solution. We obtain from (3.29) for any \(v\in L^{2}(0,T_{0};H_{0}^{2}(\varOmega))\)
We suppose there are two solutions \(u_{1}\) and \(u_{2}\). Let \(w=u_{1}-u_{2}\), thus we have \(w(0)=0\), \(w\in L^{2}(0,T_{0};H_{0}^{2}(\varOmega))\) and \(w_{t}\in L^{2}(0,T_{0};L^{2}(\varOmega))\). We set
From (3.30), it follows that
According to \(0\leq\int_{0}^{t}\int_{\varOmega}|\Delta w|^{2}\,dx\,ds\), (3.31) becomes
We construct a function \(f:{\mathbb {R}}^{*}\rightarrow\mathbb {R}\), \(f(s)=|s|^{q-2}s\log|s|\). Thus, we find that there exists a positive constant \(C_{20}\) such that
By (3.32) and (3.33),
i.e.,
Using Gronwall’s inequality and combining with (3.34), we have
So, the uniqueness is derived.
$$\begin{aligned} \int_{\varOmega}u_{t}(t)v\,dx+ \int_{\varOmega}\Delta u(t)\Delta v\,dx= \int _{\varOmega}\bigl\vert u(t) \bigr\vert ^{q-2}u(t) \log \bigl\vert u(t) \bigr\vert v\,dx. \end{aligned}$$
(3.30)
$$\begin{aligned} v(s)= \textstyle\begin{cases} u_{1}(s)-u_{2}(s),& s\in[0,t], \\ 0, &s\in[t,T_{0}]. \end{cases}\displaystyle \end{aligned}$$
$$\begin{aligned} & \int_{0}^{t} \int_{\varOmega}w_{s}w\,dx\,ds+ \int_{0}^{t} \int_{\varOmega} \vert \Delta w \vert ^{2}\,dx\,ds \\ &\quad = \int_{0}^{t} \int_{\varOmega}\bigl( \vert u_{1} \vert ^{q-2}u_{1} \log \vert u_{1} \vert - \vert u_{2} \vert ^{q-2}u_{2} \log \vert u_{2} \vert \bigr)w\,dx\,ds. \end{aligned}$$
(3.31)
$$\begin{aligned} \int_{0}^{t} \int_{\varOmega}w_{s}w\,dx\,ds \leq \int_{0}^{t} \int_{\varOmega}\bigl( \vert u_{1} \vert ^{q-2}u_{1} \log \vert u_{1} \vert - \vert u_{2} \vert ^{q-2}u_{2} \log \vert u_{2} \vert \bigr)w\,dx\,ds. \end{aligned}$$
(3.32)
$$\begin{aligned} \bigl\vert \vert u_{1} \vert ^{q-2}u_{1} \log \vert u_{1} \vert - \vert u_{2} \vert ^{q-2}u_{2} \log \vert u_{2} \vert \bigr\vert \leq C_{20} \vert w \vert . \end{aligned}$$
(3.33)
$$\begin{aligned} \int_{0}^{t} \int_{\varOmega}w_{s}w\,dx\,ds \leq C_{20} \int_{0}^{t} \int_{\varOmega}w^{2}\,dx\,ds, \end{aligned}$$
$$\begin{aligned} \frac{1}{2} \int_{\varOmega}w^{2}\,dx \leq\frac{1}{2} \int_{\varOmega}w(0)^{2}\,dx+C_{20} \int_{0}^{t} \int_{\varOmega}w^{2}\,dx\,ds \leq C_{20} \int_{0}^{t} \int_{\varOmega}w^{2}\,dx\,ds. \end{aligned}$$
(3.34)
$$\begin{aligned} \int_{\varOmega}w^{2}\,dx\leq0. \end{aligned}$$
Finally, we will study (3.2). Let \(\phi(t)\) is a nonnegative function which belongs to \(C([0,T_{0}])\). From (3.15), we can obtain
As \(m\rightarrow\infty\),
and
hold. Since \(\int_{0}^{T_{0}}J(u_{m}(t))\phi(t)\,dt\) is lower semi-continuous with respect to the weak topology of \(L^{2}(0,T_{0};H_{0}^{2}(\varOmega))\), we know that
Hence, by (3.35), it follows that
as \(m\rightarrow\infty\). \(\phi(t)\) is arbitrary nonnegative function, so we have
□
$$\begin{aligned} & \int_{0}^{T_{0}}\phi(t)\,dt \int_{0}^{t} \int_{\varOmega}u_{ms}^{2}\,dx\,ds+ \int _{0}^{T_{0}}J \bigl(u_{m}(t) \bigr) \phi(t) \,dt \\ &\quad = \int_{0}^{T_{0}}J \bigl(u_{m}(0) \bigr) \phi(t) \,dt. \end{aligned}$$
(3.35)
$$\int_{0}^{T_{0}}J \bigl(u_{m}(0) \bigr) \phi(t) \,dt\rightarrow \int_{0}^{T_{0}}J(u_{0})\phi(t)\,dt $$
$$\int_{0}^{T_{0}}\phi(t)\,dt \int_{0}^{t} \int_{\varOmega}u_{ms}^{2}\,dx\,ds\rightarrow \int _{0}^{T_{0}}\phi(t)\,dt \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds $$
$$\int_{0}^{T_{0}}J \bigl(u(t) \bigr)\phi(t)\,dt\leq \liminf_{m \to\infty} \int _{0}^{T_{0}}J \bigl(u_{m}(t) \bigr) \phi(t) \,dt. $$
$$\begin{aligned} \int_{0}^{T_{0}}\phi(t)\,dt \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds+ \int _{0}^{T_{0}}J \bigl(u(t) \bigr)\phi(t)\,dt\leq \int_{0}^{T_{0}}J(u_{0})\phi(t)\,dt, \end{aligned}$$
$$\begin{aligned} \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds+J \bigl(u(t) \bigr)\leq J(u_{0}),\quad t\in[0,T_{0}]. \end{aligned}$$
4 Global existence and decay estimates
Now as in [9], we introduce the following sets: \(\mathscr{W}_{1}=\{u\in H_{0}^{2}(\varOmega)\setminus\{0\}:J(u)< d\}\), \(\mathscr{W}_{2}=\{u\in H_{0}^{2}(\varOmega)\setminus\{0\}:J(u)=d\}\), \(\mathscr{W}_{1}^{+}=\{u\in\mathscr{W}_{1}:I(u)>0\}\), \(\mathscr{W}_{2}^{+}=\{u\in\mathscr{W}_{2}:I(u)>0\}\), \(\mathscr{W}_{1}^{-}=\{u\in\mathscr{W}_{1}:I(u)<0\}\), \(\mathscr{W}_{2}^{-}=\{u\in\mathscr{W}_{2}:I(u)<0\}\), \(\mathscr{W}=\mathscr{W}_{1}\cup\mathscr{W}_{2}\), \(\mathscr{W}^{+}=\mathscr{W}_{1}^{+}\cup\mathscr{W}_{2}^{+}\), \(\mathscr{W}^{-}=\mathscr{W}_{1}^{-}\cup\mathscr{W}_{2}^{-}\).
Definition 4.1
(Maximal existence time)
Let \(u(t)\) be a solution of problem (1.1). We define the maximal existence time \(T_{\mathrm{max}}\) as follows:
Then:
$$\begin{aligned} T_{\mathrm{max}}=\sup \bigl\{ T>0:u(t)\mbox{ exists on }[0,T] \bigr\} . \end{aligned}$$
(i)
if \(T_{\mathrm{max}}<{+\infty}\), we say that \(u(t)\) blows up in finite time and \(T_{\mathrm{max}}\) is the blow-up time;
(ii)
if \(T_{\mathrm{max}}={+\infty}\), we say that \(u(t)\) is global.
Theorem 4.1
Let
\(u_{0}\in\mathscr{W^{+}}\). Then the problem of (1.1) admits a unique global weak solution such that
and
Furthermore, if
\(u_{0}\in\mathscr{W}_{1}^{+}\), the solution
\(u(t)\)
decays exponentially.
$$\begin{aligned} u(t)\in\overline{\mathscr{W^{+}}},\quad t\in[0,\infty), \end{aligned}$$
$$\begin{aligned} \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds+J \bigl(u(t) \bigr)\leq J(u_{0}),\quad t\in[0,\infty). \end{aligned}$$
(4.1)
Proof
We will consider the following two cases.
First we address the case of the initial data \(u_{0}\in\mathscr{W}_{1}^{+}\).
Let \(\{w_{j}\}_{j=1}^{\infty}\), \(\{u_{0m}\}_{m=1}^{\infty}\), and \(\{u_{m}\} _{m=1}^{\infty}\) be the same as those stated in the proof of the local existence in the second section. Multiplying the (3.5) by \(\alpha '_{mi}(t)\), summing over i from 1 to m and integrating with respect to time variable on \([0,t]\), we have
where \(T_{\mathrm{max}}\) is the maximal existence time of solution \(u_{m}(x,t)\). We will prove that \(T_{\mathrm{max}}=\infty\).
$$\begin{aligned} \int_{0}^{t} \int_{\varOmega}u_{ms}^{2}\,dx\,ds+J \bigl(u_{m}(t) \bigr)=J \bigl(u_{m}(0) \bigr), \quad t \in[0,T_{\mathrm{max}}), \end{aligned}$$
(4.2)
By (3.3), (3.6) and the continuity of J, we have
Using (4.2) and (4.3) and combining with \(J(u_{0})< d\), we have
for sufficiently large m. Next, we will study
for sufficiently large m. We assume that (4.5) does not hold and think that there exists a smallest time \(t_{0}\) such that \(u_{m}(t_{0})\notin\mathscr{W}_{1}^{+}\). Then, we have \(u_{m}(t_{0})\in\partial {\mathscr{W}_{1}^{+}}\). So, we have
or
(4.6) contradicts with (4.4). If (4.7) holds, from (2.4) we can obtain
which contradicts with (4.4). Hence, we have (4.5), i.e., \(J(u_{m}(t))< d\), and \(I(u_{m}(t))>0\), for any \(t\in[0,T_{\mathrm{max}})\), for sufficiently large m. Then, by (2.3), we have
Using (4.8) and combining with the Poincaré inequality, we have
where \(C_{21}\) and \(C_{22}\) are the Poincaré constants. By (4.4) and (4.8), we have
Equations (4.9) and (4.10) imply that \(T_{\mathrm{max}}=\infty\). Then the rest is similar to the proof of the local existence, and we see that there exists a unique global weak solution \(u(t)\in\mathscr {W}_{1}^{+}\) of the problem (1.1), and
Now we address the case of the initial data \(u_{0}\in\mathscr{W}_{2}^{+}\).
$$\begin{aligned} J \bigl(u_{m}(0) \bigr)\rightarrow J(u_{0}) \quad \mbox{as }m\rightarrow\infty. \end{aligned}$$
(4.3)
$$\begin{aligned} \int_{0}^{t} \int_{\varOmega}u_{ms}^{2}\,dx\,ds+J \bigl(u_{m}(t) \bigr)< d,\quad t\in[0,T_{\mathrm{max}}), \end{aligned}$$
(4.4)
$$\begin{aligned} u_{m}(t)\in\mathscr{W}_{1}^{+}, \quad t \in[0,T_{\mathrm{max}}), \end{aligned}$$
(4.5)
$$\begin{aligned} J \bigl(u_{m}(t_{0}) \bigr)=d, \end{aligned}$$
(4.6)
$$\begin{aligned} I \bigl(u_{m}(t_{0}) \bigr)=0. \end{aligned}$$
(4.7)
$$\begin{aligned} J \bigl(u_{m}(t_{0}) \bigr)\geq\inf_{u\in\mathscr{N}}J(u)=d, \end{aligned}$$
$$\begin{aligned} d>J \bigl(u_{m}(t) \bigr)={}&\frac{1}{q}I \bigl(u_{m}(t) \bigr)+ \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx+ \frac{1}{q^{2}} \int_{\varOmega}\bigl\vert u_{m}(t) \bigr\vert ^{q}\,dx \\ \geq{}& \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx+ \frac{1}{q^{2}} \int_{\varOmega}\bigl\vert u_{m}(t) \bigr\vert ^{q}\,dx \\ \geq{}& \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx. \end{aligned}$$
(4.8)
$$\begin{aligned} \int_{\varOmega}\bigl\vert u_{m}(t) \bigr\vert ^{2}\,dx&\leq C_{21} \int_{\varOmega}\bigl\vert D u_{m}(t) \bigr\vert ^{2}\,dx\leq C_{21}C_{22} \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx \\ &\leq \biggl(\frac{1}{2}-\frac{1}{q} \biggr)^{-1}C_{21}C_{22} d, \end{aligned}$$
(4.9)
$$\begin{aligned} \int_{0}^{t} \int_{\varOmega}u_{ms}^{2}\,dx\,ds+ \biggl( \frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u_{m}(t) \bigr\vert ^{2}\,dx< d. \end{aligned}$$
(4.10)
$$\begin{aligned} \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds+J \bigl(u(t) \bigr)\leq J(u_{0}), \quad t\in[0,\infty). \end{aligned}$$
First, we can choose a sequence \(\{\rho_{m}\}_{m=1}^{\infty}\subset(0,1)\) and \(\lim_{m \to\infty}\rho_{m}=1\). Next, we consider the following problem:
where \(u_{0m}=\rho_{m}u_{0}\). By \(I(u_{0})>0\) and Lemma 2.1, we see that there exists a \(\overline{\lambda}_{2}>1\). Hence, \(I(u_{0m})=I(\rho _{m}u_{0})>0\) and \(J(u_{0m})=J(\rho_{m}u_{0})< J(u_{0})=d\) hold. So, we have \(u_{0m}\in\mathscr{W}_{1}^{+}\). Similar to the previous case, we see that the problem (4.11) admits that, for any positive m, there exists a unique global \(u_{m}\) which satisfies \(u_{m}\in L^{\infty}(0,\infty;H_{0}^{2}(\varOmega))\), \(u_{mt}\in L^{2}(0,\infty ;L^{2}(\varOmega))\), \(u_{m}(0)=u_{0m}=\rho_{m}u_{0}\rightarrow u_{0}\) strongly in \(H_{0}^{2}(\varOmega)\), and
for any \(w\in H_{0}^{2}(\varOmega)\), and for a.e. \(t\in[0,\infty)\). Moreover, we have
and
The remainder of the proof can be processed as the previous case.
$$\begin{aligned} \textstyle\begin{cases} u_{t}+\Delta^{2}u= \vert u \vert ^{q-2}u\log \vert u \vert ,\quad x\in\varOmega, t>0, \\ u(x,t)=\Delta u(x,t)=0,\quad x\in\partial\varOmega, t>0, \\ u(x,0)=u_{0m}(x),\quad x\in\varOmega, \end{cases}\displaystyle \end{aligned}$$
(4.11)
$$\begin{aligned} \int_{\varOmega}u_{mt}w\,dx+ \int_{\varOmega}\Delta u_{m}\Delta w\,dx= \int _{\varOmega} \vert u_{m} \vert ^{q-2}u_{m} \log \vert u_{m} \vert w\,dx, \end{aligned}$$
(4.12)
$$\begin{aligned} u_{m}(t)\in\mathscr{W}_{1}^{+}, \quad t\in[0, \infty) \end{aligned}$$
$$\begin{aligned} \int_{0}^{t} \int_{\varOmega}u_{ms}^{2}\,dx\,ds+J \bigl(u_{m}(t) \bigr)\leq J(u_{0m})< d, \quad t\in [0,\infty). \end{aligned}$$
Finally, we discuss the decay results.
Since \(u_{0}\in\mathscr{W}_{1}^{+}\), similar to the first case, we obtain \(u(t)\in\mathscr{W}_{1}^{+}\) for any \(t\in[0,\infty)\). By (2.3) and (4.1), we obtain
By \(I(u(t))>0\), (2.4) and Lemma 2.1, there exists a \(\overline{\lambda}_{3}>1\) such that \(I(\overline{\lambda}_{3}u(t))=0\). We have
Using (4.13) and (4.14), we have
which implies that
It follows from (2.2) that
Using (4.15) and (4.16), we have
which implies that
It follows from (4.15) and (4.17) that
where \(C_{23}\) and \(C_{24}\) are the Poincaré inequality constants. Hence, by (4.18), we obtain
where
Integrating the \(I(u(s))\) with respect to s over \((t,T)\) and using the embedding \(H_{0}^{2}(\varOmega)\hookrightarrow L^{2}(\varOmega)\), we obtain
where \(C_{26}\) is the embedding constant. From (4.19) and (4.20), we have
Let \(T\rightarrow\infty\) in (4.21), we can get
From Lemma 2.4, we have
The above inequality implies that the solution \(u(t)\) decays exponentially. □
$$\begin{aligned} J(u_{0})>J \bigl(u(t) \bigr)={}&\frac{1}{q}I \bigl(u(t) \bigr)+ \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx+\frac{1}{q^{2}} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\,dx \\ \geq{}& \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx+\frac{1}{q^{2}} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\,dx. \end{aligned}$$
(4.13)
$$\begin{aligned} d\leq{}& J \bigl(\overline{\lambda}_{3}u(t) \bigr) \\ ={}&\frac{1}{q}I \bigl(\overline{\lambda}_{3}u(t) \bigr) + \biggl( \frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta \bigl(\overline{ \lambda}_{3}u(t) \bigr) \bigr\vert ^{2}\,dx+ \frac{1}{q^{2}} \int _{\varOmega}\bigl\vert \overline{\lambda}_{3}u(t) \bigr\vert ^{q} \,dx \\ ={}& \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta \bigl(\overline{ \lambda}_{3}u(t) \bigr) \bigr\vert ^{2}\,dx+ \frac{1}{q^{2}} \int _{\varOmega}\bigl\vert \overline{\lambda}_{3}u(t) \bigr\vert ^{q} \,dx \\ ={}& \biggl(\frac{1}{2}-\frac{1}{q} \biggr){\overline{ \lambda}_{3}}^{2} \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx+\frac{1}{q^{2}}{ \overline{\lambda }_{3}}^{q} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\,dx \\ ={}&{\overline{\lambda}_{3}}^{q} \biggl( \biggl( \frac{1}{2}-\frac{1}{q} \biggr) {\overline{\lambda}_{3}}^{2-q} \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx+\frac{1}{q^{2}} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\,dx \biggr) \\ \leq{}&{\overline{\lambda}_{3}}^{q} \biggl( \biggl( \frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx+\frac{1}{q^{2}} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\,dx \biggr). \end{aligned}$$
(4.14)
$$\begin{aligned} d\leq{\overline{\lambda}_{3}}^{q}J(u_{0}), \end{aligned}$$
$$\begin{aligned} \overline{\lambda}_{3}\geq \biggl(\frac{d}{J(u_{0})} \biggr)^{\frac{1}{q}}. \end{aligned}$$
(4.15)
$$\begin{aligned} 0={}&I \bigl(\overline{\lambda}_{3}u(t) \bigr)= \int_{\varOmega}\bigl\vert \Delta \bigl(\overline{\lambda }_{3}u(t) \bigr) \bigr\vert ^{2}\,dx- \int_{\varOmega}\bigl\vert \overline{\lambda}_{3}u(t) \bigr\vert ^{q} \log \bigl\vert \overline{\lambda}_{3}u(t) \bigr\vert \,dx \\ ={}&{\overline{\lambda}_{3}}^{2} \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx-{\overline { \lambda}_{3}}^{q}\log\overline{ \lambda}_{3} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\,dx \\ &{}-{\overline{\lambda}_{3}}^{q} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\log \bigl\vert u(t) \bigr\vert \,dx \\ ={}&{\overline{\lambda}_{3}}^{q}I \bigl(u(t) \bigr)+{ \overline{ \lambda}_{3}}^{2} \int _{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2}\,dx-{\overline{ \lambda}_{3}}^{q} \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx \\ &{}-{\overline{\lambda}_{3}}^{q}\log\overline{ \lambda}_{3} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\,dx \\ ={}&{\overline{\lambda}_{3}}^{q}I \bigl(u(t) \bigr)+ \bigl({ \overline{\lambda }_{3}}^{2}-{\overline{ \lambda}_{3}}^{q} \bigr) \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx \\ &{}-{\overline{\lambda}_{3}}^{q}\log\overline{ \lambda}_{3} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\,dx. \end{aligned}$$
(4.16)
$$\begin{aligned} {\overline{\lambda}_{3}}^{q}I \bigl(u(t) \bigr)={}& \bigl({ \overline{\lambda}_{3}}^{q} -{\overline{ \lambda}_{3}}^{2} \bigr) \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx+{\overline { \lambda}_{3}}^{q}\log\overline{ \lambda}_{3} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\,dx \\ \geq{}& \bigl({\overline{\lambda}_{3}}^{q} -{\overline{ \lambda}_{3}}^{2} \bigr) \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx, \end{aligned}$$
$$\begin{aligned} I \bigl(u(t) \bigr)\geq \bigl(1-{\overline{\lambda}_{3}}^{2-q} \bigr) \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx. \end{aligned}$$
(4.17)
$$\begin{aligned} I \bigl(u(t) \bigr)\geq{}& \bigl(1-{\overline{ \lambda}_{3}}^{2-q} \bigr) \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx \\ \geq{}& \biggl[1- \biggl(\frac{d}{J(u_{0})} \biggr)^{{\frac{2}{q}}-1} \biggr] \int _{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2}\,dx \\ \geq{}&C_{23}^{-1} \biggl[1- \biggl(\frac{d}{J(u_{0})} \biggr)^{{\frac {2}{q}}-1} \biggr] \int_{\varOmega}\bigl\vert D u(t) \bigr\vert ^{2}\,dx \\ \geq{}&C_{23}^{-1}C_{24}^{-1} \biggl[1- \biggl(\frac{d}{J(u_{0})} \biggr)^{{\frac {2}{q}}-1} \biggr] \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{2}\,dx, \end{aligned}$$
(4.18)
$$\begin{aligned} I \bigl(u(t) \bigr)\geq{}&\frac{1}{3} \biggl[1- \biggl( \frac{d}{J(u_{0})} \biggr)^{{\frac {2}{q}}-1} \biggr] \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx \\ &{}+\frac{1}{3}C_{23}^{-1} \biggl[1- \biggl( \frac{d}{J(u_{0})} \biggr)^{{\frac {2}{q}}-1} \biggr] \int_{\varOmega}\bigl\vert D u(t) \bigr\vert ^{2}\,dx \\ &{}+\frac{1}{3}C_{23}^{-1}C_{24}^{-1} \biggl[1- \biggl(\frac{d}{J(u_{0})} \biggr)^{{\frac{2}{q}}-1} \biggr] \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{2}\,dx \\ \geq{}&C_{25} \biggl( \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx+ \int_{\varOmega}\bigl\vert D u(t) \bigr\vert ^{2}\,dx+ \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{2}\,dx \biggr) \\ ={}&C_{25} \bigl\Vert u(t) \bigr\Vert _{H_{0}^{2}(\varOmega)}^{2}, \end{aligned}$$
(4.19)
$$\begin{aligned} C_{25}={}&\max \biggl\{ \frac{1}{3} \biggl[1- \biggl( \frac{d}{J(u_{0})} \biggr)^{{\frac{2}{q}}-1} \biggr], \frac{C_{21}^{-1}}{3} \biggl[1- \biggl(\frac {d}{J(u_{0})} \biggr)^{{\frac{2}{q}}-1} \biggr], \\ &\frac{C_{21}^{-1}C_{22}^{-1}}{3} \biggl[1- \biggl(\frac{d}{J(u_{0})} \biggr)^{{\frac{2}{q}}-1} \biggr] \biggr\} . \end{aligned}$$
$$\begin{aligned} \int_{t}^{T}I \bigl(u(s) \bigr)\,ds={}&{-} \int_{t}^{T} \int_{\varOmega}u_{s}(s)u(s)\,dx\,ds =-\frac{1}{2} \int_{\varOmega}u(T)^{2}\,dx+\frac{1}{2} \int_{\varOmega}u(t)^{2}\,dx \\ \leq{}&\frac{1}{2} \int_{\varOmega}u(t)^{2}\,dx \\ \leq{}&\frac{1}{2}C_{26}^{2} \bigl\Vert u(t) \bigr\Vert _{H_{0}^{2}(\varOmega)}^{2}, \end{aligned}$$
(4.20)
$$\begin{aligned} \int_{t}^{T}C_{25} \bigl\Vert u(t) \bigr\Vert _{H_{0}^{2}(\varOmega)}^{2}\,ds \leq\frac{1}{2}C_{26}^{2} \bigl\Vert u(t) \bigr\Vert _{H_{0}^{2}(\varOmega)}^{2},\quad \mbox{for all }t\in[0,T]. \end{aligned}$$
(4.21)
$$\begin{aligned} \int_{t}^{\infty} \bigl\Vert u(t) \bigr\Vert _{H_{0}^{2}(\varOmega)}^{2} \,ds \leq\frac{1}{2}C_{25}^{-1}C_{26}^{2} \bigl\Vert u(t) \bigr\Vert _{H_{0}^{2}(\varOmega)}^{2}. \end{aligned}$$
(4.22)
$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{H_{0}^{2}(\varOmega)}^{2}\leq \bigl\Vert u(0) \bigr\Vert _{H_{0}^{2}(\varOmega)}^{2}e^{1-\frac{2t}{C_{25}^{-1}C_{26}^{2}}},\quad t\in[0, \infty). \end{aligned}$$
5 Blow up
Theorem 5.1
If
\(u_{0}\in\mathscr{W}_{1}^{-}\), the unique local weak solution
\(u(t)\)
of the problem (1.1) blows up in finite time, i.e., there exists a
\(T_{*}>0\)
such that
$$\begin{aligned} \lim_{t\to T_{*}^{-}} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{2}\,dx= \infty. \end{aligned}$$
Proof
Since \(u_{0}\in\mathscr{W}_{1}^{-}\), it follows from the local existence that there exists a unique local weak solution \(u(t)\) of the problem (1.1) such that
Next, we prove \(u(t)\in\mathscr{W}_{1}^{-}\) for \(t\in[0,T_{\mathrm{max}}]\). We assume \(u(t)\) leaves \(\mathscr{W}_{1}^{+}\) at time \(t=t_{1}\), then there exists a sequence \(\{t_{n}\}\) such that \(I(u(t_{n}))\leq0 \) as \(t_{n}\rightarrow t_{1}^{-}\). It follows from lower semicontinuity of \(L^{2}\) norm that
We have \(I(u(t_{1}))=0\) according to \(u(t_{1})\notin\mathscr{W}_{1}^{+}\). By (2.4) and (5.1), we have
which is a contradiction. So, \(u(t)\in\mathscr{W}_{1}^{-}\) for \(t\in [0,T_{\mathrm{max}}]\). Next, we will study that \(u(t)\) blows up in finite time by contradiction. Thus, we assume \(u(t)\) is global. We contract a function \(\varPhi:[0,\infty)\rightarrow\mathbb {R}^{+}\), and
We can easily obtain
By (2.2) and (5.4), we have
From \(u(t)\in\mathscr{W}_{1}^{-}\) and (5.5), we can obtain
Thus, it follows from \(u_{0}\in\mathscr{W}_{1}^{-}\) and (5.4) that
Using the Hölder inequality and combining (5.5), we have
By (2.3) and (5.5), we have
Since \(u(t)\in\mathscr{W}_{1}^{-}\), \(I(u(t))<0\). By Lemma 2.1, there exists a \(\overline{\lambda}_{4}\), \(0<\overline{\lambda}_{4}<1\) such that \(I(\overline{\lambda}_{4}u(t)))=0\). It follows from (2.3) and (2.4) that
Combining (5.9) with (5.10), we have
Using (5.3), (5.8) and (5.11), we have
We fix a \(t_{2}>0\). It follows from (5.7) that we have
Hence, by (5.12) and (5.13), we have
We choose \(T>t_{2}\) sufficiently large and contract a function \(\varPsi (t)\) as follows:
From (5.13) and (5.15), we can easily see that for any \(t\in [t_{2},T]\), \(\varPsi(t)\geq\varPhi(t)>0\) holds. It follows from (5.4) and (5.14) that, for any \(t\in [t_{2},T]\), \(\varPsi'(t)=\varPhi'(t)-\varPhi'(0)\) holds, thus we also have \(\varPsi''(t)=\varPhi''(t)>0\) from (5.6). Thus, we can obtain from (5.14)
for \(t\in[t_{2},T]\). Let \(\chi(t)=\varPsi(t)^{-\frac{q-2}{2}}\). Thus,
From (5.16) and (5.17), we have
for \(t\in[t_{2},T]\). This shows that, for any sufficiently large \(T>t_{2}\), \(\chi(t)\) is a concave function in \([t_{2},T]\). \(\chi(t_{2})>0\) and \(\chi ''(t_{2})<0\), so there exists a finite time \(T_{*}>t_{2}>0\) such that
which implies
Hence, we have
i.e.,
This is a contradiction to our assumption. □
$$\begin{aligned} \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds+J \bigl(u(t) \bigr)\leq J(u_{0})< d,\quad t\in[0,T_{\mathrm{max}}]. \end{aligned}$$
(5.1)
$$\begin{aligned} I \bigl(u(t_{1}) \bigr)\leq\liminf_{n \to\infty}I \bigl(u(t_{n}) \bigr)\leq0. \end{aligned}$$
(5.2)
$$\begin{aligned} d=\inf_{u\in\mathscr{N}}J(u)\leq J \bigl(u(t_{1}) \bigr)< d, \end{aligned}$$
$$\begin{aligned} \varPhi(t)= \int_{0}^{t} \int_{\varOmega}u^{2}\,dx\,ds. \end{aligned}$$
(5.3)
$$\begin{aligned} \varPhi'(t)= \int_{\varOmega}u^{2}\,dx. \end{aligned}$$
(5.4)
$$\begin{aligned} \varPhi''(t)=2 \int_{\varOmega}uu_{t}\,dx=2 \int_{\varOmega} \vert u \vert ^{q-2}u\log \vert u \vert \,dx-2 \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx =-2I(u). \end{aligned}$$
(5.5)
$$\begin{aligned} \varPhi''(t)>0. \end{aligned}$$
(5.6)
$$\begin{aligned} \varPhi'(t)\geq\varPhi'(0)= \int_{\varOmega}u_{0}^{2}\,dx>0. \end{aligned}$$
(5.7)
$$\begin{aligned} \frac{1}{4} \bigl(\varPhi'(t)- \varPhi'(0) \bigr)^{2}&= \frac{1}{4} \biggl( \int_{0}^{t}\varPhi''(s) \,ds \biggr)^{2} = \biggl( \int_{0}^{t} \int_{\varOmega}uu_{s}\,dx\,ds \biggr)^{2} \\ &\leq \int_{0}^{t} \int_{\varOmega}u^{2}\,dx\,ds \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds. \end{aligned}$$
(5.8)
$$\begin{aligned} \varPhi''(t)={}&{-}2I(u) =-2qJ(u) +2q \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx+ \frac{2}{q} \int_{\varOmega} \vert u \vert ^{q}\,dx \\ ={}&{-}2qJ(u_{0})+2q \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds \\ &{} +2q \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx + \frac{2}{q} \int_{\varOmega} \vert u \vert ^{q}\,dx. \end{aligned}$$
(5.9)
$$\begin{aligned} d={}&\inf_{u\in\mathscr{N}}J(u)\leq J \bigl(\overline{ \lambda}_{4}u(t) \bigr) \\ ={}&\frac{1}{q}I \bigl(\overline{\lambda}_{4}u(t) \bigr) + \biggl( \frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta \bigl(\overline{ \lambda}_{4}u(t) \bigr) \bigr\vert ^{2}\,dx + \frac{1}{q^{2}} \int_{\varOmega}\bigl\vert \overline{\lambda}_{4}u(t) \bigr\vert ^{q} \,dx \\ ={}& \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta \bigl(\overline{ \lambda}_{4}u(t) \bigr) \bigr\vert ^{2}\,dx + \frac{1}{q^{2}} \int_{\varOmega}\bigl\vert \overline{\lambda}_{4}u(t) \bigr\vert ^{q} \,dx \\ ={}&\overline{\lambda}_{4}^{2} \biggl(\frac{1}{2}- \frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx +\overline{ \lambda}_{4}^{q}\frac{1}{q^{2}} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\,dx \\ \leq{}& \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx +\frac{1}{q^{2}} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\,dx. \end{aligned}$$
(5.10)
$$\begin{aligned} \varPhi''(t) ={}&{-}2qJ(u_{0})+2q \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds +2q \biggl( \frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega} \vert \Delta u \vert ^{2}\,dx + \frac{2}{q} \int_{\varOmega} \vert u \vert ^{q}\,dx \\ ={}&{-}2qJ(u_{0})+2q \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds \\ &{}+2q \biggl[ \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \int_{\varOmega}\bigl\vert \Delta u(t) \bigr\vert ^{2} \,dx +\frac{1}{q^{2}} \int_{\varOmega}\bigl\vert u(t) \bigr\vert ^{q}\,dx \biggr] \\ \geq{}&2q \bigl(d-J(u_{0}) \bigr)+2q \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds. \end{aligned}$$
(5.11)
$$\begin{aligned} \varPhi(t)\varPhi''(t)={}& \int_{0}^{t} \int_{\varOmega}u^{2}\,dx\,ds\varPhi''(t) \\ \geq{}& \int_{0}^{t} \int_{\varOmega}u^{2}\,dx\,ds \biggl[2q \bigl(d-J(u_{0}) \bigr)+2q \int_{0}^{t} \int_{\varOmega}u_{s}^{2}\,dx\,ds \biggr] \\ \geq{}&\varPhi(t)2q \bigl(d-J(u_{0}) \bigr)+\frac{q}{2} \bigl( \varPhi'(t)-\varPhi'(0) \bigr)^{2}. \end{aligned}$$
(5.12)
$$\begin{aligned} \varPhi(t)\geq\varPhi(t_{2})= \int_{0}^{t_{2}} \int_{\varOmega}u^{2}\,dx\,ds\geq t_{2} \int_{\varOmega}u_{0}^{2}\,dx>0,\quad \mbox{for }t \in[t_{2},\infty). \end{aligned}$$
(5.13)
$$\begin{aligned} \varPhi(t)\varPhi''(t)- \frac{q}{2} \bigl(\varPhi'(t)-\varPhi'(0) \bigr)^{2} \geq{}&\varPhi(t)2q \bigl(d-J(u_{0}) \bigr) \\ \geq{}& t_{2} \int_{\varOmega}u_{0}^{2}\,dx >0, \quad \mbox{for }t \in[t_{2},\infty). \end{aligned}$$
(5.14)
$$\begin{aligned} \varPsi(t)=\varPhi(t)+(T-t) \int_{\varOmega}u_{0}^{2}\,dx,\quad t \in[t_{2},T]. \end{aligned}$$
(5.15)
$$\begin{aligned} \varPsi(t)\varPsi''(t)- \frac{q}{2}\varPsi'(t)^{2} \geq{}&\varPhi(t) \varPhi''(t)+\frac{q}{2} \bigl( \varPhi'(t)-\varPhi'(0) \bigr)^{2} \\ \geq{}&\varPhi(t)2q \bigl(d-J(u_{0}) \bigr) \\ \geq{}& t_{2} \int_{\varOmega}u_{0}^{2}\,dx>0, \end{aligned}$$
(5.16)
$$\begin{aligned} \chi'(t)={-\frac{q-2}{2}} \varPsi(t)^{-\frac{q}{2}} \varPsi'(t). \end{aligned}$$
(5.17)
$$\begin{aligned} \chi''(t)={}&{\frac{q(q-2)}{4}} \varPsi(t)^{-\frac{q+2}{2}}\varPsi'(t)^{2} {- \frac{q-2}{2}}\varPsi(t)^{-\frac{q}{2}}\varPsi''(t) \\ ={}&\frac{q-2}{2}\varPsi(t)^{-\frac{q+2}{2}} \biggl[\frac{q}{2}\varPsi '(t)^{2}-\varPsi(t)\varPsi''(t) \biggr] < 0, \end{aligned}$$
(5.18)
$$\begin{aligned} \lim_{t \to T_{*}^{-}}\chi(t)=0, \end{aligned}$$
$$\begin{aligned} \lim_{t \to T_{*}^{-}}\varPsi(t)=\infty. \end{aligned}$$
$$\begin{aligned} \lim_{t \to T_{*}^{-}} \int_{0}^{t} \int_{\varOmega}u^{2}\,dx\,ds=\infty, \end{aligned}$$
$$\begin{aligned} \lim_{t \to T_{*}^{-}} \int_{\varOmega}u^{2}\,dx=\infty. \end{aligned}$$
Acknowledgements
The authors would like to express their gratitude for the referee’s valuable suggestions for the revision and improvement of the manuscript.
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