\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2019 (2019), No. 08, pp. 1--18.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2019 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2019/08\hfil $p$-biharmonic parabolic equations] {$p$-biharmonic parabolic equations with logarithmic nonlinearity} \author[J. Wang, C. Liu \hfil EJDE-2019/08\hfilneg] {Jiaojiao Wang, Changchun Liu} \address{Jiaojiao Wang \newline School of Mathematics, Jilin University, Changchun 130012, China} \email{wjj16@mails.jlu.edu.cn} \address{Changchun Liu (corresponding author) \newline School of Mathematics, Jilin University, Changchun 130012, China} \email{liucc@jlu.edu.cn} \dedicatory{Communicated by Peter Bates} \thanks{Submitted July 26, 2018. Published January 22, 2019.} \subjclass{35K35, 35A01, 35K55} \keywords{$p$-biharmonic parabolic equation; blow-up; decay; extinction; \hfill\break\indent non-extinction} \begin{abstract} We consider an initial-boundary-value problem for a class of $p$-biharmonic parabolic equation with logarithmic nonlinearity in a bounded domain. We prove that if $20,\\ u(x,t)=\Delta u(x,t)=0, \quad x\in\partial\Omega, t>0,\\ u(x,0)=u_{0}(x), \quad x\in\Omega, \end{gathered} \end{equation} where$\Omega$is a bounded domain in$\mathbb{R}^n$with smooth boundary$\partial\Omega$,$p$,$q$are positive constants, and$u_0\in (W_0^{1,p}(\Omega)\cap W^{2,p}(\Omega))\backslash\{0\}$. The term$\Delta(|\Delta u|^{p-2}\Delta u)$is called a$p$-biharmonic operator. In the past years, there have been many contributions devoted to the higher order equation. Liu and Guo \cite{LG} considered the following$p$-biharmonic parabolic initial-boundary value problem \begin{equation} \label{1-2} \frac{\partial u}{\partial t}+\Delta(|\Delta u|^{p-2}\Delta u)+\lambda|u|^{p-2}u=0, \quad x\in\Omega, \end{equation} where$p>2$and$\lambda>0$. By using the discrete-time method and uniform estimates, they established the existence and uniqueness of weak solutions. Hao and Zhou \cite{HZ} considered a$p$-biharmonic parabolic equation \begin{equation}\label{1-3} u_t+\Delta(|\Delta u|^{p-2}\Delta u)=|u|^q-\frac1{|\Omega|}\int_\Omega|u|dx, \end{equation} where$\max\{1,\frac{2n}{n+4}\}0$. Hao and Zhou obtained results on blowup, extinction and non-extinction of the solutions. The relevant equations have also been studied in \cite{BF, L}. In this paper, we study the parabolic$p$-biharmonic equation with the logarithmic nonlinearity. The second order parabolic equation with the logarithmic nonlinearity is studied. Chen considered the semilinear heat equation with the logarithmic nonlinearity \cite{CLL} and the semilinear pseudo-parabolic equations with the logarithmic nonlinearity \cite{CT}. Ji, Yin and Cao \cite{JYC} established the existence of positive periodic solutions and discussed the instability of such solutions for the semilinear pseudo-parabolic equation with the logarithmic source. Nahn and Truong \cite{NL} studied the nonlinear equation \begin{equation}\label{1-4} u_t-\Delta u_t-\Delta_p u=|u|^{p-2}u\log(|u|). \end{equation} It is a pseudoparabolic type equation, where$\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$and$\Delta_p$is the$p$-Laplacian. By using the potential well method, Nahn and Truong obtained results of existence or nonexistence of global weak solutions, and proved the large time decay of global weak solutions and the finite time blow-up of weak solutions. Cao and Liu \cite{CL} considered equation \eqref{1-4}. They discussed two cases: global boundedness and blowing-up at$\infty$. Moreover, they proved the asymptotic behavior of solutions and gave some decay estimates and growth estimates. He, Gao and Wang \cite{HYJ} considered the pseudo-parabolic$p$-Laplacian equation \begin{equation}\label{1-5} u_t-\Delta u_t-\Delta_p u=|u|^{q-2}u\log(|u|), \end{equation} where$21$. For$u\in (W_0^{1,p}(\Omega)\cap W^{2,p}(\Omega))\backslash\{0\}$, we define the energy functional$J$and Nehari functional$I$as follows \begin{gather}\label{1-6} J(u)=\frac{1}{p}\|\Delta u\|_p^p-\frac{1}{q}\int_\Omega|u|^q\log(|u|)dx +\frac{1}{q^2}\|u\|_q^q, \\ \label{1-7} I(u)=\|\Delta u\|_p^p-\int_\Omega|u|^q\log(|u|)dx. \end{gather} Let $N=\{u\in X_0:I(u)=0\}$ be the Nehari manifold. In section 2, we will show that$N$is not empty. Thus, we can define \begin{equation} \label{1-8} d=\inf_{u\in N}J(u). \end{equation} In Section 2, we show that$d$is positive and is attained by some$u\in N$. Now as in \cite{NL}, we introduce the following sets \begin{gather*} W_1=\{u\in X_0:J(u)0\},\quad W_2^+=\{u\in W_2:I(u)>0\},\quad W^+= W_1^+\cup W_2^+, \\ W_1^-=\{u\in W_1:I(u)<0\}, \quad W_2^-=\{u\in W_2:I(u)<0\},\quad W^-= W_1^-\cup W_2^-. \end{gather*} Clearly,$ W^+\cap W^-=\emptyset$and$ W^+\cup W^-= W$. We refer to$ W$as the potential well and$d$as the depth of the well. The set$ W^+$is regarded as the good part of the well, as we will show that every weak solution exists globally in time, provided the initial data are taken from$ W^+$. On the other hand, if the initial data are taken from a part of$ W^-$, we will prove a blow-up result for weak solutions. The plan of this paper is as follows. In Section 2, we collect some properties of the energy functional$J$and the Nehari functional$I$. In Section 3, we proved that the existence of the local weak solutions and the existence of the global weak solutions. In Section 4, we establish some properties of the weak solutions, such as the finite time blow-up, extinction and non-extinction of the solutions. \section{Preliminaries} In this section, we collect some properties of the energy functional$J$and the Nehari functional$I$, which following lemmas will be used for our main results. By the Gagliardo-Nirenberg multiplicative embedding inequality that$J$and$I$are continuous. Moreover, we have \begin{equation}\label{2-1} J(u)=\frac{1}{q}I(u)+\big(\frac{1}{p}-\frac{1}{q}\big)\|\Delta u\|_p^p +\frac{1}{q^2}\|u\|_q^q. \end{equation} Let$u\in X_0$and consider the real function$j:\lambda\mapsto J(\lambda u)$for$\lambda>0$, defined as follows $j(\lambda)=J(\lambda u)=\frac{\lambda^p}{p}\|\Delta u\|_p^p -\frac{\lambda^q}{q}\int_\Omega|u|^q\log(|u|)dx-\frac{\lambda^q}{q}\log\lambda\|u\|_q^q +\frac{\lambda^q}{q^2}\|u\|_q^q.$ The following lemma shows that$j(\lambda)$has a unique positive critical point$\lambda^*=\lambda^*(u)$. \begin{lemma}\label{lem2.1} Let$u\in X_0$. Then \begin{itemize} \item[(1)]$\lim_{\lambda\to 0^+}j(\lambda)=0$and$\lim_{\lambda\to +\infty}j(\lambda)=-\infty$; \item[(2)] there exists a unique$\lambda^*=\lambda^*(u)>0$such that$j'(\lambda^*)=0$; \item[(3)]$j(\lambda)$is increasing on$(0,\lambda^*)$, decreasing on$(\lambda^*,+\infty)$and attains its maximum at$\lambda^*$; \item[(4)]$I(\lambda u)>0$for$0<\lambda<\lambda^*$,$I(\lambda u)<0$for$\lambda>\lambda^*$, and$I(\lambda^*u)=0$. \end{itemize} \end{lemma} \begin{proof} For$u\in X_0$, by the definition of$j$, we have $j(\lambda)=\frac{\lambda^p}{p}\|\Delta u\|_p^p -\frac{\lambda^q}{q}\int_\Omega|u|^q\log(|u|)dx -\frac{\lambda^q}{q}\log\lambda\|u\|_q^q +\frac{\lambda^q}{q^2}\|u\|_q^q.$ It is clearly that$(1)$holds because$20$, let$k(\lambda)=\lambda^{1-p}j'(\lambda)$, through direct calculation, we have $k'(\lambda)=-\lambda^{q-p-1}\Big((q-p)\int_\Omega|u|^q\log(|u|)dx +(q-p)\log\lambda\|u\|_q^q+\|u\|_q^q\Big).$ Hence, there exists a $\lambda_1=\exp\Big(\frac{(p-q)\int_\Omega|u|^q\log(|u|)dx +\|u\|_q^q}{(q-p)\|u\|_q^q}\Big)>0,$ such that$k'(\lambda)>0$on$(0,\lambda_1)$,$k'(\lambda)<0$on$(\lambda_1,+\infty)$and$k'(\lambda_1)=0$. Therefore,$k(\lambda)$is increasing on$(0,\lambda_1)$, decreasing on$(\lambda_1,+\infty)$. Because of$k(0)=\|\Delta u\|_p^p>0$and$\lim_{\lambda\to +\infty}k(\lambda)=-\infty$, there exactly exists a$\lambda^*>0$, such that$k(\lambda^*)=0$, i.e.$j'(\lambda^*)=0$. So (2) holds. Then$j'(\lambda)=\lambda^{p-1}k(\lambda)$is positive on$(0,\lambda^*)$, and negative on$(\lambda^*,+\infty)$. So (3) holds. The last property, (4), is only a simple corollary of the fact that $I(\lambda u)=\lambda^p\|\Delta u\|_p^p -\lambda^q\int_\Omega|u|^q\log(|u|)dx -\lambda^q\log\lambda\|u\|_q^q=\lambda j'(\lambda).$ The proof is complete. \end{proof} Consequently the Nehari manifold$N$is not empty, and the number$d$defined by \eqref{1-8} is meaningful. The blow lemma gives us that$d$is positive and is attained by some$u\in N$. \begin{lemma}\label{lem2.2}$d$is positive and there is a positive function$u\in N$such that$J(u)=d$. \end{lemma} \begin{proof} According to \eqref{2-1}, we only need to prove that there exists a positive function$u\in N$such that$J(u)=d$. Let$\{u_k\}_{k=1}^\infty\subset N$be a minimizing sequence of$J$. i.e. $\lim_{k\to \infty}J(u_k)=d.$ It is clearly that$\{|u_k|\}_{k=1}^\infty\subset N$is also a minimizing sequence of$J$. So, without loss of generality, we assume that$u_k>0$a.e.\ for all$k\in\mathbb{N}$. On the other hand, we have already observed that$J$is coercive on$N$which implies that$\{u_k\}_{k=1}^\infty$is bounded in$W_0^{1,p}(\Omega)\cap W^{2,p}(\Omega)$. Let$\mu>0$is a sufficiently small such that$q+\mu<\frac{np}{n-2p}$, so the embedding$W_0^{2,p}\hookrightarrow L^{q+\mu}$is compact, and there exists a function$u$and a subsequence of$\{u_k\}_{k=1}^\infty$, still denoted by$\{u_k\}_{k=1}^\infty$, such that \begin{gather*} u_k\rightharpoonup u, \quad \text{weakly in } W_0^{1,p}(\Omega)\cap W^{2,p}(\Omega),\\ u_k\to u, \quad\text{strongly in } L^{q+\mu}(\Omega),\\ u_k(x)\to u, \quad\text{a.e. in } \Omega. \end{gather*} Thus, we have$u\geq0$a.e. in$\Omega$. By Lebesgue dominated convergence theorem, we see that \begin{gather} \label{2-3} \int_\Omega|u|^q\log(|u|)dx=\lim_{k\to \infty}\int_\Omega|u_k|^q\log(|u_k|)dx,\\ \label{2-4} \int_\Omega|u|^qdx=\lim_{k\to \infty}\int_\Omega|u_k|^qdx. \end{gather} The weak lower semicontinuity of$\|\cdot\|_{W^{2,p}}$implies \begin{equation} \label{2-5} \|\Delta u\|_p\leq\liminf_{k\to \infty}\|\Delta u_k\|_p. \end{equation} Combining \eqref{1-6}, \eqref{1-7}, \eqref{2-3}, \eqref{2-4} and \eqref{2-5}, we deduce that \begin{gather}\label{2-6} J(u)\leq\liminf_{k\to \infty}J(u_k)=d,\\ \label{2-7} I(u)\leq\liminf_{k\to \infty}I(u_k)=0. \end{gather} Thanks to$u_k\in N$one has$u_k\in X_0$and$I(u_k)=0$. Thus, by using the fact$\log x\leq(e\mu)^{-1}x^\mu$for$x\geq1and the Sobolev embedding inequality, we obtain \begin{align*} \|\Delta u_k\|_p^p &=\int_\Omega|u_k|^q\log(|u_k|)dx \\ &=\int_{\{x\in\Omega:|u_k(x)|\geq1\}}|u_k|^q\log(|u_k|)dx +\int_{\{x\in\Omega:|u_k(x)|<1\}}|u_k|^q\log(|u_k|)dx \\ &\leq\int_{\{x\in\Omega:|u_k(x)|\geq1\}}|u_k|^q\log(|u_k|)dx \\ &\leq(e\mu)^{-1}\int_{\{x\in\Omega:|u_k(x)|\geq1\}}|u_k|^{q+\mu}dx \\ &\leq(e\mu)^{-1}\|u_k\|_{q+\mu}^{q+\mu} \leq C\|\Delta u_k\|_{q+\mu}^{q+\mu}, \end{align*} for some positive constantC$, which implies $\int_\Omega|u_k|^q\log(|u_k|)dx=\|\Delta u_k\|_p^p\geq C.$ From this inequality and \eqref{2-3}, we derive $\int_\Omega|u|^q\log(|u|)dx\geq C.$ Therefore, we have$u\in X_0$. We easily obtain$I(u)\leq0$by \eqref{2-7}, now we show that$I(u)=0$. Indeed, if it is not true, we have$I(u)<0$, then by Lemma \ref{lem2.1}, there exists a$\lambda^*$such that$0<\lambda^*<1$and$I(\lambda^*u)=0. Thus, we conclude that \begin{align*} d&\leq J(\lambda^*u) =\big(\frac{1}{p}-\frac{1}{q}\big)\|\Delta(\lambda^*u)\|_p^p +\frac{1}{q^2}\|\lambda^*u\|_q^q\\ &\leq(\lambda^*)^p \Big(\big(\frac{1}{p}-\frac{1}{q}\big)\|\Delta u\|_p^p+\frac{1}{q^2}\|u\|_q^q\Big) \\ &\leq(\lambda^*)^p\liminf_{k\to \infty} \Big(\big(\frac{1}{p}-\frac{1}{q}\big)\|\Delta u_k\|_p^p+\frac{1}{q^2}\|u_k\|_q^q\Big) \\ &\leq(\lambda^*)^p\liminf_{k\to \infty}J(u_k) =(\lambda^*)^pd0 and a unique weak solution $u(t)$ of \eqref{1-1} satisfying $u(0)=u_0$. Moreover, $u$ satisfies the energy inequality \begin{equation}\label{3-2} \int_0^t\|u'(s)\|_2^2ds+J(u(t))\leq J(u_0), ~~0\leq t\leq T. \end{equation} \end{theorem} \begin{proof} We shall employ the Galerkin's method. The proof will be divided in 3 steps. \smallskip \noindent\textbf{Step 1: Approximate problem.} In the space $W_0^{1,p}(\Omega)\cap W^{2,p}(\Omega)$, using a basis $\{\omega_j\}_{j=1}^\infty$ we define the finite dimensional space $V_m=\operatorname{span}\{\omega_1, \omega_2,\dots , \omega_m\}$. Let $u_{0m}$ be an element of $V_m$ such that \begin{equation} \label{3-3} u_{0m}=\sum_{j=1}^m a_{mj}(t)\omega_j\to u_0,\quad \text{strongly in } W_0^{1,p}(\Omega)\cap W^{2,p}(\Omega), \end{equation} as $m\to \infty$. We find the approximate solution $u_m(x,t)$ of problem \eqref{1-1} in the form \begin{equation}\label{3-4} u_m(x,t)=\sum_{j=1}^m\alpha_{mj}(t)\omega_j(x), \end{equation} where the coefficients $\alpha_{mj}(1\leq j\leq m)$ satisfy the system of ordinary differential equations \begin{equation}\label{3-5} \langle u'_m, \omega_i\rangle+\langle |\Delta u_m|^{p-2},\Delta \omega_i\rangle =\int_\Omega|u_m|^{q-2}u_m\log(|u_m|)\omega_idx, \end{equation} for $i\in\{1, 2, \dots , m\}$, with the initial conditions \begin{equation}\label{3-6} \alpha_{mj}(0)=a_{mj},\quad j\in\{1, 2, \dots , m\}. \end{equation} The standard theory of ordinary differential equations, yields that there exists a positive $T_m$ such that $\alpha_{mj}\in C^1[0,T_m]$, and therefore $u_m\in C^1([0,T_m];W_0^{1,p}(\Omega)\cap W^{2,p}(\Omega))$. \smallskip \noindent\textbf{Step 2: A priori estimates.} Multiplying \eqref{3-5} by $\alpha_{mi}(t)$, summing for $i=1,\dots ,m$, and then integrating with respect to time variable on $[0,t]$, we know that \begin{equation}\label{3-7} S_m(t)=S_m(0)+\int_0^t\int_\Omega|u_m(x,s)|^q\log(|u_m(x,s)|)\,dx\,ds, \end{equation} where \begin{equation}\label{3-8} S_m(t)=\frac{1}{2}\|u_m\|_2^2+\int_0^t\|\Delta u_m(s)\|_p^pds. \end{equation} On the other hand, for any $\mu>0$, similarly we have \begin{equation}\label{3-9} \int_\Omega|u_m(t)|^q\log(|u_m(t)|)dx\leq (e\mu)^{-1}\|u_m(t)\|_{q+\mu}^{q+\mu}, \end{equation} where $\mu$ is chosen such that $0<\mu0$ such that $0<\mu1$ because $20$ such that $(q-1+\mu)q'<\frac{np}{n-2p}$. Then by a direct calculation and using Sobolev's inequality, we have \begin{equation} \label{3-24} \begin{aligned} &\int_\Omega|\Phi_m(x,t)|^{q'}dx\\ &=\int_{\{x\in\Omega:|u_m(x,t)|\leq 1\}}|\Phi_m(x,t)|^{q'}dx +\int_{\{x\in\Omega:|u_m(x,t)|> 1\}}|\Phi_m(x,t)|^{q'}dx \\ &\leq (e(q-1))^{-q'}|\Omega|+(e\mu)^{-q'}\int_{\{x\in\Omega:|u_m(x,t)|> 1\}} |u_m(t)|^{(q-1+\mu)q'}dx \\ &\leq C_1+C_2\|\Delta u_m(t)\|_p^{(q-1+\mu)q'}\leq C, \end{aligned} \end{equation} where $\Phi_m(x,t)=|u_m(x,t)|^{q-1}\log(|u_m(x,t)|)$, and we have used the fact that $|x^{q-1}\log x|\leq(e(q-1))^{-1}$ for $01, \mu>0$. Hence, by Lions's lemma (see \cite[Lemma 1.3]{SJ}), it follows from \eqref{3-23} and \eqref{3-24} that \begin{equation} \label{3-25} |u_m|^{q-2}u_m\log(|u_m|)\to |u|^{q-2}u\log(|u|), \quad\text{weakly* in } L^\infty(0,T;L^{q'}(\Omega)). \end{equation} Passing to the limit in \eqref{3-3} and \eqref{3-5} as $m\to \infty$, by \eqref{3-19}-\eqref{3-21} and \eqref{3-23}, we can show that $u$ satisfies the initial condition $u(0)=u_0$ and \begin{equation} \label{3-26} \int_\Omega u'(t)\omega dx+\int_\Omega\mathcal{X}(t)\Delta\omega dx =\int_\Omega|u(t)|^{q-2}u(t)\log(|u(t)|)\omega dx, \end{equation} for all $\omega\in W_0^{2,p}(\Omega)$ and for almost every $t\in[0,T]$. Finally, by the well known arguments of the theory of monotone operators, we know that $\mathcal{X}=|\Delta u|^{p-2}\Delta u,$ which implies \begin{equation}\label{3-27} \langle u'(t),\omega\rangle+\langle|\Delta u|^{p-2}\Delta u,\Delta\omega\rangle =\int_\Omega|u(t)|^{q-2}u(t)\log(|u(t)|)\omega dx, \end{equation} for all $\omega\in W_0^{2,p}(\Omega)$ and for almost every $t\in[0,T]$. \smallskip \noindent\textbf{Step 4: Uniqueness.} Firstly, as a result from \eqref{3-27}, we derive that \begin{equation}\label{3-28} \langle u'(t),v(t)\rangle+\langle|\Delta u|^{p-2}\Delta u,\Delta v(t)\rangle =\int_\Omega|u(t)|^{q-2}u(t)\log(|u(t)|)v(t)dx, \end{equation} for all $v\in L^2(0,T; W_0^{2,p}(\Omega))$. Now, assume there are two solutions $u_1$ and $u_2$ to the problem \eqref{1-1} with the same initial condition $u_0\in W_0^{1,p}(\Omega)\cap W^{2,p}(\Omega)$. Let $\omega=u_1-u_2$, then $\omega(0)=0$ and $\omega\in L^2(0,T;W_0^{1,p}(\Omega)\cap W^{2,p}(\Omega)),\quad \omega'\in L^2(0,T; L^2(\Omega)).$ Let $v(s)= \begin{cases} u_1(s)-u_2(s), & s\in[0,t],\\ 0, &s\in[t,T], \end{cases}$ then, it follows from \eqref{3-28} and the monotonicity of the operator $\Delta(|\Delta u|^{p-2}\Delta u)$ that $\frac{1}{2}\|\omega(t)\|_2^2 \leq\int_0^t\langle F(u_1(s))-F(u_2(s)),u_1(s)-u_2(s)\rangle ds,$ where $F(s)=|s|^{q-2}s\log(|s|)$. As a consequence, the uniqueness is derived from the locally Lipschitz continuity of $F:\mathbb{R}^*\to \mathbb{R}$ and Gronwall's inequality. \smallskip \noindent\textbf{Step 5: Energy inequality.} Now we show that the solution $u$ satisfies the energy inequality \eqref{3-2}. For this, let $\delta\in C[0,T]$ is a nonnegative function. Then, it follows from \eqref{3-13} that \begin{equation}\label{3-29} \int_0^T\delta(t)\int_0^t\|u'_m(s)\|_2^2dsdt+\int_0^TJ(u_m(t))\delta(t)dt =\int_0^TJ(u_m(0))\delta(t)dt. \end{equation} The right hand side of \eqref{3-29} converges to $\int_0^TJ(u_0)\delta(t)dt$ as $m\to \infty$. The second term in the right hand side, $\int_0^TJ(u_m(t))\delta(t)dt$, is lower semi-continuous with respect to the weak topology of $L^2(0,T; W_0^{1,p}(\Omega)\cap W^{2,p}(\Omega))$. Hence \begin{equation}\label{3-30} \int_0^TJ(u(t))\delta(t)dt\leq\liminf_{m\to +\infty}\int_0^TJ(u_m(t))\delta(t)dt. \end{equation} Therefore, we obtain $\int_0^T\delta(t)\int_0^t\|u'(s)\|_2^2dsdt+\int_0^TJ(u(t))\delta(t)dt \leq\int_0^TJ(u_0)\delta(t)dt.$ Since $\delta$ is arbitrary nonnegative function, we obtain the energy inequality $\int_0^t\|u'(s)\|_2^2ds+J(u(t))\leq J(u_0),\quad 0\leq t\leq T.$ The proof is complete. \end{proof} Next, we state the sufficient conditions for the global existence of weak solutions to the problem \eqref{1-1}. \begin{theorem}\label{thm3.2} Let $u_0\in W^+$, there exists a unique global weak solution $u$ of \eqref{1-1} satisfying the initial condition $u(0)=u_0$. We have that $u(t)\in W^+$ holds for all $0\leq t<+\infty$, and the energy estimate \begin{equation}\label{3-31} \int_0^t\|u'(s)\|_2^2ds+J(u(t))= J(u_0),\quad 0\leq t\leq +\infty. \end{equation} Moreover, the solution decays algebraically provided $u_0\in W_1^+$. \end{theorem} To prove Theorem \ref{thm3.2}, we need the following lemma. \begin{lemma}[\cite{MP}] \label{lem3.1} Let $f:\mathbb{R}^+\to \mathbb{R}^+$ be a nonincreasing function and $\sigma$ is a positive constant such that $$\int_0^{+\infty}f^{1+\sigma}(s)ds\leq\frac{1}{\omega}f(t), \quad \forall t\geq0.$$ Then $f(t)\leq f(0)(\frac{1+\sigma}{1+\omega\sigma t})^{\frac{1}{\sigma}}$, for all $t\geq0$. \end{lemma} \begin{proof}[Proof of Theorem \ref {thm3.2}] To prove the existence of a global solution to \eqref{1-1}, we first choose a sequence $\{\gamma_m\}_{m=1}^\infty\subset(0,1)$ such that $\lim_{m\to \infty}\gamma_m=1$. Since $I(u_0)\geq0$, by Lemma \ref{lem2.1}, we have $I(\gamma_mu_0)>0$ and $J(\gamma_mu_0)0$, and $J(u_{m_{k_m}}(0))0$ and $J(u_0)0$, there exists a $\lambda_*>1$ such that $I(\lambda_*u(t))=0$. This implies that \begin{equation} \label{3-40} \begin{aligned} d&\leq J(\lambda_*u(t)) =\big(\frac{1}{p}-\frac{1}{q}\big)\|\Delta (\lambda_*u(t))\|_p^p +\frac{1}{q^2}\|\lambda_*u(t)\|_q^q \\ &\leq \lambda_*^q\big\{\big(\frac{1}{p}-\frac{1}{q}\big)\|\Delta u(t)\|_p^p +\frac{1}{q^2}\|u(t)\|_q^q\big\}. \end{aligned} \end{equation} It follows from \eqref{3-39} and \eqref{3-40} that \begin{equation}\label{3-41} \lambda_*\geq \Big(\frac{d}{J(u_0)}\Big)^{1/q}. \end{equation} On the one hand, we obtain \begin{align*} 0=I(\lambda_*u(t)) &=\lambda_*^p\|\Delta u(t)\|_p^p-\lambda_*^q \int_\Omega|u(t)|^q\log(|u(t)|)dx-\lambda_*^q\log\lambda_*\|u(t)\|_q^q \\ &=\lambda_*^qI(u)-(\lambda_*^q-\lambda_*^p)\|\Delta u(t)\|_p^p-\lambda_*^q \log\lambda_*\|u(t)\|_q^q. \end{align*} From this inequality and \eqref{3-41}, we deduce that \begin{equation} \label{3-42} I(u)\geq\Big\{1-\Big(\frac{d}{J(u_0)}\Big)^{\frac{p}{q}-1}\Big\}\|\Delta u(t)\|_p^p \geq C\|u(t)\|_{2,p}^p. \end{equation} On the other hand, by the compact embedding $W^{2,p}(\Omega)\hookrightarrow L^2(\Omega)$, we see that \begin{equation} \label{3-43} \begin{aligned} \int_t^TI(u(s))ds &=-\int_t^T\langle u'(s),u(s)\rangle ds\\ &=\frac{1}{2}\|u(t)\|_2^2-\frac{1}{2}\|u(T)\|_2^2\\ &\leq C\|u(t)\|_{2,p}^2. \end{aligned} \end{equation} By \eqref{3-42} and \eqref{3-43}, we obtain \begin{equation} \label{3-44} \int_t^TI(u(s))ds\leq\frac{1}{\omega}\Big(I(u(t))\Big)^{2/p} \leq\frac{1}{\omega}\|\Delta u(t)\|_p^2\leq\frac{1}{\omega}\|u(t)\|_{2,p}^2, \end{equation} for all $t\in[0,T]$, and where $\omega$ is a positive constant. Let $T\to +\infty$ in \eqref{3-44}, it follows that \begin{equation} \label{3-45} \int_t^{+\infty}\|u(s)\|_{2,p}^pds \leq C\int_t^{+\infty}I(u(s))ds\leq\frac{1}{\omega}\|u(t)\|_{2,p}^2. \end{equation} Since $p>2$, we can choose $f(t)=\|u(t)\|_{2,p}^2$ and $\sigma=\frac{p}{2}-1$ in Lemma \ref{lem3.1} to obtain $\|u(t)\|_{2,p}^2\leq\|u_0\|_{2,p}^2 \Big(\frac{1+\sigma}{1+\omega\sigma t}\Big)^{\frac{1}{p-2}},\quad \forall t\geq0.$ The prove is complete. \end{proof} \section{Blow-up and extinction of solutions} Firstly, we state the theorem for finite time blow-up for weak solution of problem \eqref{1-1} in when $20$, we consider $\Gamma:[0,T]\to \mathbb{R}^+$ defined by \begin{equation}\label{4-2} \Gamma(t)=\int_0^t\|u(s)\|_2^2ds. \end{equation} Then, by direct calculations, we have \begin{gather}\label{4-3} \Gamma'(t)-\Gamma'(0)=\|u(t)\|_2^2-\|u_0\|_2^2 =2\int_0^t\langle u'(s),u(s)\rangle ds, \\ \label{4-4} \Gamma''(t)=2\langle u',u\rangle=-2I(u). \end{gather} Combining \eqref{2-1} and \eqref{4-1}, we obtain \begin{equation} \label{4-5} \begin{aligned} \Gamma''(t)=&-2I(u) =-2qJ(u)+\frac{2}{q}\|u\|_q^q+\Big(\frac{2q}{p}-2\Big)\|\Delta u\|_p^p\\ &\geq-2qJ(u_0)+2q\int_0^t\|u'(s)\|_2^2ds+\frac{2}{q}\|u\|_q^q +\big(\frac{2q}{p}-2\big)\|\Delta u\|_p^p. \end{aligned} \end{equation} Since $u(t)\in W_1^-$ for $t\in[0,T_{\rm max}]$, so $I(u)<0$, then there exist a $\lambda^*\in(0,1)$ such that $I(\lambda^*u)=0$. Thus, by the definition of $d$, we have \begin{equation} \label{4-6} \big(\frac{1}{p}-\frac{1}{q}\big)\|\Delta u\|_p^p+\frac{1}{q^2}\|u\|_q^q \geq J(\lambda^*u)\geq d. \end{equation} It follows from \eqref{4-5} and \eqref{4-6} that \begin{equation}\label{4-7} \Gamma''(t)\geq 2q\int_0^t\|u'(s)\|_2^2ds+2q(d-J(u_0)). \end{equation} By \eqref{4-4} and $I(u)<0$, we know $\Gamma''(t)>0$, so we obtain \begin{equation}\label{4-8} \Gamma'(t)>\Gamma'(0)=\|u_0\|_2^2>0,\quad \forall t>0. \end{equation} From \eqref{4-3} and H\"{o}lder's inequality, we obtain \begin{equation} \label{4-9} \frac{1}{4}\left(\Gamma'(t)-\Gamma'(0)\right)^2 \leq\Big(\int_0^t\langle u'(s),u(s)\rangle ds\Big)^2 \leq\int_0^t\|u'(s)\|_2^2ds\int_0^t\|u(s)\|_2^2ds. \end{equation} Combining \eqref{4-2}, \eqref{4-7} and \eqref{4-9}, we have \begin{align*} \Gamma(t)\Gamma''(t) &\geq\int_0^t\|u(s)\|_2^2ds\Big(2q\int_0^t\|u'(s)\|_2^2ds+2q(d-J(u_0))\Big) \\ &\geq\frac{q}{2}\left(\Gamma'(t)-\Gamma'(0)\right)^2+2q(d-J(u_0))\Gamma(t). \end{align*} Now, fix $t_0>0$. The \eqref{4-8} implies \begin{equation} \label{4-10} \Gamma(t)\geq\Gamma(t_0)=\int_0^{t_0}\|u(s)\|_2^2ds\geq\|u_0\|_2^2t_0>0, \quad \forall t\geq t_0. \end{equation} Hence, \begin{equation}\label{4-11} \Gamma(t)\Gamma''(t)-\frac{q}{2}\left(\Gamma'(t)-\Gamma'(0)\right)^2 \geq2q(d-J(u_0))\|u_0\|_2^2t_0>0,\quad \forall t\geq t_0. \end{equation} We choose $T>t_0$ sufficiently large, and let $G(t)=\Gamma(t)+(T-t)\|u_0\|_2^2,~~~\forall ~t\in[0,T].$ Then $G(t)>\Gamma(t)>0$, $G'(t)=\Gamma'(t)-\|u_0\|_2^2=\Gamma'(t)-\Gamma'(0)>0$ and $G''(t)=\Gamma''(t)>0$. Thus, \eqref{4-11} implies \begin{equation} \label{4-12} G(t)G''(t)-\frac{q}{2}(G'(t))^2\geq2q(d-J(u_0))\|u_0\|_2^2t_0>0,\quad \forall t\geq t_0. \end{equation} By setting $y(t)=(G(t))^{-(q-2)/2}$, inequality \eqref{4-12} becomes $y''(t)\leq -q(q-2)(d-J(u_0))\|u_0\|_2^2t_0(G(t))^{-\frac{q+2}{2}}<0, \quad \forall t\in[t_0,T].$ This inequality implies that $y$ is a concave function in $[t_0,T]$, for each $T>t_0$. Because of $y(t_0)>0$ and $y'(t)=-\frac{q-2}{2}(G(t))^{-\frac{q}{2}}G'(t)<0$, for all $t$, there exists a finite time $T_*$ such that $\lim_{t\to T_*^-}y(t)=0$ if we choose $T$ sufficiently large. Consequently, $\lim_{t\to T_*^-}G(t)=+\infty$. This implies that $\lim_{t\to T_*^-}\int_0^t\|u(s)\|_2^2ds=+\infty$. Hence, we see that $\lim_{t\to T_*^-}\|u(t)\|_2^2=+\infty$ which contradicts the assumption of $u(t)$ being global. The proof is complete. \end{proof} Next, we discuss the finite time blow-up, extinction and non-extinction of the weak solution to the problem \eqref{1-1} in the case of $\max\{1,\frac{2n}{n+4}\}0$. Before showing these results, we claim that the local existence of the weak solution to the problem \eqref{1-1} can be obtained by using Galerkin approximation method. Let $u(x,t)$ be the weak solution to the problem \eqref{1-1}. We introduce some functionals and notations as follows: \begin{gather}\label{4-13} E(t)=\frac{1}{p}\|\Delta u\|_p^p-\frac{1}{q}\int_\Omega|u|^q\log(|u|)dx +\frac{1}{q^2}\|u\|_q^q, \\ \label{4-14} M(t)=\frac{1}{2}\int_\Omega u^2dx,~~H(t)=\int_0^tM(s)ds. \end{gather} Since the embedding $W^{2,p}(\Omega)\hookrightarrow L^2(\Omega)$ holds if $\max\{1,\frac{2n}{n+4}\}2$ and $E(0)\leq0$, then \begin{equation}\label{4-19} q\left(H'(t)-H'(0)\right)^2\leq 2H(t)H''(t). \end{equation} \end{lemma} \begin{proof} By H\"{o}lder's inequality and \eqref{4-17}, we obtain \begin{align*} H'(t)-H'(0) &=M(t)-M(0)\\ &=\int_0^tM'(s)ds=\int_0^t\int_\Omega uu_s\,dx\,ds\\ &\leq\Big(\int_0^t\int_\Omega |u|^2\,dx\,ds\Big)^{1/2} \Big(\int_0^t\int_\Omega |u_s|^2\,dx\,ds\Big)^{1/2}\\ &\leq\big(\frac{2}{q}\big)^{1/2}(H(t))^{1/2}(M'(t))^{1/2}\\ &=\big(\frac{2}{q}\big)^{1/2}(H(t))^{1/2}(H''(t))^{1/2}. \end{align*} Moreover, by \eqref{4-17} again, we have $H'(t)-H'(0)=\int_0^tM'(s)ds \geq q\int_0^t\int_0^s\int_\Omega |u_\tau|^2dxd\tau ds\geq0.$ Then the conclusion follows from the two inequalities above. The proof is complete. \end{proof} \begin{lemma}[{\cite[Lemma 1.2]{GB}}] \label{lem4.3} Suppose that $\theta>0$, $\alpha>0$, $\beta>0$ and $h(t)$ is a nonnegative and absolutely continuous function satisfying $h'(t)+\alpha h^\theta(t)\geq\beta$, then for $00,\quad \beta\varphi^{r-l}(0)<\alpha\,. \end{gathered} \end{equation} Then \begin{gather*} \varphi(t)\leq [-\alpha_0(1-l)t+\varphi^{1-l}(0)]^{\frac{1}{1-l}},\quad 00$ and $T_0=\alpha_0^{-1}(1-l)^{-1}\varphi^{1-l}(0)$. \end{lemma} \begin{theorem} \label{thm4.2} Assume that $p2$, $E(0)\leq0$ and $\|u_0\|_2>0$. Then the solution to problem \eqref{1-1} blows up in the finite time. \end{theorem} \begin{proof} We will give the proof by contradiction. Suppose that the solution $u(x,t)$ to the problem \eqref{1-1} exists for all $t>0$. Then by the definition of weak solution, we know that $u\in C([0,+\infty);L^2(\Omega))$. For any $t_0>0$, we claim that \begin{equation} \label{4-21} \int_0^{t_0}\int_\Omega|u_s|^2\,dx\,ds>0. \end{equation} Otherwise, there exists a $\hat{t}_0>0$ such that $\int_0^{\hat{t}_0}\int_\Omega|u_s|^2\,dx\,ds=0$, and hence $u_t(x,t)=0$ for a.e. $(x,t)\in \Omega\times(0,\hat{t}_0)$. Thus it follows from \eqref{4-18} that $\int_\Omega|\Delta u|^pdx=\int_\Omega|u|^q\log(|u|)dx$ for a.e.\ $t\in(0,\hat{t}_0)$, and then we obtain from \eqref{4-16} that $E(t)=\frac{q-p}{pq}\int_\Omega|\Delta u|^pdx+\frac{1}{q^2}\int_\Omega|u|^qdx$ for a.e. $t\in(0,\hat{t}_0)$, which combines $E(t)\leq E(0)\leq 0$ and $p0$. Then \eqref{4-21} holds. Now, fix $t_0>0$, and let $\rho=\int_0^{t_0}\int_\Omega|u_s|^2\,dx\,ds$. By \eqref{4-21} we know that $\rho$ is a positive constant. Integrating \eqref{4-17} over $(t_0,t)$, we obtain \begin{equation} \label{4-22} \begin{aligned} M(t)&\geq M(t_0)+q\int_{t_0}^t\int_0^s\int_\Omega|u_\tau|^2dxd\tau ds\\ &\geq\int_{t_0}^t\int_0^{t_0}\int_\Omega|u_\tau|^2dxd\tau ds\geq \rho(t-t_0). \end{aligned} \end{equation} Hence, \begin{equation}\label{4-23} \lim_{t\to +\infty}H'(t)=\lim_{t\to +\infty}M(t)=+\infty. \end{equation} Combining \eqref{4-23} and the fact that $q>2$, we have $\lim_{t\to +\infty}\frac{(H'(t))^2}{[H'(t)-H'(0)]^2}=1<\frac{4q}{3q+2}.$ Therefore, there exists $t^*>t_0$ such that \frac{3q+2}{4}(H'(t))^20, we obtain z(t) cannot converge to 0 as t\to +\infty. However, since \lim_{t\to +\infty}H(t)=+\infty, we obtain from the definition of z(t) and that z(t) is convergent to 0 as t\to +\infty, which is a contraction. The proof is complete. \end{proof} \begin{theorem} \label{thm4.3} Assume that p>q and E(0)<0. Then the solution to problem \eqref{1-1} does not go extinct in finite time. \end{theorem} \begin{proof} Recall the M(t) defined in \eqref{4-14}. According to \eqref{4-14} and \eqref{4-16}, we obtain \begin{equation} \label{4-24} \begin{aligned} &M'(t) \\ &=-\int_\Omega|\Delta u|^pdx+\int_\Omega|u|^q\log(|u|)dx\\ &=\int_\Omega|u|^q\log(|u|)dx-pE(t)-\frac{p}{q}\int_\Omega|u|^q\log(|u|)dx +\frac{1}{q^2}\int_\Omega|u|^qdx\\ &=\frac{q-p}{q}\int_\Omega|u|^q\log(|u|)dx-pE(0)+\frac{1}{q^2}\int_\Omega|u|^qdx +p\int_0^t\int_\Omega|u_s|^2\,dx\,ds\\ &\geq\frac{q-p}{q}\int_\Omega|u|^q\log(|u|)dx-pE(0). \end{aligned} \end{equation} \noindent\textbf{Case 1: p>q.} By p\leq2, we obtain q<2. Then there exists \mu>0 such that q+\mu<2, hence \begin{equation} \label{4-25} \begin{aligned} \frac{q-p}{q}\int_\Omega|u|^q\log(|u|)dx &\geq\frac{q-p}{e\mu q}\int_\Omega|u|^{q+\mu}dx\\ &\geq\frac{q-p}{e\mu q}\Big(\int_\Omega|u|^2dx\Big)^{\frac{{q+\mu}}{2}} |\Omega|^{\frac{2-q-\mu}{2}}\\ &=AM^{\frac{{q+\mu}}{2}}(t), \end{aligned} \end{equation} where A=\frac{q-p}{e\mu q}2^{\frac{{q+\mu}}{2}}|\Omega|^{\frac{2-q-\mu}{2}}>0. So, by \eqref{4-24} and \eqref{4-25}, we see that \[ M'(t)\geq-AM^{\frac{{q+\mu}}{2}}-pE(0). By Lemma \ref{lem4.3} and $E(0)<0$, we have $M(t)\geq\min\Big\{M(0),\Big(\frac{-pE(0)}{A}\Big)^{\frac{2}{q+\mu}}\Big\}, \quad t>0.$ Since $M(0)=\frac{1}{2}\|u_0\|_2^2>0$, $A>0$ and $E(0)<0$, we derive $M(t)>0$ for all $t>0$. \smallskip \noindent\textbf{Case 2: $p=q$.} Since $E(0)<0$, from \eqref{4-24} we obtain $M'(t)\geq-pE(0)>0$. Hence, we have $M(t)\geq M(0)-pE(0)>0,\quad t>0.$ Again as in Case 1, we obtain $M(t)>0$ for all $t>0$. The two cases above imply $\|u(\cdot,t)\|_2=\sqrt{2M(t)}>0$ for all $t>0$. Then for any $s>1$, by the interpolation inequality, we have $\|u\|_2\leq \|u\|_s^{1/2}\|u\|_{s'}^{1/2},$ where $s'=s/(s-1)>1$. Combining the above inequality with $\|u(\cdot,t)\|_2>0$, we know that $\forall s>1$, there does not exist $T^*>0$ such that $\lim_{t\to T^*}\|u\|_s=0$. The proof is complete. \end{proof} \begin{theorem} \label{thm4.4} Assume that $p0$ is sufficiently small such that $q+\mu<2$. \end{theorem} \begin{proof} Multiplying the first equation of \eqref{1-1} by $u$ and integrating over $\Omega$, we have $\frac{1}{2}\int_\Omega u^2dx+\int_\Omega|\Delta u|^pdx =\int_\Omega|u|^q\log(|u|)dx.$ Recall the $M(t)$ defined in \eqref{4-14}, then the above equation is equivalent to the inequality \begin{equation} \label{4-26} M'(t)+\int_\Omega|\Delta u|^pdx\leq\int_\Omega|u|^q\log(|u|)dx. \end{equation} Then \eqref{4-15}, \eqref{4-26} and H\"{o}lder's inequality imply $M'(t)+2^{p/2}B^{-p}M^{p/2}(t) \leq\frac{1}{e\mu}2^{\frac{q+\mu}{2}}|\Omega|^{\frac{2-q-\mu}{2}} M^{\frac{q+\mu}{2}}(t);$ that is, $M'(t)+\alpha M^{p/2}(t)\leq \beta M^{\frac{q+\mu}{2}}(t),$ where $\alpha=2^{p/2}B^{-p}>0$, $\beta=\frac{1}{e\mu}2^{\frac{q+\mu}{2}}|\Omega|^{\frac{2-q-\mu}{2}}>0$, and $0<\frac{p}{2}<\frac{q+\mu}{2}\leq1$. By Lemma \ref{lem4.4} and the assumption $0<\|u_0\|_2^{q+\mu-p}0$ and $T_*=\alpha_0^{-1}\frac{2}{2-p}M^{\frac{2-p}{2}}(0)$. Then the conclusion follows by $\|u(\cdot,t)\|_2=\sqrt{2M(t)}$. The proof is complete. \end{proof} \subsection*{Acknowledgements} This work is supported by the Jilin Scientific and Technological Development Program (number 20170101143JC). \begin{thebibliography}{99} \bibitem{BF} F. Bernis; \emph{Qualitative properties for some nonlinear higher order degenerate parabolic equations}, Houston J. Math. 14(3) (1988), 319-352. \bibitem{CL}Y. Cao, C. Liu; \emph{Initial boundary value problem for a mixed pseudo- parabolic p-Laplacian type equation with logarithmic nonlinearity}, Electronic Journal of Differential Equations, 2018 (116) (2018), 1-19. \bibitem{CLL} H. Chen, P. Luo, G. Liu; \emph{Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity}, J. Math. Anal. Appl., 422 (2015), 84-98. \bibitem{CT} H. Chen, S. Tian; \emph{Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity}, J. Differential Equations, 258 (2015), 4424-4442. \bibitem{GB} B. Guo, W. J. Gao; \emph{Non-extinction of the solutions to a fast diffusive $p$-Laplace equation with Neumann boundary conditions}, J. Math. Anal. Appl., 422 (2015), 1527-1531. \bibitem{HZ} A. J. Hao, J. Zhou; \emph{Blowup, extinction and non-extinction for a nonlocal $p$-biharmonic parabolic equation}, Appl. Math. Lett., 64 (2017), 198-204. \bibitem{HYJ} Y. J. He, H. H. Gao, H. Wang; \emph{Blowup and decay for a class of pseudo-parabolic $p$-Laplacian equation with logarithmic nonlinearity}, Comput. Math. Appl., 75 (2018), 459-469. \bibitem{JYC} S. Ji, J. Yin, Y. Cao; \emph{Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity}, J. Differential Equations, 261(2016), 5446-5464. \bibitem{L} C. Liu; \emph{A sixth order degenerate equation with the higher order p-Laplacian operator}, Mathematica Slovaca, 60(6)(2010), 847-864. \bibitem{LG} C. Liu, J. Guo; \emph{Weak solutions for a fourth order degenerate parabolic equation}, Bulletin of the Polish Academy of Sciences, Mathematics, 54(1) (2006), 27-39. \bibitem{MP} P. Martinez; \emph{A new method to obtain decay rate estimates for dissipative systems}, ESAIM Control Optim. Calc. Var., 4(1999), 419-444. \bibitem{NL} L. C. Nhan, L. X. Truong; \emph{Global solution and blow-up for a class of pseudo $p$-Laplacian evolution equations with logarithmic nonlinearity}, Comput. Math. Appl., 73(9) (2017), 2076-2091. \bibitem{SJ} J. Simon; \emph{Compact sets in the space $L^p(0,T;B)$}, Ann. Mat. Pura. Appl., 146(1987), 65-96. \end{thebibliography} \end{document}