Qualitative analysis of solutions for a parabolic type Kirchhoff equation with logarithmic nonlinearity

1. Introduction

In this work, we investigate the existence of global, decay and finite time blow up of solutions for the parabolic type Kirchhoff equation with logarithmic source term

where \(\Omega \) is a bound domain in \( \mathbb{R}^{n}(n\geq 1)\) with smooth boundary \(\partial \Omega \). Also, \( M(s)=1+s^{\gamma },\) \(\left( \gamma >0\right) ,\) \(\oint\nolimits_{\Omega }u_{0}dx=\frac{1}{\left\vert \Omega \right\vert }\int\nolimits_{\Omega }u_{0}dx=0\) and \[ \begin{cases} 2\gamma +2\leq k\leq +\infty ,& n=1,2, \\ 2\gamma +2\leq k\leq \frac{2n}{n-2},& n\geq 3. \end{cases} \] Many other authors studied the problem (1) in a more general form \[ \begin{cases} u_{t}-\Delta u=f(u)-\oint\nolimits_{\Omega }f(u)dx,& x\in \Omega,\ \ t>0, \\ \frac{\partial u(x,t)}{\partial \eta }=0,& x\in \partial \Omega ,\ \ t>0, \\ u(x,0)=u_{0}(x),& x\in \Omega , \end{cases} \] where \(\Omega \) is a bound domain in \(\mathbb{R}^{n}\) \((n\geq 1)\) with \(\left\vert \Omega \right\vert \) denoting its Lebesgue measure, the function \(f(u)\) is usually taken to be a power of \(u\), and \(n\) is the outer normal vector of \(\partial \Omega \). Studies of logarithmic nonlinearity have a long history in physics as it occurs naturally in different areas of physics such as supersymmetric field theories, optics, quantum mechanics and inflationary cosmology [1,2]. When \(M\equiv 1\) and \(k=2,\) (1) become semilinear pseudo-parabolic equation as follow Chen and Tian [3] obtained the global existence, behavior of vacuum isolation and blow-up of solutions at \(+\infty \) of the Equation (2). Without \(\Delta u,\) (2) become the following semilinear parabolic equation Chen et al., [4] studied the global existence, decay and blow-up at \( +\infty \) of solutions of the Equation (3). Yan and Yang [5] studied nonlocal parabolic equation with logarithmic nonlinearity \[ u_{t}-\Delta u_{t}=u\log \left\vert u\right\vert -\oint\nolimits_{\Omega }u\log \left\vert u\right\vert dx. \] Recently, they obtained the results under appropriate conditions on blow-up and existion of the solutions. Toualbia et al., [6] studied the initial boundary value problem of a nonlocal parabolic equation with logarithmic nonlinearity \[ u_{t}-div(\left\vert \nabla u\right\vert ^{p-2}\nabla u)=\left\vert u\right\vert ^{p-2}u\log \left\vert u\right\vert -\oint\nolimits_{\Omega }\left\vert u\right\vert ^{p-2}u\log \left\vert u\right\vert dx. \] They obtained the decay, blow up and nonextinction of solutions under appropriate condition. Also, recently some authors studied the parabolic and hyperbolic type equation with logarithmic source term (see [7,8,9,10,11,12,13,14,15]).

The organization of the remaining part of this paper is as follows: In the next Section 2, we introduce some lemmas which will be needed later. In Section 3, under some conditions, we get the unique global weak solution of the problem (1). Moreover, we find that the decay of solutions. In the lastly, we get the blow up theorem.

2. Preliminaries

Throughout this work, we adopt the following abbreviations \[ \text{ }\left\Vert u\right\Vert _{s}=\left\Vert u\right\Vert _{L^{s}(\Omega )},\text{ }\left\Vert u\right\Vert _{H_{0}^{1}(\Omega )}=\left( \left\Vert \nabla u\right\Vert ^{2}+\left\Vert u\right\Vert ^{2}\right) ^{\frac{1}{2}}, \] for \(1< s< \infty .\)

The energy functional \(J\) and Nehari functional \(I\) defined on \( H_{0}^{1}(\Omega )\backslash \{0\}\) as follow

and By (4) and (5), we obtain Let \[ \mathcal{N}=\{u\in H_{0}^{1}(\Omega )\backslash \{0\}:I(u)=0\}, \] be the Nehari manifold. Thus, we may define \(d\) is positive and is obtained by some \(u\in \mathcal{N}.\)

Lemma 1. Let \(u\in H_{0}^{1}(\Omega )\backslash \{0\},\) we consider the function \( j:\lambda \to J(\lambda u)\) for \(\lambda >0\). Then we possess

• (i)   \(\lim_{\lambda \to 0^{+}}j(\lambda )=0\) and \(\lim_{\lambda \to +\infty }j(\lambda )=-\infty ;\)
• (ii)   there is a unique \(\lambda ^{\ast }>0\) such that \(j^{\prime }(\lambda ^{\ast })=0;\)
• (iii)   \(j(\lambda )\) is strictly increasing on \((0,\lambda ^{\ast }),\) strictly decreasing on \((\lambda ^{\ast },+\infty )\) and takes the maximum at \(\lambda ^{\ast };\) \(I(\lambda u)=\lambda j^{\prime }(\lambda )\) and \[ I(\lambda u)\left\{ \begin{array}{l} >0,\text{ }0< \lambda < \lambda ^{\ast }, \\ =0,\text{ }\lambda =\lambda ^{\ast }, \\ < 0,\text{ }\lambda ^{\ast }< \lambda < +\infty . \end{array} \right. \]

Proof. For \(u\in H_{0}^{1}(\Omega )\backslash \{0\},\) by the definition of \(j,\) we get

It is clear that (i) holds due to \(\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx\neq 0.\) We get \begin{align*} \frac{d}{d\lambda }j(\lambda ) =&\lambda \left\Vert \nabla u\right\Vert ^{2}+\lambda ^{2\gamma +1}\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right) }-\lambda ^{k-1}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert u\right\vert dx \\ &-\lambda ^{k-1}\ln \lambda \int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx-\frac{\lambda ^{k-1}}{k}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx+\frac{\lambda ^{k-1}}{k}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx \\ =&\lambda \left\Vert \nabla u\right\Vert ^{2}+\lambda ^{2\gamma +1}\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right) }-\lambda ^{k-1}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert u\right\vert dx-\lambda ^{k-1}\ln \lambda \int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx \\ =&\lambda \left( \left\Vert \nabla u\right\Vert ^{2}+\lambda ^{2\gamma }\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right) }-\lambda ^{k-2}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert u\right\vert dx-\lambda ^{k-2}\ln \lambda \int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx\right) . \end{align*} Let \begin{equation*} \varphi \left( \lambda \right) :=\lambda ^{2\gamma }\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right) }-\lambda ^{k-2}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert u\right\vert dx-\lambda ^{k-2}\ln \lambda \int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx. \end{equation*} Then from \(2\gamma +2\leq k\) we can conclude that \(\lim_{\lambda \to \infty }\varphi \left( \lambda \right) =-\infty ,\) \(\varphi \left( \lambda \right) \) is monotone decreasing when \(\lambda >\lambda ^{\ast }\) and there exists a unique \(\lambda ^{\ast }\) such that \(\varphi \left( \lambda \right) \mid _{\lambda =\lambda ^{\ast }=0}\). Hence, we obtain there is a \(\lambda ^{\ast }>0\) such that \(\left\Vert \nabla u\right\Vert ^{2}+\varphi \left( \lambda \right) =0\), which means \(\frac{d}{ d\lambda }J\left( \lambda u\right) \mid _{\lambda =\lambda ^{\ast }=0}\). The conclusion (iii) directly follows from the proof of the conclusion (ii) and \begin{align*} I(\lambda u) =&\left\Vert \nabla (\lambda u)\right\Vert ^{2}+\left\Vert \nabla (\lambda u)\right\Vert ^{2\left( \gamma +1\right) }-\int\nolimits_{\Omega }\left\vert \lambda u\right\vert ^{k}\ln \left\vert \lambda u\right\vert dx \\ =&\lambda ^{2}\left\Vert \nabla u\right\Vert ^{2}+\lambda ^{2\left( \gamma +1\right) }\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right) }-\lambda ^{k}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert u\right\vert dx-\lambda ^{k}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \lambda dx \\ =&\lambda ^{2}\left\Vert \nabla u\right\Vert ^{2}+\lambda ^{2\left( \gamma +1\right) }\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right) }-\lambda ^{k}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert u\right\vert dx-\lambda ^{k}\ln \lambda \int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx \\ =&\lambda \left( \lambda \left\Vert \nabla u\right\Vert ^{2}+\lambda ^{2\gamma +1}\left\Vert \nabla u\right\Vert ^{2\left( \gamma +1\right) }-\lambda ^{k-1}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}\ln \left\vert u\right\vert dx-\lambda ^{k-1}\ln \lambda \int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx\right) \\ =&\lambda j^{\prime }(\lambda ). \end{align*} Thus, \(I(\lambda u)>0\) for \(0< \lambda < \lambda ^{\ast },\ I(\lambda u)< 0\) for \(\lambda ^{\ast }< \lambda < +\infty \) and \(I(\lambda ^{\ast }u)=0.\) The proof is complete.

Lemma 2. [16]

• a)   For any function \(u\in W_{0}^{1,p}(\Omega )\) such that \( \left\Vert u\right\Vert _{q}\leq B_{q,p}\left\Vert \nabla u\right\Vert _{p,} \) for all \(1\leq q\leq p^{\ast }\) where \(p^{\ast }=\frac{np}{n-p}\) if \(n>p\) and \(p^{\ast }=\infty \) if \(n\leq p.\) The best constant \(B_{q,p}\) depends only on \(\Omega ,\) \(n,\) \(p\) and \(q.\) We will denote the constant \(B_{p,p}\) by \(B_{p}.\)
• b)   For any \(u\in W_{0}^{1,p}(\Omega ),\) \(p\geq 1,\) and \(r\geq 1,\) the inequality \[ \left\Vert u\right\Vert _{q}\leq C\left\Vert \nabla u\right\Vert _{p}^{\theta }\left\Vert u\right\Vert _{r}^{1-\theta }, \] is valid, where \( \theta =\left( \frac{1}{r}-\frac{1}{q}\right) \left( \frac{1}{n}-\frac{1}{p} + \frac{1}{r}\right) ^{-1} \) and for \(p\geq n=1,\) \(r\leq q\leq \infty ;\) for \(n>1\) and \(p< n,\) \(q\in \lbrack r,p^{\ast }]\) if \(r< p^{\ast }\) 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 3. [17] Assume that \(f:R^{+} \to R^{+}\) be a nonincreasing function and \(\sigma \) is a nonnegative constant such that \[ \int\nolimits_{t}^{+\infty }f^{1+\sigma }(s)ds\leq \frac{1}{\omega } f^{\sigma }(0)f(t), \  \  \forall t\geq 0. \] Hence

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

3. Main results

Now as in [18], we introduce the follows sets: \begin{align*} \mathcal{W}_{1} &=\{u\in H_{0}^{1}(\Omega )\backslash \{0\}:J(u)< d\},\\ \mathcal{W}_{2}&=\{u\in H_{0}^{1}(\Omega )\backslash \{0\}:J(u)=d\},\\ \mathcal{W}&=\mathcal{W}_{1}\cup \mathcal{W}_{2}, \\ \mathcal{W}_{1}^{+} &=\{u\in \mathcal{W}_{1}:I(u)>0\},\\ \mathcal{W} _{2}^{+}&=\{u\in \mathcal{W}_{2}:I(u)>0\},\\ \mathcal{W}^{+}&=\mathcal{ W }_{1}^{+}\cup \mathcal{W}_{2}^{+}, \\ \mathcal{W}_{1}^{-} &=\{u\in \mathcal{W}_{1}:I(u)< 0\},\\ \mathcal{W} _{2}^{-}&=\{u\in \mathcal{W}_{2}:I(u)< 0\},\\ \mathcal{W}^{-}&=\mathcal{ W }_{1}^{-}\cup \mathcal{W}_{2}^{-}. \end{align*}

Definition 4. (Maximal existence time). Assume that \(u\) be weak solutions of problem (1). We define the maximal existence time \(T_{\max }\) as follows \[ T_{\max }=\sup \{T>0:u(t)\text{ exists on }[0,T]\}. \] Then

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

Definition 5. (Weak solution). We define a function \(u\in L^{\infty }(0,T;H_{0}^{1}(\Omega ))\) with \(u_{t}\in L^{2}(0,T;H_{0}^{1}(\Omega ))\) to be a weak solution of problem (1) over \([0,T],\) if it satisfies the initial condition \( u(0)=u_{0}\in H_{0}^{1}(\Omega )\backslash \{0\},\) and \begin{align*} & < u_{t},w>+< \nabla u,\nabla w> +< \left\Vert \nabla u\right\Vert ^{2\gamma }\nabla u,\nabla w>+< \nabla u_{t},\nabla w> \\ & =\int\nolimits_{\Omega }u^{k-1}\ln \left\vert u\right\vert wdx-\int\nolimits_{\Omega }\oint\nolimits_{\Omega }u^{k-1}\ln \left\vert u\right\vert wdsdx,\text{ } \end{align*} for all \(w\in H_{0}^{1}(\Omega ),\) and for a.e. \(t\in \lbrack 0,T].\)

Theorem 6. (Global existence). Let \(u_{0}\in \) \(\mathcal{W}^{+},\) \(0< J(u_{0})< d\) and \( I(u)>0.\) Hence, there is a unique global weak solution \(u\) of (1) satisfying \(u(0)=u_{0}.\) We obtain \(u(t)\in \mathcal{W}^{+}\) holds for all \( 0\leq t< +\infty ,\) and the energy estimate \[ \int\nolimits_{0}^{t}\left\Vert u_{s}(s)\right\Vert _{H_{0}^{1}(\Omega )}^{2}ds+J(u(t))\leq J(u_{0}),\text{ }0\leq t\leq +\infty . \] Also, the solution decay exponentially provided \(u_{0}\in \mathcal{W} _{1}^{+}.\)

Proof. (Global existence) In the space \(H_{0}^{1}(\Omega ),\) we take a Galerkin bases \(\{w_{j}\}_{j=1}^{\infty }\) and define the finite dimensional space

\begin{equation*} V_{m}=span\{w_{1},w_{2},...,w_{m}\}. \end{equation*} Let \(u_{0m}\) be an element of \(V_{m}\) such that as \(m\to \infty .\) We can find the approximate solution \(u_{m}(x,t)\) of the problem (1) in the form 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 Now, multiplying (11) by \(a_{mi}^{\prime },\) summing over \(i\) from \(1\) to \(m\) and integrating with related to time variable on \([0,t],\) we obtain where \(T_{\max }\) is the maximal existence time of solution \(u_{m}(t).\) By (9), (13) and the continuity of \(J\) that with \(J(u_{0})< d\) and the continuity of functional \(J,\) by (14), we have \begin{equation*} J(u_{0m})< d,\text{ for sufficiently large }m. \end{equation*} And therefore, from (13), we obtain for sufficiently large \(m.\) Next, we will show that for sufficiently large \(m.\) We suppose that (16) does not hold and think that there exists a smallest time \(t_{0}\) such that \( u_{m}(t_{0})\notin \mathcal{W}_{1}^{+}.\) Then, by continuity of \( u_{m}(t_{0})\in \partial \mathcal{W}_{1}^{+}.\) So, we obtain and It is clear that (17) could not occur by (15) while if (18) holds then, by definition of \(d,\) we get \begin{equation*} J(u_{m}(t_{0}))\geq \underset{u\in \mathcal{N}}{\inf }J(u)=d, \end{equation*} which contradicts with (15). Thus, we get (16), i.e., \( J(u_{m}(t))< d\) and \(I(u_{m}(t))>0,\) for any \(t\in \lbrack 0,T_{\max }),\) for sufficiently large \(m.\) Hence, by (6), we get \begin{align*} d &>J(u_{m}(t)) \\ &=\frac{1}{k}I(u_{m}(t))+\frac{k-2}{2k}\left\Vert \nabla u_{m}(t)\right\Vert ^{2}+\frac{k-2\gamma -2}{2k(\gamma +1)}\left\Vert \nabla u_{m}(t)\right\Vert ^{2\left( \gamma +1\right) }+\frac{1}{k^{2}} \int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx \\ &\geq \frac{k-2}{2k}\left\Vert \nabla u_{m}(t)\right\Vert ^{2}+\frac{\gamma }{2(\gamma +1)}\left\Vert \nabla u_{m}(t)\right\Vert ^{2\left( \gamma +1\right) }+\frac{1}{k^{2}}\int\nolimits_{\Omega }\left\vert u\right\vert ^{k}dx \geq \frac{k-2}{2k}\left\Vert \nabla u_{m}(t)\right\Vert ^{2}. \end{align*} And therefore, we deduce from (15) that \begin{equation*} \int\nolimits_{0}^{t}\left\Vert u_{ms}(s)\right\Vert _{H_{0}^{1}(\Omega )}^{2}ds+\frac{k-2}{2k}\left\Vert \nabla u_{m}(t)\right\Vert ^{2}< d. \end{equation*} This inequality implies \(T_{\max }=+\infty .\) Further, by the logarithmic inequality, we get \begin{align*} \left\Vert \nabla u_{m}(t)\right\Vert ^{2} =&2J(u_{m}(t))+\frac{2}{k} \int\nolimits_{\Omega }\left\vert u_{m}(t)\right\vert ^{k}\ln \left\vert u_{m}(t)\right\vert dx -\frac{2}{k^{2}}\left\Vert u_{m}(t)\right\Vert ^{2}-\frac{1}{\left( \gamma +1\right) }\left\Vert \nabla u_{m}(t)\right\Vert ^{2\left( \gamma +1\right) } \\ \leq &2J(u_{m}(0))+\frac{2}{k}\int\nolimits_{\Omega }\left\vert u_{m}(t)\right\vert ^{k}\ln \left\vert u_{m}(t)\right\vert dx. \end{align*} This implies that \begin{equation*} \left\Vert \nabla u_{m}(t)\right\Vert ^{2}\leq 2J(u_{m}(0))+\frac{2}{k} \int\nolimits_{\Omega }\left\vert u_{m}(t)\right\vert ^{k}\ln \left\vert u_{m}(t)\right\vert dx. \end{equation*} We deduce that \begin{equation*} \left\Vert \nabla u_{m}(t)\right\Vert ^{2}\leq C_{d},\text{ }\forall t\in \lbrack 0,T_{\max }). \end{equation*} (Decay estimates) Thanks to \(u_{0}\in \mathcal{W}_{1}^{+}\), we deduce from (6) that By \(I(u(t))>0,\) (7) and Lemma 1, there exists a \(\lambda ^{\ast }>1\) such that \(I(\lambda ^{\ast }u(t))=0.\) We have Using (19) and (20), we have \( d\leq (\lambda ^{\ast })^{k}J(u_{0}), \) which implies that By (5), we get Using (21) and (22), we have \begin{align*} (\lambda ^{\ast })^{k}I(u(t)) =&\left[ (\lambda ^{\ast })^{k}-(\lambda ^{\ast })^{2}\right] \left\Vert \nabla u(t)\right\Vert ^{2}+\left[ (\lambda ^{\ast })^{k}-(\lambda ^{\ast })^{2\left( \gamma +1\right) }\right] \left\Vert \nabla u(t)\right\Vert ^{2\left( \gamma +1\right) } +(\lambda ^{\ast })^{k}\ln (\lambda ^{\ast })\int\nolimits_{\Omega }\left\vert u\left( t\right) \right\vert ^{k}dx \\ \geq &\left[ (\lambda ^{\ast })^{k}-(\lambda ^{\ast })^{2}\right] \int\nolimits_{\Omega }\left\vert u\left( t\right) \right\vert ^{k}dx, \end{align*} which implies that It follows from (21) and (23) that where \(C\) is constant. Hence, by (24), we get where \begin{equation*} C_{1}=\max \left\{ \frac{1}{2}\left[ 1-\left( \frac{d}{J(u_{0})}\right) ^{ \frac{2-k}{k}}\right] ,\frac{C}{2}\left[ 1-\left( \frac{d}{J(u_{0})}\right) ^{\frac{2-k}{k}}\right] \right\} . \end{equation*} Integrating the \(I(u(s))\) with respect to \(s\) over \((t,T)\), we get From (25) and (26), we have Let \(T\to +\infty \) in (27), we can have \begin{equation*} \int\nolimits_{t}^{\infty }\left\Vert u(t)\right\Vert _{H_{0}^{1}\left( \Omega \right) }^{2}ds\leq \frac{1}{2C_{1}}\left\Vert u(t)\right\Vert _{H_{0}^{1}\left( \Omega \right) }^{2}. \end{equation*} From Lemma 3, we get \begin{equation*} \left\Vert u(t)\right\Vert _{H_{0}^{1}\left( \Omega \right) }^{2}\leq \left\Vert u(0)\right\Vert _{H_{0}^{1}\left( \Omega \right) }^{2}e^{1-2C_{1}t}. \end{equation*} The above inequality satisfies that the solution \(u\) decays exponentially.

Theorem 7. (Blow up). Let \(u_{0}\in W_{1}^{-}\) and assume that \(u(t)\) be a unique weak solution to the problem (1). Then \(u(t)\) blows up in finite time, that is, there exists \(T_{\ast }>0\) such that \[ \underset{t\to T_{\ast }}{\lim }\left\Vert u(t)\right\Vert _{H_{0}^{1}(\Omega )}^{2}=\infty . \]

Proof. We show that \(u(t)\) blows up at a finite time. Using contradiction, we suppose that \(u(t)\) is global. We contract a function \(\phi :[0,\infty )\to\mathbb{R}^{+},\) and

Then, thorough direct calculation, we have By (5) and (29), we have By (30) and \(I(u)< 0,\) we get \(\phi ^{\prime \prime }(t)>0\), hence From the Hölder inequality and combining (30), we get It follows from (6) and (30) that Using (28), (32) and (33), we get By (34), we get Choose \(T>t_{0}\) sufficiently large and let \begin{equation*} \psi (t)=\phi (t)+(T-t)\left\Vert u_{0}\right\Vert _{H_{0}^{1}(\Omega )}^{2}t_{0},\text{ }\forall t\in \lbrack 0,T]. \end{equation*} Hence, \(\mu (t)\geq \phi (t)>0,\) \(\mu ^{\prime }(t)=\phi ^{\prime }(t)-\phi ^{\prime }(0)\) and \(\mu ^{\prime \prime }(t)=\phi ^{\prime \prime }(t)>0,\) so (35) implies Let \(\psi (t)=\mu (t)^{-\frac{k-2}{2}}.\) Thus, From (36) and (37), we get \begin{equation*} \psi ^{\prime \prime }(t)=\frac{k-2}{2}\mu (t)^{-\frac{k+2}{2}}\left[ \frac{k }{2}\mu ^{\prime }(t)^{2}-\mu (t)\mu ^{\prime \prime }(t)\right] < 0,\text{ for all }t\in \lbrack t_{0},T]. \end{equation*} Therefore, for any sufficiently large \(T>t_{0},\) \(\psi (t)\) is a concave function in \([t_{0},T].\) Since \(\psi (t_{0})>0\) and \(\psi ^{\prime }(t_{0})< 0,\) there exists a finite time \(T_{\ast }\) such that \begin{equation*} \underset{t\to T_{\ast }^{-}}{\lim }\psi (t)=0. \end{equation*} Consequently, \begin{equation*} \underset{t\to T_{\ast }^{-}}{\lim }\mu (t)=\infty , \end{equation*} which satisfies \begin{equation*} \underset{t\to T_{\ast }^{-}}{\lim }\int\nolimits_{0}^{t}\left\Vert u(s)\right\Vert _{H_{0}^{1}(\Omega )}^{2}ds=\infty , \end{equation*} therefore, we have \begin{equation*} \underset{t\to T_{\ast }^{-}}{\lim }\left\Vert u(s)\right\Vert _{H_{0}^{1}(\Omega )}^{2}=\infty . \end{equation*} This contradicts with \(u(t)\) being a global solution.

Author Contributions

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

Conflicts of Interest

The authors declare no conflict of interest.