Global existence and decay of solutions for p-biharmonic parabolic equation with logarithmic nonlinearity

1. Introduction

In this paper, we investigate the existence of global and decay of solutions for the p-biharmonic parabolic equation with logarithmic nonlinearity

where \(\Omega \) is bounded domain \( \mathbb{R} ^{n}\) with smooth boundary \(\partial \Omega ,\) \(p,\) \(q\) are positive constants, \(2< p< q< p\left( 1+\frac{4}{n}\right) ,\) and \(u_{0}\in \left( W_{0}^{1,p}\left( \Omega \right) \cap W^{2,p}\left( \Omega \right) \right) \backslash \left\{ 0\right\} .\) The term \(\Delta \left( \left\vert \Delta u\right\vert ^{p-2}\Delta u\right) \) is called a \(p\)-biharmonic operator.

Studies of logarithmic nonlinearity have a long history in physics as it occurs naturally in different areas of physics such as supersymmetric field theories, inflationary cosmology, nuclear physics, optics and quantum mechanics [1,2]. Peng and Zhou [3] studied the following heat equation with logarithmic nonlinearity

\begin{equation*} u_{t}-\Delta u_{t}=\left\vert u\right\vert ^{p-2}u\ln \left\vert u\right\vert . \end{equation*} They obtained the global existence and blow-up of solutions. Also, they discussed the upper bound of blow-up time under suitable conditions. Nhan and Truong [4] studied the following nonlinear pseudo-parabolic equation \begin{equation*} u_{t}-\Delta u_{t}-div\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) =\left\vert u\right\vert ^{p-2}u\ln \left\vert u\right\vert . \end{equation*} They obtained results as regard the existence or non-existence of global solutions. Also, He et al., [5] proved the decay and the finite time blow-up for weak solutions of the equation. Cao and Liu [6] studied the following nonlinear evolution equation with logarithmic source \begin{equation*} u_{t}-\Delta u_{t}-div\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) -k\Delta u_{t}=\left\vert u\right\vert ^{p-2}u\ln \left\vert u\right\vert . \end{equation*} They established the existence of global weak solutions. Moreover, they considered global boundedness and blowing-up at \(\infty \).

Wang and Liu [7] considered the following p-biharmonic parabolic equation with the logarithmic nonlinearity

\begin{equation*} u_{t}+\Delta \left( \left\vert \Delta u\right\vert ^{p-2}\Delta u\right) =\left\vert u\right\vert ^{q-2}u\ln \left\vert u\right\vert \end{equation*} They studied existence of weak solutions by potential well method, blow up at finite time by concative method.

Recently some authors studied the hyperbolic and parabolic equation with logarithmic source term (see [8,9,10,11,12,13,14,15,16,17,18,19,20]). This paper is organized as follows: In the §2, we introduce some lemma which will be needed later. In §3, under some conditions, we obtain the unique global weak solution of the problem (1). Meanwhile, we find that the solution is decay polynomially.

It is necessary to note that prence of the logarithmic nonlinearity causes some difficulties in deploying the potantial well method. In order to handle this situation we need the following logarithmic Sobolev inequality which was introduced by ([4,21,22]).

Proposition 1. Let \(u\) be any function in \(H^{1}\left( \mathbb{R} ^{n}\right) \) and \(\mu >0\) be any number. Then

where \begin{equation*} \mathcal{L} _{p}=\frac{p}{n}\left( \frac{p-1}{e}\right) ^{p-1}\pi ^{-\frac{p }{2}}\left[ \frac{\Gamma \left( \frac{\pi }{2}+1\right) }{\Gamma \left( n \frac{p-1}{p}+1\right) }\right] ^{\frac{p}{n}}. \end{equation*}

2. Preliminaries

For simplicity, we denote \begin{equation*} \text{ }\left\Vert u\right\Vert _{s}=\left\Vert u\right\Vert _{L^{s}(\Omega )},\text{ }\left\Vert u\right\Vert _{W_{0}^{2,p}\left( \Omega \right) }=\left\Vert u\right\Vert _{2,s}=\left( \left\Vert \Delta u\right\Vert _{s}^{s}+\left\Vert \nabla u\right\Vert _{s}^{s}+\left\Vert u\right\Vert _{s}^{s}\right) ^{\frac{1}{s}}, \end{equation*} for \(1< s< \infty \) (see [23,24], for details). We also use notation \(X_{0}\) to denote \(\left( W_{0}^{1,p}\left( \Omega \right) \cap W^{2,p}\left( \Omega \right) \right) \backslash \left\{ 0\right\} \) and \(W^{-2,p^{\prime }}\left( \Omega \right) \) to denote the dual space of \(W^{2,s}\left( \Omega \right) \), where \( s^{\prime }\) is Hölder conjugate functional of \(s>1.\)

Let us introduce the energy functional \(J\) and Nehari functional \(I\) defined on \(X_{0}\) as follow

and By (3) and (4), we get Let \begin{equation*} \mathcal{N}=\{u\in X_{0}:I(u)=0\}, \end{equation*} be the Nehari manifold. Thus, we may define \(d\) is positive and is obtained by some \(u\in \mathcal{N}.\) Then it is obvious that \begin{equation*} M=\frac{1}{p^{2}}\left( \frac{p^{2}e}{n\mathcal{L} _{p}}\right) ^{\frac{n}{p} }. \end{equation*} From [4], we know \(d\geq M.\)

The local existence of the weak solutions can be obtained via the standard parabolic theory. It is easy to obtain the following equality

Lemma 1. Let \(u\in X_{0}\). Then we possess

• (i) \(\lim_{\lambda \to 0^{+}}j(\lambda )=0\) and \(\lim_{\lambda \to +\infty }j(\lambda )=-\infty ;\)
• (ii) there is a unique \(\lambda ^{\ast }>0\) such that \(j^{\prime }(\lambda ^{\ast })=0;\)
• (iii) \(j(\lambda )\) is increasing on \((0,\lambda ^{\ast }),\) decreasing on \( (\lambda ^{\ast },+\infty )\) and attains the maximum at \(\lambda ^{\ast };\)
• (iv) \(I(\lambda u)>0\) for \(0< \lambda < \lambda ^{\ast },\) \(I(\lambda u)< 0\) for \(\lambda ^{\ast }< \lambda < +\infty \) and \(I(\lambda ^{\ast }u)=0.\)

Proof. For \(u\in X_{0},\) by the definition of \(j,\) we get

It is clear that (i) holds due to \(\int\nolimits_{\Omega }\left\vert u\right\vert ^{q}dx\neq 0.\) We have \begin{eqnarray*} \frac{d}{d\lambda }j(\lambda ) &=&\lambda ^{p-1}\left\Vert \Delta u\right\Vert _{p}^{p}-\lambda ^{q-1}\int\nolimits_{\Omega }\left\vert u\right\vert ^{q}\ln \left\vert u\right\vert dx-\lambda ^{q-1}\ln \lambda \left\Vert u\right\Vert _{q}^{q}, \\ &=&\lambda ^{p-1}\left( \left\Vert \Delta u\right\Vert _{p}^{p}-\lambda ^{q-p}\int\nolimits_{\Omega }\left\vert u\right\vert ^{q}\ln \left\vert u\right\vert dx-\lambda ^{q-p}\ln \lambda \left\Vert u\right\Vert _{q}^{q}\right) . \end{eqnarray*} Since \(\lambda >0,\) let \(\varphi \left( \lambda \right) =\lambda ^{1-p}j^{\prime }(\lambda ),\) through direct calculation, we get \begin{equation*} \varphi ^{\prime }(\lambda )=-\lambda ^{q-p-1}\left( \left( q-p\right) \int\nolimits_{\Omega }\left\vert u\right\vert ^{q}\ln \left\vert u\right\vert dx+\left( q-p\right) \ln \lambda \left\Vert u\right\Vert _{q}^{q}+\left\Vert u\right\Vert _{q}^{q}\right) . \end{equation*} Hence, there exists a \begin{equation*} \lambda ^{\ast }=\exp \left( \frac{\left( p-q\right) \int\nolimits_{\Omega }\left\vert u\right\vert ^{q}\ln \left\vert u\right\vert dx-\left\Vert u\right\Vert _{q}^{q}}{\left( q-p\right) \left\Vert u\right\Vert _{q}^{q}} \right) >0, \end{equation*} such that \(\varphi ^{\prime }(\lambda )>0\) on \((0,\lambda ^{\ast }),\) \( \varphi ^{\prime }(\lambda )< 0\) on \((\lambda ^{\ast },+\infty )\) and on \( \varphi ^{\prime }(\lambda )=0.\) So, \(\varphi (\lambda )\) is increasing on \( (0,\lambda ^{\ast }),\) decreasing on \((\lambda ^{\ast },+\infty ).\) Since \( \lim_{\lambda \to 0^{+}}\) \(\varphi (\lambda )=\left\Vert \nabla u\right\Vert ^{2}>0,\) \(\lim_{\lambda \to +\infty }\) \(\varphi (\lambda )=-\infty ,\) there exists a unique \(\lambda ^{\ast }>0\) such that \( \varphi (\lambda ^{\ast })=0,\) i.e., \(j^{\prime }(\lambda ^{\ast })=0.\) So (ii) holds. Then, \(j^{\prime }(\lambda )=\lambda \varphi (\lambda )\) is positive on \((0,\lambda ^{\ast }),\) negative on \((\lambda ^{\ast },+\infty ).\) Thus, \(j(\lambda )\) is increasing on \((0,\lambda ^{\ast }),\) decreasing on \((\lambda ^{\ast },+\infty )\) and attains the maximum at \( \lambda ^{\ast }.\) So (iii) holds. The last property, (iv) , is only a simple corallary of the fact that \begin{eqnarray*} I(\lambda u) &=&\left\Vert \Delta \left( \lambda u\right) \right\Vert _{p}^{p}-\int\nolimits_{\Omega }\left\vert \lambda u\right\vert ^{q}\ln \left\vert \lambda u\right\vert dx \\ &=&\lambda ^{p}\left\Vert \Delta u\right\Vert _{p}^{p}-\lambda ^{q}\int\nolimits_{\Omega }\left\vert u\right\vert ^{q}\ln \left\vert u\right\vert dx-\lambda ^{q}\ln \lambda \left\Vert u\right\Vert _{q}^{q} \\ &=&\lambda j^{\prime }(\lambda ). \end{eqnarray*} Thus, \(I(\lambda u)>0\) for \(0< \lambda < \lambda ^{\ast },\ I(\lambda u)< 0\) for \(\lambda ^{\ast }< \lambda < +\infty \) and \(I(\lambda ^{\ast }u)=0.\) So (iv) holds. The proof is complete.

Next we denote

\begin{equation*} R:=\left( \frac{p^{2}e}{n\mathcal{L} _{p}}\right) ^{n/p^{2}}. \end{equation*}

Lemma 2.

• (i) if \(I(u)>0\) then \(0< \left\Vert u\right\Vert _{p}< R,\)
• (ii) if \(I(u)< 0\) then \(\left\Vert u\right\Vert _{p}>R,\)
• (iii) if \(I(u)=0\) then \(\left\Vert u\right\Vert _{p}\geq R.\)

Proof. By the definition of \(I(u)\), we get \begin{eqnarray*} I(u) &=&\left\Vert \Delta u\right\Vert _{p}^{p}-\int\nolimits_{\Omega }\left\vert u\right\vert ^{q}\ln \left\vert u\right\vert dx \\ &\geq &\left\Vert \Delta u\right\Vert _{p}^{p}-\int\nolimits_{\Omega }\left\vert u\right\vert ^{p}\left( \ln \frac{\left\vert u\right\vert }{ \left\Vert u\right\Vert _{p}}+\ln \left\Vert u\right\Vert _{p}\right) dx \\ &\geq &\left( 1-\frac{\mu }{p}\right) \left\Vert \Delta u\right\Vert _{p}^{p}+\left( \frac{n}{p^{2}}\ln \left( \frac{p\mu e}{n\mathcal{L} _{p}} \right) -\ln \left\Vert u\right\Vert _{p}\right) \left\Vert u\right\Vert _{p}^{p}. \end{eqnarray*} Choosing \(\mu =p,\) we have \begin{equation*} I(u)\geq \left( \frac{n}{p^{2}}\ln \left( \frac{p^{2}e}{n\mathcal{L} _{p}} \right) -\ln \left\Vert u\right\Vert _{p}\right) \left\Vert u\right\Vert _{p}^{p}. \end{equation*} (i) if \(I(u)>0,\) then \begin{equation*} \ln \left\Vert u\right\Vert _{p}< \ln \left( \frac{p^{2}e}{n\mathcal{L} _{p}} \right) ^{\frac{n}{p^{2}}}, \end{equation*} that's mean \begin{equation*} \left\Vert u\right\Vert _{p}< \left( \frac{p^{2}e}{n\mathcal{L} _{p}}\right) ^{\frac{n}{p^{2}}}=R, \end{equation*} and (ii) if \(I(u)< 0,\) we obtain \begin{equation*} \left\Vert u\right\Vert _{p}>\left( \frac{p^{2}e}{n\mathcal{L} _{p}}\right) ^{\frac{n}{p^{2}}}=R, \end{equation*} property (iii) we can argue similarly the proof of (ii).

The proof of lemma is complete.

Lemma 3. [25] For any \(u\in W_{0}^{1,p}(\Omega )\), \(p\geq 1\), \(r\geq 1\) and \(p_{\ast } =\frac{np}{n-p}\), the inequality \begin{equation*} \left\Vert u\right\Vert _{q}\leq C\left\Vert \nabla u\right\Vert _{p}^{\theta }\left\Vert u\right\Vert _{r}^{1-\theta }, \end{equation*} is valid, where \begin{equation*} \theta =\left( \frac{1}{r}-\frac{1}{q}\right) \left( \frac{1}{n}-\frac{1}{p} + \frac{1}{r}\right) ^{-1}, \end{equation*} and for \(p\geq n=1,\) \(r\leq q\leq \infty ;\) for \(n>1\) and \(p< n,\) \(q\in \lbrack r,p_{\ast }]\) if \(r< p_{\ast }\) and \(q\in \lbrack p_{\ast },r]\) if \( r\geq p_{\ast }\) for \(p=n>1,\) \(r\leq q\leq \infty ;\) for \(p>n>1,\) \(r\leq q\leq \infty .\)

Here, the constant \(C\) depends on \(n,p,q\) and \(r.\)

Lemma 4. [26] Let \(f:R^{+}\to R^{+}\) be a nonincreasing function and \(\sigma \) is a nonnegative constant such that \begin{equation*} \int\nolimits_{t}^{+\infty }f^{1+\sigma }(s)ds\leq \frac{1}{\omega } f^{\sigma }(0)f(t),\text{ }\forall t\geq 0. \end{equation*} Hence

• (a) \(f(t)\leq f(0)e^{1-\omega t},\) for all \(t\geq 0,\) whenever \(\sigma =0,\)
• (b) \(f(t)\leq f(0)\left( \frac{1+\sigma }{1+\omega \sigma t}\right) ^{\frac{1 }{\sigma }},\) for all \(t\geq 0,\) whenever \(\sigma >0.\)

3. Main results

Now as in ([4]), we introduce the follows sets: \begin{eqnarray*} \mathcal{W}_{1} &=&\{u\in X_{0}:J(u)< d\},\text{ }\mathcal{W}_{2}=\{u\in X_{0}:J(u)=d\},\text{ }\mathcal{W}=\mathcal{W}_{1}\cup \mathcal{W}_{2}, \\ \mathcal{W}_{1}^{+} &=&\{u\in \mathcal{W}_{1}:I(u)>0\},\text{ }\mathcal{W} _{2}^{+}=\{u\in \mathcal{W}_{2}:I(u)>0\},\text{ }\mathcal{W}^{+}=\mathcal{W }_{1}^{+}\cup \mathcal{W}_{2}^{+}, \\ \mathcal{W}_{1}^{-} &=&\{u\in \mathcal{W}_{1}:I(u)< 0\},\text{ }\mathcal{W} _{2}^{-}=\{u\in \mathcal{W}_{2}:I(u)< 0\},\text{ }\mathcal{W}^{-}=\mathcal{W }_{1}^{-}\cup \mathcal{W}_{2}^{-}. \end{eqnarray*}

Definition 1. (Maximal Existence Time). Assume that \(u\) be weak solutions of problem (1). We define the maximal existence time \(T_{\max }\) as follows \begin{equation*} T_{\max }=\sup \{T>0:u(t)\text{ exists on }[0,T]\}. \end{equation*} Then

• (i) If \(T_{\max }< \infty ,\) we say that \(u\) blows up in finite time and \( T_{\max }\) is the blow-up time;
• (ii) If \(T_{\max }=\infty ,\) we say that \(u\) is global.

Definition 2. (Weak solution). We define a function \(u\in L^{\infty }(0,T;X_{0})\) with \( u_{t}\in L^{2}(0,T;H_{0}^{1}(\Omega ))\) to be a weak solution of problem (1) over \([0,T],\) if it satisfies the initial condition \( u(0)=u_{0}\in X_{0},\) and \begin{equation*} \left\langle u_{t},w\right\rangle +\left\langle \left\vert \Delta u\right\vert ^{p-1},\Delta w\right\rangle +\left\langle \nabla u_{t},\nabla w\right\rangle=\int\nolimits_{\Omega }u\left\vert u\right\vert ^{q-2}\ln \left( \left\vert u\right\vert \right) wdx,\text{ } \end{equation*} for all \(w\in X_{0},\) and for a.e. \(t\in \lbrack 0,T].\)

Theorem 1. (Global Existence). Let \(u_{0}\in \) \(\mathcal{W}^{+},\) \(0< J(u_{0})< M\) and \( I(u)>0.\) Then there is a unique global weak solution \(u\) of (1) satisfying \(u(0)=u_{0}.\) We have \(u(t)\in \mathcal{W}^{+}\)holds for all \( 0\leq t< +\infty ,\) and the energy estimate \begin{equation*} \int\nolimits_{0}^{t}\left\Vert u_{s}(s)\right\Vert _{H_{0}^{1}(\Omega )}^{2}ds+J(u(t))\leq J(u_{0}),\text{ }0\leq t\leq +\infty . \end{equation*} Also, the solution decay polynomially provided \(u_{0}\in \mathcal{W} _{1}^{+}. \)

Proof. The Faedo-Galerkin's methods is used. In the space \(W_{0}^{1,p}\left( \Omega \right) \cap W^{2,p}\left( \Omega \right) ,\) we take a bases \( \{w_{j}\}_{j=1}^{\infty }\) and define the finite orthogonal space \begin{equation*} V_{m}=span\{w_{1},w_{2},...,w_{m}\}. \end{equation*} Let \(u_{0m}\) be an element of \(V_{m}\) such that

as \(m\to \infty .\) We construct the following approximate solution \( u_{m}(x,t)\) of the problem (1) where the coefficients \(a_{mj}\) \((1\leq j\leq m)\) satisfy the ordinary differential equations for \(i\in \{1,2,...,m\},\) with the initial condition We multiply both sides of (11) by \(a_{mi}^{\prime },\) sum for \( i=1,...,m\) and integrating with respect to time variable on \([0,t],\) we get where \(T_{\max }\) is the maximal existence time of solution \(u_{m}(t).\) We shall prove that \(T_{\max }=+\infty .\) From (9), (13) and the continuity of \(J\), we obtain Thanks to \(J(u_{0})< d\) and the continuity of functional \(J,\) it follows from (14) that \begin{equation*} J(u_{0m})< d,\text{ for sufficiently large }m. \end{equation*} And therefore, from (13), we obtain for sufficiently large \(m.\) Next, we will study for sufficiently large \(m.\) We assume that (16) does not process and think that there exists a sufficiently small time \(t_{0}\) such that \( u_{m}(t_{0})\notin \mathcal{W}_{1}^{+}.\) Then, by continuity of \( u_{m}(t_{0})\in \partial \mathcal{W}_{1}^{+}.\) So, we get and Nevertheless, by definition of \(d,\) we see that (17) could not consist by (15) while if (18) holds then, we get \begin{equation*} J(u_{m}(t_{0}))\geq \underset{u\in \mathcal{N}}{\inf }J(u)=d, \end{equation*} which also contradicts with (15). Moreover, we have (16), i.e., \(J(u_{m}(t))< d,\) and \(I(u_{m}(t))>0,\) for any \(t\in \lbrack 0,T_{\max }),\) for sufficiently large \(m.\) Then, from (5), we obtain \begin{eqnarray*} d &>&J(u_{m}(t)) \\ &=&\frac{1}{q}I(u_{m})+\left( \frac{1}{p}-\frac{1}{q}\right) \left\Vert \Delta u_{m}\right\Vert _{p}^{p}+\frac{1}{q^{2}}\left\Vert u_{m}\right\Vert _{q}^{q} \\ &\geq &\left( \frac{1}{p}-\frac{1}{q}\right) \left\Vert \Delta u_{m}\right\Vert _{p}^{p}+\frac{1}{q^{2}}\left\Vert u_{m}\right\Vert _{q}^{q}, \end{eqnarray*} which gives and Since \(u_{m}(x,t)\in \mathcal{W}_{1}^{+}\) for \(m\) large enough, it follows from (5) that \(J(u_{m})\geq 0\) for \(s\) large enough. So, by (15 ) it follows for \(m\) large enough By (20), we know that \begin{equation*} T_{\max }=+\infty . \end{equation*} It follows from (19) and (21) that there exist a function \( X_{0}\) and a subsequence of \(\{u_{m}\}_{j=1}^{\infty }\) is indicated by \( \{u_{m}\}_{j=1}^{\infty }\) such that \begin{equation*} \left\vert \Delta u\right\vert ^{p-2}\Delta u\to \chi \text{ weakly in }L^{\infty }\left( 0,\infty ;W^{-2,p^{\prime }}\left( \Omega \right) \right). \end{equation*} By (22), (23) and Aubin-Lions compactness theorem, we obtain \begin{equation*} u_{m}\to u\text{ strongly in }C([0,+\infty ];L^{2}(\Omega )). \end{equation*} This yields that Moreover, since \begin{equation*} \alpha ^{r-1}\ln \alpha =-(e\left( r-1\right) )^{-1}\text{ for }\alpha >1, \end{equation*} and \begin{equation*} \ln \alpha =2\ln \left( \alpha ^{\frac{1}{2}}\right) \leq 2\alpha ^{\frac{1}{ 2}}\text{ for }\alpha >0. \end{equation*} By (19), we have where \begin{equation*} \Omega _{1}=\{x\in \Omega :\left\vert u_{m}(t)\right\vert \leq 1\},\text{ and }\Omega _{2}=\{x\in \Omega :\left\vert u_{m}(t)\right\vert \geq 1\}. \end{equation*} Hence, it follows from (24) and (25) that \begin{equation*} u_{m}\left\vert u_{m}\right\vert ^{q-2}\ln \left\vert u_{m}\right\vert \to u\left\vert u\right\vert ^{q-2}\ln \left\vert u\right\vert \text{ weakly* in }L^{\infty }(0,+\infty ;L^{\frac{2q}{2q-1}}(\Omega ))\text{ }. \end{equation*} Then integrating (11) respect to \(t\) for \(0\leq t< \infty ,\) we obtain \begin{equation*} \left\langle u_{t},w\right\rangle +\left\langle \chi (t),\Delta w\right\rangle +\left\langle \nabla u_{t},\nabla w\right\rangle =\int\nolimits_{\Omega }u\left\vert u\right\vert ^{p-2}\ln \left( \left\vert u\right\vert \right) wdx, \end{equation*} for all \(w\in W_{0}^{2,p}\left( \Omega \right) \) and for almost every \(t\in \left[ 0,\infty \right] .\) Finally, well known arguments of the theory of monotone operators implied \begin{equation*} \chi =\left\vert \Delta u\right\vert ^{p-2}\Delta u, \end{equation*} which yields \begin{equation*} \left\langle u_{t},w\right\rangle +\left\langle \left\vert \Delta u\right\vert ^{p-1},\Delta w\right\rangle +\left\langle \nabla u_{t},\nabla w\right\rangle =\int\nolimits_{\Omega }u\left\vert u\right\vert ^{p-2}\ln \left\vert u\right\vert wdx. \end{equation*} for all \(w\in W_{0}^{2,p}\left( \Omega \right) \) and for a.e. \(t\in \left[ 0,\infty \right] .\)

Finally, we discuss the decay results.

Thanks to \(u(t)\in \mathcal{W}_{1}^{+},\) we deduce from (13) that

\begin{equation*} \left( \frac{1}{p}-\frac{1}{q}\right) \left\Vert \Delta u\right\Vert _{p}^{p}+\frac{1}{q^{2}}\left\Vert u\right\Vert _{q}^{q}\leq J(u(t))\leq J(u_{0}),\text{ }t\in \lbrack 0,T]. \end{equation*} By using (5) and Proposition 1, we put \(p\left( \frac{J(u_{0})}{M} \right) ^{\frac{p}{n}}< \mu < p,\) we know \begin{eqnarray*} I(u(t)) &\geq &\left( 1-\frac{\mu }{p}\right) \left\Vert \Delta u\right\Vert _{p}^{p}+\left( \frac{n}{p^{2}}\ln \left( \frac{p\mu e}{n\mathcal{L} _{p}} \right) -\ln \left\Vert u(t)\right\Vert _{p}\right) \left\Vert u(t)\right\Vert _{p}^{p} \\ &\geq &\left( 1-\frac{\mu }{p}\right) \left\Vert \Delta u\right\Vert _{p}^{p}+\frac{1}{p}\ln \left( \frac{M}{J(u_{0})}\left( \frac{\mu }{p} \right) ^{\frac{n}{p}}\right) \left\Vert u(t)\right\Vert _{p}^{p} \\ &=&C_{1}\left\Vert u(t)\right\Vert _{W^{2,p}\left( \Omega \right) }^{p}. \end{eqnarray*} Integrating the \(I(u(s))\) with respect to \(s\) over \((t,T)\), we obtain where \(C_{2}\) stand by the best constant in the embedding \(W^{2,p}\left( \Omega \right) \hookrightarrow \to H_{0}^{1}\left( \Omega \right) \) From (26), we have Let \(T\to +\infty \) in (27), we can get \begin{equation*} \int\nolimits_{t}^{\infty }\left\Vert u(t)\right\Vert _{W^{2,p}\left( \Omega \right) }^{p}ds\leq \frac{1}{\omega }\left\Vert u(t)\right\Vert _{W^{2,p}\left( \Omega \right) }^{2}. \end{equation*} From Lemma 5, we have \(f(t)=\left\Vert u(t)\right\Vert _{W^{2,p}\left( \Omega \right) }^{2},\) \(\sigma =\frac{p}{2}-1,\) \(f(0)=1\) \begin{equation*} \left\Vert u(t)\right\Vert _{W^{2,p}\left( \Omega \right) }\leq \left\Vert u_{0}\right\Vert _{W^{2,p}\left( \Omega \right) }\left( \frac{p}{2+\omega \left\Vert u_{0}\right\Vert _{W_{0}^{2,p}\left( \Omega \right) }^{p-2}\left( p-2\right) t}\right) ^{\frac{1}{p-2}},\text{ }t\geq 0. \end{equation*} The above inequality implies that the solution \(u\) decays polynomially.

Acknowledgments :

The author would like to thank Prof. Charles N. Moore of Washington State University, USA for his valuable suggestions on this article.

Author Contributions:

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

''The authors declares no conflict of interest.''

Data Availability:

All data required for this research is included within this paper.

Funding Information:

No funding is available for this research.