Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level

Abstract

In this paper, we continue to study the initial boundary value problem of the quasi-linear pseudo-parabolic equation

$$ u_{t}-\triangle u_{t}-\triangle u-\operatorname{div}\bigl(| \nabla u|^{2q}\nabla u\bigr)=u^{p} $$

which was studied by Peng et al. (Appl. Math. Lett. 56:17–22, 2016), where the blow-up phenomena and the lifespan for the initial energy $J(u_{0})<0$ were obtained. We establish the finite time blow-up of the solution for the initial data at arbitrary energy level and the lifespan of the blow-up solution. Furthermore, as a product, we obtain the blow-up rate and refine the lifespan when $J(u_{0})<0$.

1 Introduction

In this paper, we investigate the initial boundary value problem of the following quasi-linear pseudo-parabolic equation:

$$ \textstyle\begin{cases} u_{t}-\triangle u_{t}-\triangle u-\operatorname{div}(|\nabla u|^{2q}\nabla u)=u^{p}, & (x, t)\in\Omega\times(0,T), \\ u(x,t)=0, &(x, t)\in\partial\Omega\times(0,T), \\ u(x,t)=u_{0}(x), &x\in\Omega, \end{cases} $$

(1.1)

where $\Omega\subset\mathbb{R}^{n}$ ($n\geq3$) is a bounded domain with sufficiently smooth boundary ∂Ω, $p>1$ and $0\leq 2q< p-1$. $T\in(0, \infty]$ denotes the maximal existence time of the solution.

Problem (1.1) describes a variety of important physical and biological phenomena such as the aggregation of population [1], the unidirectional propagation of nonlinear, dispersive, long waves [2], and the nonstationary processes in semiconductors [3]. In the absence of the term $\operatorname{div}(|\nabla u|^{2q}\nabla u)$, Eq. (1.1) reduces to the following semilinear pseudo-parabolic equation:

$$ u_{t}-\triangle u_{t}-\triangle u=u^{p},\quad (x, t)\in\Omega\times(0,T). $$

(1.2)

There are many results for Eq. (1.2) such as the existence and uniqueness in [4], blow-up in [5–8], asymptotic behavior in [6, 9], and so on. Using the integral representation and the semigroup, Cao et al.[10] obtained the critical global existence exponent and the critical Fujita exponent for Eq. (1.2). Chen et al. [11] considered Eq. (1.2) with the logarithmic nonlinearity source term by the potential well methods.

Recently, Peng et al. [12] considered the blow-up phenomena on problem (1.1). By the way, Payne et al. [13] considered the blow-up phenomena of solutions on the initial boundary problem of the nonlinear parabolic equation

$$ u_{t}-\operatorname{div} \bigl(\rho\bigl(|\nabla u|^{2} \bigr)\nabla u \bigr)=f(u). $$

In addition, Long et al. [14] investigated the blow-up phenomena for a nonlinear pseudo-parabolic equation with nonlocal source

$$ u_{t}-\triangle u_{t}-\operatorname{div} \bigl(|\nabla u|^{2q}\nabla u \bigr)=u^{p}(x,t) \int _{\Omega}k(x,y)u^{p+1}(y,t)\,dy. $$

Finally, we mention some interesting works concerning quasi-linear or degenerate parabolic equations. For example, Winkert and Zacher [15] considered a generate class of quasi-linear parabolic problems and established global a priori bounds for the weak solutions of such problems; Fragnelli and Mugnai [16] established Carleman estimates for degenerate parabolic equations with interior degeneracy and non-smooth coefficients.

Throughout this paper, we use $\|\cdot\|_{p}= (\int_{\Omega}|\cdot |^{p}\, dx )^{\frac{1}{p}}$ and $\|\cdot\|_{W_{0}^{1,p}}= (\int_{\Omega }(|\cdot|^{p}+|\nabla\cdot|^{p})\,dx )^{\frac{1}{p}}$ as the norms on the Banach spaces $L^{p}=L^{p}(\Omega)$ and $W_{0}^{1, p}=W_{0}^{1, p}(\Omega)$, respectively. As in [12], we define the energy functional and the Nehari functional of (1.1), respectively, by

$$\begin{aligned}& J(u):=\frac{1}{2}\|\nabla u\|_{2}^{2}+ \frac{1}{2q+2}\|\nabla u\| _{2q+2}^{2q+2}-\frac{1}{p+1}\|u \|_{p+1}^{p+1}, \end{aligned}$$

(1.3)

$$\begin{aligned}& I(u):=\bigl(J'(u), u\bigr)=\|\nabla u \|_{2}^{2}+\|\nabla u\|_{2q+2}^{2q+2}-\|u \|_{p+1}^{p+1}. \end{aligned}$$

(1.4)

Let $\lambda_{1}$ be the first nontrivial eigenvalue of −△ operator in Ω with homogeneous Dirichlet condition, then we have

$$ \lambda_{1}\|u\|_{2}^{2}\leq\|\nabla u\|_{2}^{2},\qquad \|\nabla u\|_{2}^{2} \geq\frac {\lambda_{1}}{1+\lambda_{1}}\|u\|_{H_{0}^{1}}^{2},\quad u\in H_{0}^{1}(\Omega). $$

(1.5)

In order to compare with our work, in this paper, we summarize the blow-up results obtained in [12] as follows.

(RES1) If $0\leq2q< p-1$, $J(u_{0})<0$, and u is a nonnegative solution of (1.1), then u blows up at some finite time T, where T is bounded by

$$ T\leq T_{1}:=\frac{\|u_{0}\|^{2}_{H_{0}^{1}}}{(1-p^{2})J(u_{0})}. $$

(1.6)

From the above (RES1), we notice that (1) the blow-up rate is not given when $J(u_{0})<0$; (2) the blow-up phenomena and the lifespan are still unsolved when $J(u_{0})\geq0$.

Motivated by the above-mentioned facts, we investigate these two problems in this paper. Firstly, we state the local existence theorem of problem (1.1) by Faedo–Galerkin method (see Theorem 2.1 in [12]).

(RES2) For any $u_{0}\in W_{0}^{1,2q+2}(\Omega)$, there exists $T>0$ such that problem (1.1) has a unique local weak solution $u\in L^{\infty}(0,T;W_{0}^{1, 2q+2}(\Omega))$ with $u_{t}\in L^{2}(0,T; H_{0}^{1}(\Omega))$ which satisfies

$$ \langle u_{t}, v\rangle+\langle\nabla u_{t}, \nabla v \rangle+\langle\nabla u,\nabla v\rangle+\bigl\langle |\nabla u|^{2q}\nabla u, \nabla v\bigr\rangle =\bigl\langle u^{p}, v\bigr\rangle $$

for all $v\in W_{0}^{1, 2q+2}(\Omega)$.

Our main result of this paper can be stated as the following theorem.

Theorem 1.1

For all $0\leq2q< p-1$, the nonnegative solution u of problem (1.1) blows up at finite time in $H_{0}^{1}$-norm provided that

$$ J(u_{0})< \frac{(p-1)\lambda_{1}}{2(p+1)(1+\lambda_{1})}\|u_{0} \|^{2}_{H_{0}^{1}}. $$

(1.7)

Furthermore, the lifespan T can be estimated by

$$ T\leq T_{2}:=\frac{8(p+1)(1+\lambda_{1})\|u_{0}\| ^{2}_{H_{0}^{1}}}{(p-1)^{2}[(p-1)\lambda_{1}\|u_{0}\|^{2}_{H_{0}^{1}}-2(p+1)(1+\lambda_{1})J(u_{0})]}. $$

(1.8)

Remark 1.1

For the case $J(u_{0})<0$, the initial data condition given in (1.7) is obviously satisfied. Noticing the values of $T_{1}$ and $T_{2}$ given in (1.6) and (1.8), we can refine the lifespan T as

$$ T\leq\min\{T_{1}, T_{2}\}= \textstyle\begin{cases} T_{2}, & \mbox{if }-\frac{(p-1)^{2}\lambda_{1}\|u_{0}\| ^{2}_{H_{0}^{1}}}{2(p+1)(3p+5)(1+\lambda_{1})}\leq J(u_{0})< 0; \\ T_{1}, & \mbox{if }J(u_{0})< -\frac{(p-1)^{2}\lambda_{1}\|u_{0}\| ^{2}_{H_{0}^{1}}}{2(p+1)(3p+5)(1+\lambda_{1})}. \end{cases} $$

2 Proof of Theorem 1.1

In this section, we prove Theorem 1.1 by using the following lemma (see [17]).

Lemma 2.1

Suppose that a nonnegative, twice-differentiable function $\theta(t)$ satisfies the inequality

$$\theta''(t)\theta(t)-(1+r) \bigl( \theta'(t)\bigr)^{2} \geq0, \quad t>0, $$

where $r>0$ is some constant. If $\theta(0) > 0$ and $\theta'(0)>0$, then there exists $0 < t_{1}\leq\frac{\theta(0)}{r\theta'(0)}$ such that $\theta(t)\rightarrow+\infty$ as $t\rightarrow t_{1}^{-}$.

Proof of Theorem 1.1

We give the proof in the following two steps.

Step 1: Blow-up. Let $u(t)$ be the solution of problem (1.1) with the initial data satisfying (1.7). We may assume $J(u(t))\geq0$; otherwise, there exists some $t_{0}\geq0$ such that $J(u(t_{0}))<0$, then $u(t)$ will blow up in finite time by (RES1), the proof of this step is complete. So, in the following, we give our proof by contradiction and assume that $u(t)$ exists globally and $J(u(t))\geq0$ for all $t\geq0$.

Differentiating (1.3) and making use of (1.1) and (1.4), we have the following equalities:

$$\begin{aligned}& \frac{d}{dt}J\bigl(u(t)\bigr)=-\|u_{t} \|_{2}^{2}-\|\nabla u_{t}\|_{2}^{2}=- \|u_{t}\|^{2}_{H_{0}^{1}}, \end{aligned}$$

(2.1)

$$\begin{aligned}& \begin{aligned}[b] \frac{d}{dt} \biggl(\frac{1}{2} \bigl\Vert u(t) \bigr\Vert _{H^{1}_{0}}^{2} \biggr)&=-\| \nabla u \|_{2}^{2}-\|\nabla u\|_{2q+2}^{2q+2}+\|u \|_{p+1}^{p+1} \\ &=-I\bigl(u(t)\bigr). \end{aligned} \end{aligned}$$

(2.2)

Since

$$ \int_{0}^{t} \bigl\Vert u_{s}(s) \bigr\Vert _{H_{0}^{1}}\,ds\geq \biggl\Vert \int_{0}^{t}u_{s}(s)\,ds \biggr\Vert _{H_{0}^{1}}= \bigl\Vert u(t)-u_{0} \bigr\Vert _{H_{0}^{1}} \geq \bigl\Vert u(t) \bigr\Vert _{H_{0}^{1}}- \Vert u_{0} \Vert _{H_{0}^{1}}, \quad t\geq0, $$

by Hölder’s inequality, (2.1), and $J(u_{0})\geq J(u(t))\geq0$, we obtain that

$$ \begin{aligned}[b] \bigl\Vert u(t) \bigr\Vert _{H_{0}^{1}}&\leq \|u_{0}\|_{H_{0}^{1}}+t^{\frac{1}{2}}\biggl[ \int_{0}^{t} \bigl\Vert u_{s}(s) \bigr\Vert _{H_{0}^{1}}\,ds\biggr]^{\frac{1}{2}} \\ &= \|u_{0}\|_{H_{0}^{1}}+t^{\frac{1}{2}}\bigl[J(u_{0})-J \bigl(u(t)\bigr)\bigr]^{\frac{1}{2}} \\ &\leq \|u_{0}\|_{H_{0}^{1}}+t^{\frac{1}{2}}\bigl(J(u_{0}) \bigr)^{\frac{1}{2}},\quad t\geq0. \end{aligned} $$

(2.3)

Combining (1.5) and Hölder’s inequality, we deduce that

$$ \bigl\Vert \nabla u(t) \bigr\Vert _{2q+2}^{2q+2}\geq| \Omega|^{-q} \biggl(\frac{\lambda _{1}}{1+\lambda_{1}} \biggr)^{q+1} \bigl\Vert u(t) \bigr\Vert _{H_{0}^{1}}^{2q+2}. $$

On the other hand, by (1.3), (1.4), (2.2), and $0\leq 2q< p-1$, we obtain

$$\begin{aligned} \frac{d}{dt} \biggl(\frac{1}{2} \bigl\Vert u(t) \bigr\Vert _{H^{1}_{0}}^{2} \biggr)&=\frac{p-1}{2} \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}+\frac{p-2q-1}{2q+2} \bigl\Vert \nabla u(t) \bigr\Vert _{2q+2}^{2q+2}-(p+1)J\bigl(u(t)\bigr) \\ &\geq\frac{(p-1)\lambda_{1}}{2(1+\lambda_{1})} \bigl\Vert u(t) \bigr\Vert ^{2}_{H_{0}^{1}}+ \frac {p-2q-1}{2q+2} \biggl(\frac{\lambda_{1}}{1+\lambda_{1}} \biggr)^{q+1}|\Omega |^{-q} \bigl\Vert u(t) \bigr\Vert _{H_{0}^{1}}^{2q+2} \\ &\quad {}-(p+1)J \bigl(u(t)\bigr) \\ &\geq\frac{(p-1)\lambda_{1}}{1+\lambda_{1}} \biggl[\frac{1}{2} \bigl\Vert u(t) \bigr\Vert _{H^{1}_{0}}^{2}-\frac{(p+1)(1+\lambda_{1})}{(p-1)\lambda_{1}}J\bigl(u(t)\bigr) \biggr] . \end{aligned}$$

Since $\frac{d}{dt}(J(u(t)))\leq0$, it follows from the above inequality that

$$\begin{aligned}& \frac{d}{dt} \biggl[\frac{1}{2} \bigl\Vert u(t) \bigr\Vert _{H^{1}_{0}}^{2}-\frac{(p+1)(1+\lambda _{1})}{(p-1)\lambda_{1}}J\bigl(u(t)\bigr) \biggr] \\& \quad \geq \frac{(p-1)\lambda_{1}}{1+\lambda _{1}} \biggl[\frac{1}{2} \bigl\Vert u(t) \bigr\Vert _{H^{1}_{0}}^{2}-\frac{(p+1)(1+\lambda _{1})}{(p-1)\lambda_{1}}J\bigl(u(t)\bigr) \biggr]. \end{aligned}$$

Let

$$H(t)=\frac{1}{2} \bigl\Vert u(t) \bigr\Vert _{H^{1}_{0}}^{2}- \frac{(p+1)(1+\lambda _{1})}{(p-1)\lambda_{1}}J\bigl(u(t)\bigr), $$

then

$$\frac{d}{dt}H(t)\geq\frac{(p-1)\lambda_{1}}{1+\lambda_{1}}H(t) $$

for all $t\geq0$. By using Gronwall’s inequality, we get

$$H(t)\geq e^{\frac{(p-1)\lambda_{1}}{1+\lambda_{1}}t}H(0). $$

Noticing that $H(0)>0$ via (1.7) and the assumption $J(u(t))\geq 0$ for $t\geq0$, we deduce

$$ \bigl\| u(t)\bigr\| _{H_{0}^{1}}\geq\sqrt{2H(0)}e^{\frac{(p-1)(\lambda_{1}}{2(1+\lambda _{1})}t},\quad t\geq0, $$

which is a contradiction with (2.3) for t sufficiently large. Hence, $u(t)$ blows up at some finite time, i.e., $T<\infty$.

Step 2: Lifespan. We will find an upper bound for T. Firstly, we claim that

$$ I\bigl(u(t)\bigr)= \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla u(t) \bigr\Vert _{2q+2}^{2q+2}- \bigl\Vert u(t) \bigr\Vert _{p+1}^{p+1}< 0,\quad t\in[0, T). $$

(2.4)

Indeed, combining (1.3) and (1.4), after a simple calculation, we get

$$\begin{aligned} J\bigl(u(t)\bigr) =&\frac{p-1}{2(p+1)} \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}+\frac {p-2q-1}{2(q+1)(p+1)} \bigl\Vert \nabla u(t) \bigr\Vert _{2q+2}^{2q+2} \\ &{}+\frac {1}{p+1}I\bigl(u(t) \bigr),\quad t\in[0,T). \end{aligned}$$

(2.5)

It follows from (1.5), (1.7), and (2.5) that

$$ \frac{(p-1)\lambda_{1}}{2(p+1)(1+\lambda_{1})}\|u_{0}\|^{2}_{H_{0}^{1}}>J(u_{0}) \geq \frac{p-1}{2(p+1)}\frac{\lambda_{1}}{1+\lambda_{1}}\|u_{0}\|_{H_{0}^{1}}^{2}+ \frac {1}{p+1}I(u_{0}), $$

where we also use $0\leq2q< p-1$, which implies $I(u_{0})<0$. Hence, if (2.4) does not hold, there must exist $t_{0}\in(0,T)$ such that $I(u(t_{0}))=0$, $I(u(t))<0$ for $t\in[0,t_{0})$. Then, by (2.2), we obtain that $\|u(t)\|_{H_{0}^{1}}^{2}$ is strictly increasing on $[0, t_{0})$. Then it follows from (1.7) that

$$ \begin{aligned}[b] J(u_{0})&< \frac{(p-1)\lambda_{1}}{2(p+1)(1+\lambda_{1})} \Vert u_{0} \Vert ^{2}_{H_{0}^{1}} \\ &< \frac{(p-1)\lambda_{1}}{2(p+1)(1+\lambda_{1})} \bigl\Vert u(t_{0}) \bigr\Vert ^{2}_{H_{0}^{1}}. \end{aligned} $$

(2.6)

On the other hand, combining (2.1) and (2.5), we get

$$\begin{aligned} J(u_{0})&\geq J\bigl(u(t_{0})\bigr)=\frac{p-1}{2(p+1)} \bigl\Vert \nabla u(t_{0}) \bigr\Vert _{2}^{2}+ \frac {p-2q-1}{2(q+1)(p+1)} \bigl\Vert \nabla u(t_{0}) \bigr\Vert _{2q+2}^{2q+2}+\frac {1}{p+1}I\bigl(u(t_{0})\bigr) \\ &\geq\frac{(p-1)\lambda_{1}}{2(p+1)(1+\lambda_{1})} \bigl\Vert u(t_{0}) \bigr\Vert ^{2}_{H_{0}^{1}}, \end{aligned}$$

which is a contradiction with (2.6). Hence, $I(u(t))<0$ and $\| u(t)\|_{H_{0}^{1}}^{2}$ is strictly increasing on $[0, T)$.

We define the functional

$$ F(t)= \int_{0}^{t} \bigl\Vert u(s) \bigr\Vert _{H_{0}^{1}}^{2}\,ds+(T-t) \Vert u_{0} \Vert _{H_{0}^{1}}^{2}+\beta(t+\gamma )^{2}, \quad t\in[0,T), $$

with two positive constants β, γ to be chosen later. Since $\|u(t)\|_{H_{0}^{1}}^{2}$ is strictly increasing, we get

$$ \begin{aligned}[b] F'(t)&= \bigl\Vert u(t) \bigr\Vert _{H_{0}^{1}}^{2}- \Vert u_{0} \Vert _{H_{0}^{1}}^{2}+2\beta (t+\gamma) \\ &= \int_{0}^{t}\frac{d}{ds} \bigl\Vert u(s) \bigr\Vert _{H_{0}^{1}}^{2}\,ds+2\beta(t+\gamma)\geq2\beta (t+ \gamma)>0 \end{aligned} $$

(2.7)

and

$$ \begin{aligned}[b] F''(t)&= \frac{d}{dt} \bigl\Vert u(t) \bigr\Vert _{H_{0}^{1}}^{2}+2 \beta \\ &=(p-1) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}+ \frac{p-2q-1}{q+1} \bigl\Vert \nabla u(t) \bigr\Vert _{2q+2}^{2q+2}-2(p+1)J \bigl(u(t)\bigr)+2\beta \\ &\geq\frac{(p-1)\lambda_{1}}{1+\lambda_{1}} \bigl\Vert u(t) \bigr\Vert _{H_{0}^{1}}^{2}+2(p+1) \int _{0}^{t} \Vert u_{s} \Vert _{H_{0}^{1}}^{2}\,ds-2(p+1)J(u_{0}). \end{aligned} $$

(2.8)

Noticing that

$$F(0)=T\|u_{0}\|^{2}_{H_{0}^{1}}+\beta \gamma^{2}>0 $$

and

$$F'(0)=2\beta\gamma>0, $$

by using Young’s inequality, Hölder’s inequality, and the element algebraic inequality

$$ab+cd\leq\sqrt{a^{2}+c^{2}}\sqrt{b^{2}+d^{2}}, $$

we can deduce

$$ \begin{aligned} \xi(t)&:= \biggl( \int_{0}^{t} \bigl\Vert u(s) \bigr\Vert _{H_{0}^{1}}^{2}\,ds+\beta(t+\gamma)^{2} \biggr) \biggl( \int_{0}^{t} \Vert u_{s} \Vert _{H_{0}^{1}}^{2}\,ds+\beta \biggr) \\ &\quad {}- \biggl( \int_{0}^{t}\frac{1}{2}\frac {d}{ds} \bigl\Vert u(s) \bigr\Vert _{H_{0}^{1}}^{2}\,ds+\beta(t+\gamma) \biggr)^{2} \\ &\geq0. \end{aligned} $$

Hence, it follows from the above inequality and (2.7) that

$$ \begin{aligned} -\bigl(F'(t)\bigr)^{2}&=-4 \biggl[\frac{1}{2} \int_{0}^{t}\frac{d}{ds} \bigl\Vert u(s) \bigr\Vert _{H_{0}^{1}}^{2}\,ds+2\beta(t+\gamma) \biggr]^{2} \\ &=4 \biggl(\xi(t)- \bigl(F(t)-(T-t) \Vert u_{0} \Vert _{H_{0}^{1}}^{2} \bigr) \biggl( \int_{0}^{2}\frac {d}{ds} \bigl\Vert u(s) \bigr\Vert _{H_{0}^{1}}^{2}\,ds+\beta \biggr) \biggr) \\ &\geq-4F(t) \biggl( \int_{0}^{t}\frac{d}{ds} \bigl\Vert u(s) \bigr\Vert _{H_{0}^{1}}^{2}\,ds+\beta \biggr). \end{aligned} $$

By the above equality, (2.8), and the fact that $\|u(t)\| _{H_{0}^{1}}^{2}$ is strictly increasing, we have

$$ \begin{aligned} F(t)F''(t)- \frac{p+1}{2}\bigl(F'(t)\bigr)^{2}&\geq F(t) \biggl[F''(t)-2(p+1) \biggl( \int _{0}^{t}\frac{d}{ds} \bigl\Vert u(s) \bigr\Vert _{H_{0}^{1}}^{2}\,ds+\beta \biggr) \biggr] \\ &\geq2(p+1)F(t) \biggl[\frac{(p-1)\lambda_{1}}{2(p+1)(1+\lambda_{1})} \Vert u_{0} \Vert _{H_{0}^{1}}^{2}-J(u_{0})-\beta \biggr]. \end{aligned} $$

From (1.7), we can choose β sufficiently small such that

$$ 0< \beta\leq\beta_{0}:=\frac{(p-1)\lambda_{1}}{2(p+1)(1+\lambda_{1})} \|u_{0}\| _{H_{0}^{1}}^{2}-J(u_{0}). $$

(2.9)

Then the conditions of Lemma 2.1 are satisfied with $r=\frac{p-1}{2}$, so we have

$$ T\leq\frac{2F(0)}{(p-1)F'(0)}=\frac{\|u_{0}\|_{H_{0}^{1}}^{2}}{(p-1)\beta\gamma }T+\frac{\gamma}{p-1}. $$

(2.10)

Fixing arbitrary β satisfying (2.9), then let γ be sufficiently large such that

$$ \frac{\|u_{0}\|_{H_{0}^{1}}^{2}}{(p-1)\beta}< \gamma< +\infty, $$

then it follows from (2.10) that

$$ T\leq\frac{\beta\gamma^{2}}{(p-1)\beta\gamma-\|u_{0}\|_{H_{0}^{1}}^{2}}. $$

(2.11)

Define a function $T_{\beta}(\gamma)$ by

$$ T_{\beta}(\gamma)=\frac{\beta\gamma^{2}}{(p-1)\beta\gamma-\|u_{0}\| _{H_{0}^{1}}^{2}},\quad \gamma\in \biggl( \frac{\|u_{0}\|_{H_{0}^{1}}^{2}}{(p-1)\beta}, +\infty \biggr). $$

It is easy to prove that the function $T_{\beta}(\gamma)$ has a unique minimum at

$$\gamma_{\beta}:=\frac{2\|u_{0}\|_{H_{0}^{1}}^{2}}{(p-1)\beta}\in\biggl(\frac{\|u_{0}\| _{H_{0}^{1}}^{2}}{(p-1)\beta}, +\infty \biggr). $$

Then it follows from (2.11) that

$$ T\leq\inf_{\gamma\in (\frac{\|u_{0}\|_{H_{0}^{1}}^{2}}{(p-1)\beta}, +\infty )}T_{\beta}(\gamma)=T_{\beta}( \gamma_{\beta})=\frac{4\|u_{0}\| _{H_{0}^{1}}^{2}}{(p-1)^{2}\beta} $$

for any β satisfying (2.9). Finally, we obtain

$$ T\leq\inf_{\beta\in(0,\beta_{0}]}\frac{4\|u_{0}\|_{H_{0}^{1}}^{2}}{(p-1)^{2}\beta }=\frac{4\|u_{0}\|_{H_{0}^{1}}^{2}}{(p-1)^{2}\beta_{0}} = \frac{8(p+1)(1+\lambda_{1})\|u_{0}\|^{2}_{H_{0}^{1}}}{(p-1)^{2}[(p-1)\lambda_{1}\| u_{0}\|^{2}_{H_{0}^{1}}-2(p+1)(1+\lambda_{1})J(u_{0})]}. $$

This completes the proof of Theorem 1.1. □

Corollary 2.1

For all $0\leq2q< p-1$ and any $M>0$, there exists initial $u_{0M}\in W_{0}^{1, 2q+2}(\Omega)$ such that the weak solution for corresponding problem (1.1) will blow up in finite time.

Proof

Let $M>0$, and $\Omega_{1}$ and $\Omega_{2}$ be two arbitrary disjoint open subdomains of Ω. We assume that $v\in W_{0}^{1, 2q+2}(\Omega _{1})\subset W_{0}^{1, 2q+2}(\Omega)\subset H_{0}^{1}(\Omega)$ is an arbitrary nonzero function, then we can take $\alpha_{1}>0$ sufficiently large such that

$$\begin{aligned} \|\alpha_{1}v\|_{H_{0}^{1}}^{2}&=\alpha_{1}^{2} \int_{\Omega}|v^{2}|\,dx+\alpha_{1}^{2} \int _{\Omega}|\nabla v|^{2}\,dx=\alpha_{1}^{2} \int_{\Omega_{1}}|v^{2}|\,dx+\alpha_{1}^{2} \int _{\Omega_{1}}|\nabla v|^{2}\,dx \\ &>\frac{2(p+1)(1+\lambda_{1})}{(p-1)\lambda_{1}}M. \end{aligned}$$

We claim that there exist $w\in W_{0}^{1, 2q+2}(\Omega_{2})\subset W_{0}^{1, 2q+2}(\Omega)$ and $\alpha>\alpha_{1}$ such that $J(w)=M-J(\alpha v)$.

In fact, we choose a function $w_{k}\in C_{0}^{1}(\Omega_{2})$ such that $\| \nabla w_{k}\|_{2}\geq k$ and $\|w_{k}\|_{\infty}\leq c_{0}$. Hence,

$$\begin{aligned} &\frac{1}{2} \int_{\Omega_{2}}|\nabla w_{k}|^{2}\,dx+ \frac{1}{2q+2} \int_{\Omega _{2}}|\nabla w_{k}|^{2q+2}\,dx- \frac{1}{p+1} \int_{\Omega_{2}}|w_{k}|^{p+1}\,dx \\ &\quad \geq\frac{1}{2} \int_{\Omega_{2}}|\nabla w_{k}|^{2}\,dx+ \frac{1}{2q+2}|\Omega _{2}|^{-q} \biggl( \int_{\Omega_{2}}|\nabla w_{k}|^{2}\,dx \biggr)^{q+1}-\frac {1}{p+1}c_{0}^{p+1}| \Omega_{2}|. \end{aligned}$$

On the other hand, since $0\leq2q< p-1$, it holds that

$$\begin{aligned} M-J(\alpha v)&=M- \frac{\alpha^{2}}{2} \int_{\Omega_{1}}|\nabla v|^{2}\,dx-\frac {\alpha^{2q+2}}{2q+2} \int_{\Omega_{1}}|\nabla|^{2q+2}\,dx \\ &\quad {}+\frac{\alpha^{p+1}}{p+1} \int_{\Omega_{1}}|v|^{p+1}\,dx\rightarrow+\infty , \quad \mbox{as }\alpha\rightarrow+\infty. \end{aligned}$$

Hence, there exist $k>0$ and $\alpha>\alpha_{1}$ both sufficiently large such that

$$ M-J(\alpha v)=\frac{1}{2} \int_{\Omega_{2}}|\nabla w_{k}|^{2}\,dx+ \frac {1}{2q+2} \int_{\Omega_{2}}|\nabla w_{k}|^{2q+2}\,dx- \frac{1}{p+1} \int_{\Omega _{2}}|w_{k}|^{p+1}\,dx. $$

Then we choose $w=w_{k}$ and denote $u_{0M}:=\alpha v+w$. Hence, we have

$$\begin{aligned} \|u_{0M}\|_{H_{0}^{1}}^{2}&= \int_{\Omega} \bigl\vert u_{0M}^{2} \bigr\vert \,dx+ \int_{\Omega} \vert \nabla u_{0M} \vert ^{2}\,dx\geq\alpha^{2} \int_{\Omega_{1}} \bigl\vert v^{2} \bigr\vert \,dx+ \alpha^{2} \int_{\Omega _{1}} \vert \nabla v \vert ^{2}\,dx \\ &>\frac{2(p+1)(1+\lambda_{1})}{(p-1)\lambda_{1}}M \end{aligned}$$

and

$$M=J(\alpha v)+J(w)=J(u_{0M})< \frac{(p-1)\lambda_{1}}{2(p+1)(1+\lambda _{1})}\|u_{0M} \|^{2}_{H_{0}^{1}}. $$

The proof is complete. □

Remark 2.1

In this remark, we establish the blow-up rate for $J(u_{0})<0$. We define the functionals $\varphi(t)=\|u(t)\|_{H_{0}^{1}}^{2}$ and $\psi(t)=-2(p+1)J(u(t))$ as these in [12]. It was shown in (4.8) of [12] that

$$ \frac{\varphi'(t)}{[\varphi(t)]^{\frac{p+1}{2}}}\geq\frac{\psi (0)}{[\varphi(0)]^{\frac{p+1}{2}}}. $$

Now, we integrate the inequality from t to T, noticing $\lim_{t\rightarrow T^{-}}\varphi(t)=+\infty$ (by (RES1)), we obtain

$$ \varphi(t)\leq \biggl[\frac{(p-1)\psi(0)}{2[\varphi(0)]^{\frac {p+1}{2}}} \biggr]^{\frac{2}{1-p}}. $$

Then it follows from the definitions of $\varphi(t)$ and $\psi(t)$ that

$$ \bigl\| u(t)\bigr\| _{H_{0}^{1}}\leq \biggl[\frac{(1-p^{2})J(u_{0})}{\|u_{0}\| _{H_{0}^{1}}^{p+1}} \biggr]^{\frac{1}{1-p}}(T-t)^{-\frac{1}{p-1}}. $$