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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 $$J(u_{0})<0$$.

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 [58], 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

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

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

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

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Acknowledgements

The authors would like to thank the referees for the careful reading of this paper and for the valuable suggestions to improve the presentation and the style of the paper.

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This work was supported by Key Scientific Research Foundation of the Higher Education Institutions of Henan Province, China (Grant No. 19A110004).

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Correspondence to Gongwei Liu.

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Keywords

• Blow-up
• Lifespan
• Blow-up rate
• Quasi-linear pseudo-parabolic equation