Evolutionary p-Laplacian with convection and reaction under dynamic boundary condition

Abstract

We study the global existence and blow-up phenomenon of solutions to an evolutionary p-Laplacian with convection and reaction under dynamic boundary condition.

1 Introduction

In this paper, we are concerned with the following evolutionary p-Laplacian under dynamic boundary condition:

$$\begin{aligned}& \frac{\partial u}{\partial t}=\operatorname {div}\bigl(\vert \nabla u\vert ^{p-2}\nabla u\bigr)-\overset{\rightarrow }{g}(u)\cdot\nabla u+f(u), \quad x\in \Omega, t>0, \end{aligned}$$

(1.1)

$$\begin{aligned}& \sigma u_{t}+\vert \nabla u\vert ^{p-2} \nabla u\cdot\nu=0, \quad x\in\partial \Omega, t>0, \end{aligned}$$

(1.2)

$$\begin{aligned}& u(x,0)=u_{0}(x), \quad x\in\Omega, \end{aligned}$$

(1.3)

where $p>1$, $\overset{\rightarrow}{g}:\mathbb{R}\to\mathbb{R}^{N}$, $f:\mathbb{R}\to\mathbb{R}$, $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with smooth boundary ∂Ω, and $\nu:\partial\Omega\to\mathbb{R}^{N}$ is the outer unit normal vector.

The quasilinear parabolic problems with dynamic boundary conditions of type (1.1)-(1.3) arise in numerous areas such as heat conduction, chemical reactor theory, colloid chemistry and population growth, see [1, 2] and the references therein. Many reaction-diffusion equations under dynamic boundary conditions have been considered in the past years. An early study of problem (1.1)-(1.3) with $p=2$ and $\overset{\rightarrow}{g}=\vec{0}$ was carried out by Below and Mailly [3] who showed a complete result about the blow-up phenomena as well as the lower and upper bounds for the blow-up time. Moreover, some of the techniques were also applied to the porous medium equation with reaction. Later on, for the evolutionary p-Laplacian with $p\ge2N/(N+2)$, where N is the dimension of the domain, Gal and Warma [4] considered the following equation without convection:

$$\frac{\partial u}{\partial t}-\operatorname {div}\bigl(\vert \nabla u\vert ^{p-2}\nabla u\bigr)+f(u)=g(x), \quad x\in \Omega, t>0, $$

coupled with dynamic boundary conditions. The well-posedness and the existence of a global attractor results were established. More recently, Mailly and Rault [2] studied the nonlinear convection problem (1.1)-(1.3) with $p=2$ and proved the global existence and blow-up phenomena of local solutions. For other results about the solvability of quasilinear parabolic equations with dynamic boundary conditions, we refer the readers to [5–7], etc.

Throughout this paper, we suppose that the dissipativity condition holds

$$ \sigma>0, \quad \sigma\in C^{1}\bigl(\partial\Omega\times[0,+ \infty)\bigr), $$

(1.4)

and the functions in problem (1.1)-(1.3) are smooth

$$ f\in C^{1}(\mathbb{R}),\qquad f(s)\ge0 \quad \mbox{for } s\ge0,\qquad \overset{\rightarrow}{g}\in C^{1}\bigl(\mathbb{R},\mathbb{R}^{N} \bigr), $$

(1.5)

the initial data is non-negative and satisfies

$$ u_{0}\ge0,\quad u_{0}\in L^{\infty}(\Omega) \cap W^{1,p}(\Omega). $$

(1.6)

In Section 2 we develop the comparison principle for a regularized problem and the local existence of weak and strong solutions of problem (1.1)-(1.3). In Section 3 we derive the global existence of the strong solutions, while in Section 4 we prove the blow-up phenomenon of strong solutions by formulating a family of radially symmetric lower solutions.

2 Comparison principle and local existence

In this section, we use the regularization method and compactness theorems to prove the local existence of the solutions to problem (1.1)-(1.3).

Consider the following regularized problem:

$$\begin{aligned}& \frac{\partial u}{\partial t}=\operatorname {div}\biggl(\biggl(\frac{1}{n}+ \vert \nabla u\vert ^{2}\biggr)^{\frac {p-2}{2}}\nabla u \biggr) -\overset{\rightarrow}{g}(u_{+})\cdot\nabla u+f_{M}(u), \quad x\in \Omega, t>0, \end{aligned}$$

(2.1)

$$\begin{aligned}& \sigma u_{t}+\biggl(\frac{1}{n}+\vert \nabla u \vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla u\cdot\nu=0, \quad x\in \partial\Omega, t>0, \end{aligned}$$

(2.2)

$$\begin{aligned}& u(x,0)=u_{0,n}(x), \quad x\in\Omega, \end{aligned}$$

(2.3)

where $f_{M}(s)=\min\{f_{+}(s), M\}$, $s_{+}=\max\{s,0\}$, $M>0$, $n\in \mathbb{Z}^{+}$, $u_{0,n}\in C^{\infty}(\overline{\Omega})$ satisfies

$$\inf_{\Omega}u_{0}\le u_{0,n}\le\sup _{\Omega}u_{0}, \qquad \Vert u_{0,n}\Vert _{W^{1,p}}\le2\Vert u_{0}\Vert _{W^{1,p}}, \qquad \lim _{n\to\infty} \Vert u_{0,n}-u_{0}\Vert _{W^{1,p}}=0. $$

Since $f,\overset{\rightarrow}{g}\in C^{1}$, we can verify that $f_{M}$, $\overset{\rightarrow}{g}(s_{+})$ are locally Lipschitz continuous.

Hereafter, we suppose that the regularized problem (2.1)-(2.3) has a classical solution $u_{n,M}\in C^{2,1}(\overline{\Omega}\times[0,T_{n,M}))$ with the maximal existence time $0< T_{n,M}\le+\infty$. Let $Q_{T}=\overline{\Omega}\times(0,T)$ for $T>0$ and define

$$\begin{aligned}& F_{n,M}[u]=F_{n,M}(u,\nabla u)=\operatorname {div}\biggl(\biggl( \frac{1}{n}+\vert \nabla u\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla u \biggr) -\overset{\rightarrow}{g}(u_{+})\cdot\nabla u+f_{M}(u), \\& B_{n}[u]=\sigma u_{t}+\biggl(\frac{1}{n}+\vert \nabla u\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla u\cdot \nu. \end{aligned}$$

Notice that, in the dynamic boundary condition, $B_{n}[u]$ is nonlinear with respect to ∇u. First, we need the following comparison principles which are simple variations of the comparison principles in [8].

Lemma 2.1

Let $u,v\in C^{2,1}(Q_{T})\cap C(\overline{Q}_{T})$ satisfying

$$\begin{aligned}& u_{t}-F_{n,M}[u] >v_{t}-F_{n,M}[v],\quad (x,t)\in Q_{T}, \\& B_{n}[u] >B_{n}[v], \quad (x,t)\in\partial\Omega \times(0,T), \\& u(x,0) >v(x,0), \quad x\in\overline{\Omega}. \end{aligned}$$

Then

$$u(x,t)>v(x,t), \quad (x,t)\in Q_{T}. $$

Proof

Suppose that there exists $(x_{0},t_{0})\in Q_{T}$ such that $u(x_{0},t_{0})\le v(x_{0},t_{0})$. Let

$$t^{*}=\sup\bigl\{ \tau\in(0,T);u(x,t)>v(x,t), \forall(x,t)\in Q_{\tau}\bigr\} . $$

Then $t^{*}\in(0,t_{0}]\subset(0,T)$ and $\min_{\overline{Q}_{t^{*}}}\{u-v\}=0$. Thus, $u-v$ attains its minimum 0 at some point $(x^{*},t^{*})$ with $x^{*}\in\overline{\Omega}$. If $x^{*}\in\Omega$, then

$$u=v,\qquad u_{t}\le v_{t},\qquad \nabla u=\nabla v, \qquad D^{2}u\ge D^{2}v \quad \mbox{at } \bigl(x^{*},t^{*}\bigr), $$

which contradicts $u_{t}-F_{n,M}[u]>v_{t}-F_{n,M}[v]$. If $x^{*}\in\partial\Omega$, then

$$u_{t}\le v_{t},\qquad \frac{\partial u}{\partial\nu}\le\frac{\partial v}{\partial\nu},\qquad \frac{\partial u}{\partial\mu_{i}}=\frac{\partial v}{\partial\mu_{i}} \quad \mbox{at } \bigl(x^{*},t^{*}\bigr), $$

where $\frac{\partial}{\partial\mu_{i}}$, $i=1,2,\dots,N-1$, are the tangential derivatives in the local coordinates at $(x^{*},t^{*})$. We can verify that

$$\biggl(\frac{1}{n}+\vert \nabla u\vert ^{2} \biggr)^{\frac{p-2}{2}}\nabla u\cdot\nu= \Biggl(\frac{1}{n}+\sum _{i=1}^{N-1}\biggl\vert \frac{\partial u}{\partial\mu _{i}} \biggr\vert ^{2}+\biggl\vert \frac{\partial u}{\partial\nu}\biggr\vert ^{2} \Biggr)^{\frac{p-2}{2}}\frac{\partial u}{\partial\nu}, $$

which is increasing with respect to $\frac{\partial u}{\partial\nu}$ since $p>1$. Therefore, $B_{n}[u]\le B_{n}[v]$. We arrive at another contradiction. □

Using Lemma 2.1, we can prove the following comparison principle, which is similar to Theorem 2.2 in [8], but without the global one-side Lipschitz condition.

Lemma 2.2

Let $u,v\in C^{2,1}(Q_{T})\cap C(\overline{Q}_{T})$ satisfying

$$\begin{aligned}& u_{t}-F_{n,M}[u] \ge v_{t}-F_{n,M}[v],\quad (x,t)\in Q_{T}, \\& B_{n}[u] \ge B_{n}[v], \quad (x,t)\in\partial\Omega \times(0,T), \\& u(x,0) \ge v(x,0),\quad x\in\overline{\Omega}. \end{aligned}$$

Then

$$u(x,t)\ge v(x,t),\quad (x,t)\in Q_{T}. $$

Proof

For any given $T>0$, $\varepsilon>0$, since $u,v\in C(\overline{\Omega}\times[0,T])$, by the continuities and $u(x,0)\ge v(x,0)$, there exists $\delta=\delta_{\varepsilon}>0$ such that

$$u(x,t)>v(x,t)-\varepsilon,\quad x\in\overline{\Omega}, t\in (0,\delta]. $$

Notice that $v\in C^{2,1}(\overline{\Omega}\times[\delta,T-\varepsilon])$, $v_{+}(x,t)\in[0,\max_{\overline{Q}_{T}}v]$, and $\overset{\rightarrow}{g}\in C^{1}([0,\max_{\overline{Q}_{T}}v])$. There exists a constant $K>0$ such that

$$\begin{aligned}& \bigl\vert \overset{\rightarrow}{g}\bigl((v-s)_{+}\bigr)\cdot\nabla v- \overset{\rightarrow }{g}(v_{+})\cdot\nabla v\bigr\vert \le \sup _{\overline{\Omega}\times[\delta,T-\varepsilon]}\vert \nabla v\vert \cdot\bigl\vert \overset{ \rightarrow}{g}\bigl((v-s)_{+}\bigr)-\overset{\rightarrow}{g}(v_{+})\bigr\vert \le \frac{K}{2}s, \\& \bigl\vert f_{M}(v-s)-f_{M}(v)\bigr\vert \le \frac{K}{2}s,\quad s\ge0. \end{aligned}$$

Define $\varphi=v-\varepsilon e^{(K+1)(t-\delta)}$. Thus,

$$\begin{aligned}& \begin{aligned} \varphi_{t}-F_{n,M}[\varphi]&\le v_{t}-(K+1)\varepsilon e^{(K+1)(t-\delta )}-F_{n,M}[v] +K \varepsilon e^{(K+1)(t-\delta)} \\ &< v_{t}-F_{n,M}[v]\le u_{t}-F_{n,M}[u],\quad (x,t)\in\Omega\times[\delta,T-\varepsilon], \end{aligned} \\& B_{n}[\varphi]=B_{n}[v]-(K+1)\sigma\varepsilon e^{(K+1)(t-\delta )}< B_{n}[v]\le B_{n}[u],\quad (x,t)\in\partial \Omega\times(\delta,T-\varepsilon), \\& \varphi(x,\delta)=v(x,\delta)-\varepsilon< u(x,\delta),\quad x\in \overline{\Omega}. \end{aligned}$$

Lemma 2.1 implies $u(x,t)\ge\varphi(x,t)$ for $(x,t)\in\Omega\times[\delta_{\varepsilon},T-\varepsilon]$. Therefore, $u(x,t)\ge\min\{v(x,t)-\varepsilon,v(x,t)-\varepsilon e^{(K+1)(t-\delta_{\varepsilon})}\}$ for $(x,t)\in\Omega\times(0,T-\varepsilon]$. By the arbitrariness of $\varepsilon>0$, we deduce $u(x,t)\ge v(x,t)$ for $(x,t)\in\Omega\times(0,T)$. □

Lemma 2.3

There exists at most one classical solution of problem (2.1)-(2.3).

Proof

Lemma 2.2 yields the uniqueness of classical solutions of problem (2.1)-(2.3). □

Lemma 2.4

The solution $u_{n,M}$ of problem (2.1)-(2.3) satisfies

$$ \inf_{\Omega}u_{0}\le u_{n,M}(x,t)\le\sup_{\Omega}u_{0}+Mt, \quad (x,t)\in \overline{\Omega}\times(0,T_{n,M}). $$

(2.4)

Thus, the maximal existence time $T_{n,M}=+\infty$.

Proof

For any given $T\in(0,T_{n,M})$ and $\varepsilon>0$, define $\underline{u}_{\varepsilon}=\inf_{\Omega }u_{0}-\varepsilon-\varepsilon t$, $\overline{u}_{\varepsilon}=\sup_{\Omega}u_{0}+\varepsilon +(M+\varepsilon)t$. Then

$$\begin{aligned}& \frac{\partial u_{n,M}}{\partial t}-F_{n,M}[u_{n,M}]=0>-\varepsilon \ge \frac{\partial\underline{u}_{\varepsilon}}{\partial t}-F_{n,M}[\underline{u}_{\varepsilon}], \\& B_{n}[u_{n,M}]=0>-\sigma\varepsilon=B_{n}[ \underline{u}_{\varepsilon}], \\& u_{n,M}(x,0)=u_{0,n}(x)>u_{0}(x)-\varepsilon\ge \underline{u}_{\varepsilon}. \end{aligned}$$

Lemma 2.1 implies $u_{n,M}\ge\underline{u}_{\varepsilon}$. Since $\varepsilon>0$ is arbitrary, we have $u_{n,M}\ge\inf_{\Omega}u_{0}$. The proof of $u_{n,M}\le\sup_{\Omega}u_{0}+Mt$ follows similarly. □

Lemma 2.5

For $M_{1}\ge M_{2}>0$, there holds

$$u_{n,M_{1}}\ge u_{n,M_{2}},\quad x\in\Omega, t>0. $$

Proof

For any given $T>0$, we see that $u_{n,M_{1}}, u_{n,M_{2}}\in C^{2,1}(Q_{T})\cap C(\overline{Q}_{T})$ and $f_{M_{1}}(s)\ge f_{M_{2}}(s)$ for $s\in\mathbb{R}$. Thus,

$$\frac{\partial u_{n,M_{1}}}{\partial t}-F_{n,M_{2}}[u_{n,M_{1}}]\ge\frac {\partial u_{n,M_{1}}}{\partial t}-F_{n,M_{1}}[u_{n,M_{1}}]=0 =\frac{\partial u_{n,M_{2}}}{\partial t}-F_{n,M_{2}}[u_{n,M_{2}}]. $$

Using Lemma 2.2, we complete this proof. □

Lemma 2.6

There exist constants $\delta_{0}, M_{0}>0$ independent of n, M such that

$$\sup_{n,M}u_{n,M}(x,t)\le\sup_{\Omega}u_{0}+M_{0} \delta_{0}, \quad x\in\overline{\Omega}, t\in(0,\delta_{0}]. $$

Proof

Let $\underline{u}_{0}=\inf_{\Omega}u_{0}$ and $\overline{u}_{0}=\sup_{\Omega}u_{0}$. Set $M_{0}=2\max_{s\in\{\underline{u}_{0},\overline{u}_{0}\} }f(s)+1$ and define

$$h(t)=\max_{\underline{u}_{0}\le s\le\overline{u}_{0}+M_{0}t}\bigl\vert f(s)\bigr\vert . $$

Since $f\in C^{1}([\underline{u}_{0}, \overline{u}_{0}+M_{0}])$, we see that h is Lipschitz continuous on $[0,1]$ and $h(0)=\max_{s\in\{\underline{u}_{0},\overline{u}_{0}\}}f(s)< M_{0}$. Thus, there exists a constant $0<\delta_{0}<1$ such that $h(t)< M_{0}$ for all $t\in[0,\delta_{0}]$. By Lemma 2.4, $u_{n,M_{0}}\in[\underline{u}_{0}, \overline{u}_{0}+M_{0}t]$. Therefore,

$$ f\bigl(u_{n,M_{0}}(x,t)\bigr)\le h(t)< M_{0},\quad x\in \overline{\Omega}, t\in [0,\delta_{0}], $$

(2.5)

and

$$f_{M_{0}}\bigl(u_{n,M_{0}}(x,t)\bigr)=\min\bigl\{ f \bigl(u_{n,M_{0}}(x,t)\bigr),M_{0}\bigr\} =f\bigl(u_{n,M_{0}}(x,t) \bigr),\quad (x,t)\in\overline{\Omega}\times(0,\delta_{0}]. $$

If $M'\le M_{0}$, Lemma 2.5 implies

$$u_{n,M'}\le u_{n,M_{0}}\le\sup_{\Omega}u_{0}+M_{0} \delta_{0}, \quad (x,t)\in\overline{\Omega}\times(0,\delta_{0}]. $$

If $M'>M_{0}$, since $u_{n,M'}\in C(\overline{Q}_{\delta_{0}})$ and $u_{n,M'}(x,0)=u_{0,n}(x)\in[\underline{u}_{0}, \overline{u}_{0}]$, we have

$$f\bigl(u_{n,M'}(x,0)\bigr)\le h(0)< M_{0},\quad x\in\overline{\Omega}, $$

and there exists a constant $\delta_{M'}>0$ such that

$$ f\bigl(u_{n,M'}(x,t)\bigr)< M_{0}, \quad (x,t)\in \overline{\Omega}\times(0,\delta_{M'}]. $$

(2.6)

Thus,

$$f_{M_{0}}\bigl(u_{n,M'}(x,t)\bigr)=f\bigl(u_{n,M'}(x,t) \bigr)=f_{M'}\bigl(u_{n,M'}(x,t)\bigr),\quad (x,t)\in\overline{\Omega}\times(0,\delta_{M'}]. $$

We see that $u_{n,M_{0}}$, $u_{n,M'}$ are two classical solutions of problem (2.1)-(2.3) with $M=M_{0}$ on $\overline{\Omega}\times(0,\delta_{M'}]$. According to the uniqueness, Lemma 2.3, we have

$$u_{n,M_{0}}(x,t)=u_{n,M'}(x,t), \quad (x,t)\in\overline{\Omega}\times (0, \delta_{M'}]. $$

By the continuity of $u_{n,M'}(x,t)$ and inequality (2.5), we can take $\delta_{M'}=\delta_{0}$ in inequality (2.6). Then we have

$$u_{n,M'}(x,t)=u_{n,M_{0}}(x,t)\le\sup_{\Omega}u_{0}+M_{0} \delta_{0},\quad (x,t)\in\overline{\Omega}\times(0,\delta_{0}]. $$

We arrive at a locally uniform bound of $u_{n,M}$. □

Remark

Lemma 2.4 shows that $u_{n,M}\le\sup_{\Omega}u_{0}+Mt$. However, the family $\{\sup_{\Omega}u_{0}+Mt\}_{M>0}$ is not uniformly bounded on any interval $(0,\delta]$, $\delta>0$. Lemma 2.6 provides the locally uniform bound of $u_{n,M}$.

Next, we derive some estimates on the solution $u_{n,M}$.

Lemma 2.7

Suppose that σ does not depend on time. For any given $T>0$, $M>0$, there exists a constant $C=C(M,T)$ independent of n such that

$$\int_{\Omega}u_{n,M}^{2}(x,t)\,dx,\qquad \int_{\partial\Omega} \sigma u_{n,M}^{2}(x,t)\,dS, \qquad \int_{Q_{T}}\vert \nabla u_{n,M}\vert ^{p} \,dx\,dt \le C,\quad t\in(0,T). $$

Moreover, if $T=\delta_{0}$ (the constant in Lemma  2.6), then the constant $C=C(\delta_{0})$ is independent of n, M.

Proof

We write $u=u_{n,M}$ in this proof for the sake of convenience. Since $u\in C^{2,1}(Q_{T})\cap C(\overline{Q}_{T})$, multiplying equation (2.1) by u and integrating by parts over $Q_{\tau}$, $\tau\in(0,T]$, we have

$$\begin{aligned} &\int_{Q_{\tau}}uu_{t} \,dx\,dt+\int _{Q_{\tau}}\biggl(\frac{1}{n}+\vert \nabla u\vert ^{2}\biggr)^{\frac{p-2}{2}}\vert \nabla u\vert ^{2}\,dx\,dt \\ &\qquad {}-\int_{0}^{\tau}\int _{\partial\Omega}u\biggl(\frac{1}{n}+\vert \nabla u\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla u\cdot\nu\,dS\,dt \\ &\quad =-\int_{Q_{\tau}}\bigl(\overset{\rightarrow}{g}(u_{+})\cdot \nabla u\bigr)u \,dx\, dt+\int_{Q_{\tau}}f_{M}(u)u \,dx \,dt. \end{aligned}$$

Using the dynamic boundary condition (2.2), we conclude

$$\begin{aligned} &\int_{Q_{\tau}}uu_{t} \,dx\,dt+\int _{Q_{\tau}}\biggl(\frac{1}{n}+\vert \nabla u\vert ^{2}\biggr)^{\frac{p-2}{2}}\vert \nabla u\vert ^{2}\,dx\,dt +\int_{0}^{\tau}\int _{\partial\Omega}\sigma uu_{t} \,dS\,dt \\ &\quad =-\int_{Q_{\tau}}\bigl(\overset{\rightarrow}{g}(u_{+})\cdot \nabla u\bigr)u \,dx\, dt+\int_{Q_{\tau}}f_{M}(u)u \,dx \,dt. \end{aligned}$$

That is,

$$\begin{aligned} &\frac{1}{2}\int_{\Omega}u^{2}(x, \tau)\,dx +\int_{Q_{\tau}}\biggl(\frac{1}{n}+\vert \nabla u \vert ^{2}\biggr)^{\frac{p-2}{2}}\vert \nabla u\vert ^{2}\, dx\,dt +\frac{1}{2}\int_{\partial\Omega}\sigma u^{2}(x,\tau)\,dS \\ &\quad =\frac{1}{2}\int_{\Omega}u_{0,n}^{2}(x) \,dx+\frac{1}{2}\int_{\partial \Omega }\sigma u_{0,n}^{2}(x) \,dS -\int_{Q_{\tau}}\bigl(\overset{\rightarrow}{g}(u_{+})\cdot\nabla u \bigr)u \,dx\, dt+\int_{Q_{\tau}}f_{M}(u)u \,dx\,dt. \end{aligned}$$

Notice that $u_{0,n}\le\sup_{\Omega}u_{0}$,

$$\biggl\vert \int_{Q_{\tau}}\bigl(\overset{\rightarrow}{g}(u_{+}) \cdot\nabla u\bigr)u \, dx\,dt \biggr\vert \le \frac{1}{4}\int _{Q_{\tau}} \vert \nabla u\vert ^{p} \,dx\,dt+C\int _{Q_{\tau}}\bigl(\bigl\vert \overset {\rightarrow}{g}(u_{+})\bigr\vert u\bigr)^{\frac{p}{p-1}}\,dx\,dt, $$

and

$$\begin{aligned} &\vert \nabla u\vert ^{p}\le\biggl(\frac{1}{n}+\vert \nabla u\vert ^{2}\biggr)^{\frac{p-2}{2}}\vert \nabla u\vert ^{2},\quad p\ge2, \\ &\vert \nabla u\vert ^{p}\le2^{\frac{2-p}{2}}\biggl( \frac{1}{n}+\vert \nabla u\vert ^{2}\biggr)^{\frac{p-2}{2}}\vert \nabla u\vert ^{2}+\biggl(\frac{1}{n} \biggr)^{\frac{p}{2}} \\ &\hphantom{\vert \nabla u\vert ^{p}} \le2\biggl(\frac{1}{n}+\vert \nabla u\vert ^{2}\biggr)^{\frac{p-2}{2}}\vert \nabla u\vert ^{2}+1, \quad 1< p< 2. \end{aligned}$$

Lemma 2.4 implies $\vert u\vert \le\sup_{\Omega }u_{0}+MT$ for $(x,t)\in Q_{T}$. Therefore,

$$\int_{\Omega}u^{2}(x,t)\,dx,\qquad \int_{\partial\Omega} \sigma u^{2}(x,t)\,dS, \qquad \int_{Q_{T}}\vert \nabla u\vert ^{p} \,dx\,dt\le C(M,T). $$

If $T=\delta_{0}$, Lemma 2.6 shows $\vert u\vert \le\sup_{\Omega}u_{0}+M_{0}\delta_{0}$ for $(x,t)\in Q_{\delta_{0}}$, which is a uniform bound independent of n, M. □

Lemma 2.8

Suppose that σ does not depend on time and $p\ge2$. For any given $T>0$, $M>0$, there exists a constant $C=C(M,T)$ independent of n such that

$$\int_{\Omega} \vert \nabla u_{n,M}\vert ^{p} \,dx,\qquad \int_{Q_{T}} \biggl\vert \frac{\partial u_{n,M}}{\partial t} \biggr\vert ^{2}\,dx\,dt, \qquad \int_{0}^{T} \int_{\Omega}\sigma \biggl\vert \frac{\partial u_{n,M}}{\partial t} \biggr\vert ^{2}\,dS\,dt \le C. $$

Moreover, if $T=\delta_{0}$ (the constant in Lemma  2.6), then the constant $C=C(\delta_{0})$ is independent of n, M.

Proof

We write $u=u_{n,M}$ in this proof for the sake of convenience. Since $f_{M}$, $g(s_{+})$ are Lipschitz continuous, the classical regularity results in [9] imply that $u_{t}\in C^{1,0}(\overline{\Omega}\times(0,T))$. Multiplying equation (2.1) by $u_{t}$ and integrating over $Q_{\tau}$, we have

$$\begin{aligned} &\int_{Q_{\tau}}u_{t}^{2}\,dx\,dt+ \int_{Q_{\tau}}\biggl(\frac{1}{n}+\vert \nabla u\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla u\cdot\nabla u_{t} \,dx\,dt +\int_{0}^{\tau}\int_{\partial\Omega} \sigma u_{t}^{2}\,dS\,dt \\ &\quad =-\int_{Q_{\tau}}\bigl(\overset{\rightarrow}{g}(u_{+})\cdot \nabla u\bigr)u_{t} \, dx\,dt+\int_{Q_{\tau}}f_{M}(u)u_{t} \,dx\,dt. \end{aligned}$$

Next, we show that

$$\begin{aligned} &\int_{Q_{\tau}}\biggl(\frac{1}{n}+\vert \nabla u\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla u\cdot \nabla u_{t} \,dx\,dt \\ &\quad =\int_{Q_{\tau}}\frac{1}{2}\frac{\partial}{\partial t} \int_{0}^{\vert \nabla u(x,t)\vert ^{2}}\biggl(s+\frac {1}{n} \biggr)^{\frac{p-2}{2}}\,ds\,dx\,dt \\ &\quad =\int_{Q_{\tau}} \frac{1}{p}\frac{\partial}{\partial t} \biggl(\biggl(\frac {1}{n}+\bigl\vert \nabla u(x,t)\bigr\vert ^{2}\biggr)^{\frac{p}{2}}-\biggl( \frac{1}{n}\biggr)^{\frac{p}{2}} \biggr)\,dx\,dt \\ &\quad =\frac{1}{p}\int_{\Omega}\biggl( \frac{1}{n}+\bigl\vert \nabla u(x,\tau)\bigr\vert ^{2} \biggr)^{\frac{p}{2}}\,dx- \frac{1}{p}\int_{\Omega}\biggl(\frac{1}{n}+\bigl\vert \nabla u_{0,n}(x)\bigr\vert ^{2}\biggr)^{\frac{p}{2}}\,dx \\ &\quad \ge\frac{1}{p}\int_{\Omega}\bigl\vert \nabla u(x,\tau)\bigr\vert ^{p} \,dx -\frac{1}{p}2^{\frac{p}{2}}\int_{\Omega} \vert \nabla u_{0,n}\vert ^{p} \,dx-\frac {1}{p}2^{\frac{p}{2}}\vert \Omega \vert . \end{aligned}$$

Young’s inequality yields

$$\begin{aligned} &\biggl\vert -\int_{Q_{\tau}}\bigl(\overset{ \rightarrow}{g}(u_{+})\cdot\nabla u\bigr)u_{t} \,dx\,dt \biggr\vert \le \frac{1}{4}\int_{Q_{\tau}}u_{t}^{2}\,dx \,dt+\int_{Q_{\tau}}\bigl\vert \overset {\rightarrow}{g}(u_{+})\bigr\vert ^{2}\vert \nabla u\vert ^{2}\,dx\,dt \\ &\hphantom{\biggl\vert -\int_{Q_{\tau}}\bigl(\overset{ \rightarrow}{g}(u_{+})\cdot\nabla u\bigr)u_{t} \,dx\,dt \biggr\vert }\le\frac{1}{4}\int_{Q_{\tau}}u_{t}^{2} \,dx\,dt+C(M,T)\int_{Q_{\tau}} \vert \nabla u\vert ^{p} \,dx\,dt,\quad p\ge2, \\ &\biggl\vert \int_{Q_{\tau}}f_{M}(u)u_{t} \,dx\,dt \biggr\vert \le\frac{1}{4}\int_{Q_{\tau}}u_{t}^{2} \,dx\,dt+\int_{Q_{\tau}}f_{M}^{2}(u)\,dx\,dt. \end{aligned}$$

We conclude the estimate. □

Now, we define the following two types of weak solutions of problem (1.1)-(1.3).

Definition 2.1

A function $u\in L^{p}((0,T);W^{1,p}(\Omega))$ is called a local weak solution of problem (1.1)-(1.3) if the integral equality

$$\begin{aligned} &{-}\int_{\Omega}u_{0}\varphi\,dx-\int _{Q_{T}}u\varphi_{t} \,dx\,dt+\int_{Q_{T}} \vert \nabla u\vert ^{p-2}\nabla u\cdot\nabla\varphi\,dx\,dt -\int _{0}^{T}\int_{\partial\Omega}u(\sigma \varphi)_{t} \,dS\,dt \\ &\quad =-\int_{Q_{T}}\bigl(\overset{\rightarrow}{g}(u) \cdot\nabla u\bigr)\varphi\,dx\, dt+\int_{Q_{T}}f(u)\varphi\,dx \,dt \end{aligned}$$

(2.7)

holds for any $\varphi\in C^{\infty}(\overline{Q}_{T})$ that satisfies $\varphi(x,T)=0$ for $x\in\overline{\Omega}$, $\varphi(x,0)=0$ for $x\in \partial\Omega$.

Definition 2.2

A function $u\in L^{p}((0,T);W^{1,p}(\Omega))$ is called a local strong solution of problem (1.1)-(1.3) if $u_{t}\in L^{2}(Q_{T})$, u is the a.e. limit function of a subsequence $\{u_{n_{k},M_{k}}\}$ of classical solutions to the regularized problem (2.1)-(2.3), and the integral equality (2.7) holds for any $\varphi\in C^{\infty}(\overline{Q}_{T})$ that satisfies $\varphi(x,T)=0$ for $x\in\overline{\Omega}$, $\varphi(x,0)=0$ for $x\in \partial\Omega$.

Theorem 2.1

Suppose that σ does not depend on time. Problem (1.1)-(1.3) admits at least one local weak solution.

Proof

For any $T>0$, $\varphi\in C^{\infty}(\overline{Q}_{T})$ that satisfies $\varphi(x,T)=0$ for $x\in\overline{\Omega}$, $\varphi(x,0)=0$ for $x\in \partial\Omega$, multiplying (2.1) by φ, integrating over $Q_{T}$, we have

$$\begin{aligned} &\int_{Q_{T}}\frac{\partial u_{n,M}}{\partial t}\varphi\,dx\,dt +\int _{Q_{T}}\biggl(\frac{1}{n}+\vert \nabla u_{n,M} \vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla u_{n,M}\cdot \nabla\varphi\,dx\,dt \\ &\qquad {}-\int_{0}^{T}\int_{\partial\Omega} \biggl(\frac{1}{n}+\vert \nabla u_{n,M}\vert ^{2} \biggr)^{\frac{p-2}{2}}\nabla u_{n,M}\cdot\nu\varphi\,dS\,dt \\ &\quad =-\int_{Q_{T}}\bigl(\overset{\rightarrow}{g} \bigl((u_{n,M})_{+}\bigr)\cdot\nabla u_{n,M}\bigr)\varphi\,dx\,dt +\int_{Q_{T}}f_{M}(u_{n,M})\varphi\,dx\,dt. \end{aligned}$$

By the dynamic boundary condition (2.2), we obtain

$$\begin{aligned} &-\int_{0}^{T}\int_{\partial\Omega} \biggl(\frac{1}{n}+\vert \nabla u_{n,M}\vert ^{2} \biggr)^{\frac{p-2}{2}}\nabla u_{n,M}\cdot\nu\varphi\,dS\,dt \\ &\quad =\int_{0}^{T}\int_{\partial\Omega} \sigma\frac{\partial u_{n,M}}{\partial t}\varphi\,dS\,dt =-\int_{0}^{T} \int_{\partial\Omega}u_{n,M}(\sigma\varphi)_{t} \,dS \,dt. \end{aligned}$$

Thus,

$$\begin{aligned} &{-}\int_{\Omega}u_{0,n}\varphi\,dx-\int _{Q_{T}}u_{n,M}\varphi_{t} \,dx\,dt +\int _{Q_{T}}\biggl(\frac{1}{n}+\vert \nabla u_{n,M} \vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla u_{n,M}\cdot \nabla\varphi\,dx\,dt \\ &\quad =\int_{0}^{T}\int_{\partial\Omega}u_{n,M}( \sigma\varphi)_{t} \,dS\,dt -\int_{Q_{T}}\bigl( \overset{\rightarrow}{g}(u_{n,M})\cdot\nabla u_{n,M}\bigr)\varphi \,dx\,dt+\int_{Q_{T}}f_{M}(u_{n,M})\varphi \,dx\,dt. \end{aligned}$$

By the uniform estimates in Lemma 2.6 and Lemma 2.7, there exist a subsequence $\{u_{n_{k},M_{k}}\}$ ($n_{k}\to\infty$, $M_{k}\to \infty$, as $k\to\infty$) and a function $u\in L^{p}((0,\delta_{0});W^{1,p}(\Omega))$ such that $u_{n_{k},M_{k}}$ converges weakly to u in $L^{2}(Q_{\delta_{0}})$, $\nabla u_{n_{k},M_{k}}$ converges weakly to ∇u in $L^{p}(Q_{\delta_{0}})$, and $u_{n_{k},M_{k}}$ converges weakly to u in $L^{p}(\partial\Omega \times (0,\delta_{0}))$ in the sense of trace. Hence the above integral equality converges to (2.7) for $T=\delta_{0}$ and u is a local weak solution to problem (1.1)-(1.3). □

Theorem 2.2

Suppose that σ does not depend on time and $p\ge2$. Problem (1.1)-(1.3) admits at least one local strong solution.

Proof

By the uniform estimates in Lemma 2.6, Lemma 2.7, and Lemma 2.8, the norms $\Vert u_{n,M}\Vert _{H^{1}(Q_{\delta_{0}})}$, $\Vert \nabla u_{n,M}\Vert _{L^{p}(Q_{\delta _{0}})}$ are uniformly bounded. There exist a subsequence $\{u_{n_{k},M_{k}}\}$ ($n_{k}\to\infty$, $M_{k}\to \infty$, as $k\to\infty$) and a function $u\in L^{p}((0,\delta_{0});W^{1,p}(\Omega))$, $u_{t}\in L^{2}(Q_{\delta_{0}})$ such that $\{u_{n_{k},M_{k}}\}$ converges weakly to u in $H^{1}(Q_{\delta_{0}})$, $\nabla u_{n_{k},M_{k}}$ converges weakly to ∇u in $L^{p}(Q_{\delta_{0}})$. Hence $u_{n_{k},M_{k}}\to u$ almost everywhere and the integral equality (2.7) holds. □

Remark

For any given $M>0$, by the estimates in Lemma 2.7 and Lemma 2.8, using the diagonal procedure, we can choose a subsequence $\{u_{n_{k},M}\}$ ($\{n_{k}\}$ might depend on M) and a function $u_{M}$ such that $u_{n_{k},M}$ converges to $u_{M}$ on $Q_{T}$ for any $T>0$ in the manner stated in the proof of Theorem 2.2. Furthermore, we can verify that $u_{M}$ is the global strong solution to the following equation:

$$\frac{\partial u}{\partial t}=\operatorname {div}\bigl(\vert \nabla u\vert ^{p-2}\nabla u\bigr)-\overset{\rightarrow }{g}(u)\cdot\nabla u+f_{M}(u) $$

coupled with the initial-boundary value conditions (1.2)-(1.3). Using the diagonal procedure again, we can choose a subsequence $\{n_{k}\}$ independent of M and then choose $u_{M}$ such that $u_{n_{k},M}$ converges to $u_{M}$ for any $M\in \mathbb{Z}^{+}$ in the same manner. Lemma 2.5 implies $u_{n_{k},M_{1}}\ge u_{n_{k},M_{2}}$ for $M_{1}\ge M_{2}$. Thus, $\{u_{M}\}_{M\in\mathbb{Z}^{+}}$ is monotone with respect to M. Define

$$T^{*}=\sup\Bigl\{ T>0;\sup_{M\in\mathbb{Z}^{+}}\sup_{(x,t)\in\overline{\Omega}\times (0,T)}u_{M}(x,t)< \infty\Bigr\} , $$

and

$$u(x,t)=\sup_{M\in\mathbb{Z}^{+}}u_{M}(x,t), \quad (x,t)\in\overline{\Omega}\times\bigl(0,T^{*}\bigr). $$

Lemma 2.6 shows $T^{*}\ge\delta_{0}$. Similar to the proof of Theorem 2.1 and Theorem 2.2, we can prove that u is a strong solution to problem (1.1)-(1.3) with maximal existence time $T^{*}$.

3 Global existence

In this section, we study the global existence of local strong solutions to problem (1.1)-(1.3) defined in Section 2. We need to find an appropriate upper-solution to the regularized problem (2.1)-(2.3) which is independent of n, M and exists globally. If $p=2$, the p-Laplacian is reduced to Laplacian, so we only consider $p>2$ in this section.

Lemma 3.1

Let $\alpha=\frac{p-1}{p-2}$, $p>2$, $K>0$, $\eta\in C^{1}([0,+\infty))$. For a fixed integer $1\le j\le N$, define $\underline{x}_{j}=\min_{\overline{\Omega}}x_{j}$, $\overline{x}_{j}=\max_{\overline{\Omega}}x_{j}$, and

$$U(x,t)=\frac{1}{\alpha}\bigl(Ke^{\eta(t)}+x_{j}-\overline{x}_{j}\bigr)^{\alpha},\quad x\in\overline{\Omega}, t\ge0. $$

Then U is an upper solution of the regularized problem (2.1)-(2.3) provided

$$\begin{aligned}& Ke^{\eta(0)}+\underline{x}_{j}-\overline{x}_{j}\ge1,\qquad \frac{1}{\alpha}\bigl(Ke^{\eta(0)}+\underline{x}_{j}-\overline{x}_{j}\bigr)^{\alpha}\ge\sup_{\Omega}u_{0}, \\& \eta'(t)\ge\alpha2^{p}, \qquad \sigma(x,t)\eta'(t) \ge2^{p},\quad x\in \partial\Omega, t\ge0, \end{aligned}$$

and

$$g_{j}(s)s^{\frac{1}{p-1}}\ge f(s),\qquad s\ge\frac{1}{\alpha} \bigl(Ke^{\eta(0)}+\underline{x}_{j}-\overline{x}_{j} \bigr)^{\alpha}. $$

Proof

By a simple computation, we have

$$\begin{aligned} &\operatorname {div}\biggl(\biggl(\frac{1}{n}+\vert \nabla u \vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla u\biggr) \\ &\quad =\biggl( \frac{1}{n}+\vert \nabla u\vert ^{2}\biggr)^{\frac{p-2}{2}}\Delta u +(p-2) \biggl(\frac{1}{n}+\vert \nabla u\vert ^{2}\biggr)^{\frac{p-2}{2}-1}\frac{\partial u}{\partial x_{i}}\frac{\partial u}{\partial x_{j}} \frac{\partial^{2} u}{\partial x_{i}\,\partial x_{j}}. \end{aligned}$$

Notice that $\alpha>1$ and $(\alpha-1)(p-1)=\alpha$. We show that

$$\begin{aligned}& \vert \nabla U\vert =\frac{\partial U}{\partial x_{j}}=\bigl(Ke^{\eta (t)}+x_{j}- \overline{x}_{j}\bigr)^{\alpha-1}\ge1, \\& U_{t}=\bigl(Ke^{\eta(t)}+x_{j}-\overline{x}_{j}\bigr)^{\alpha-1}Ke^{\eta(t)}\eta '(t)\ge \bigl(Ke^{\eta(t)}+x_{j}-\overline{x}_{j} \bigr)^{\alpha}\eta'(t), \\& \biggl\vert \biggl(\frac{1}{n}+\vert \nabla U\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla U\cdot\nu\biggr\vert \le 2^{\frac{p-2}{2}}\vert \nabla U\vert ^{p-1} \le2^{p}\bigl(Ke^{\eta(t)}+x_{j}-\overline{x}_{j}\bigr)^{\alpha}, \\& \operatorname {div}\biggl(\biggl(\frac{1}{n}+\vert \nabla U\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla U\biggr) \le2^{\frac{p-2}{2}}\vert \nabla U\vert ^{p-2}\Delta U+(p-2) \max\bigl\{ 2^{\frac {p-2}{2}-1},1\bigr\} \vert \nabla U\vert ^{p-2} \frac{\partial^{2} U}{\partial x_{j}^{2}} \\& \hphantom{\operatorname {div}\biggl(\biggl(\frac{1}{n}+\vert \nabla U\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla U\biggr)}\le\alpha2^{p}\bigl(Ke^{\eta(t)}+x_{j}- \overline{x}_{j}\bigr)^{\alpha-1}. \end{aligned}$$

Thus,

$$\begin{aligned}& B_{n}[U]=\sigma U_{t}-\biggl(\frac{1}{n}+\vert \nabla U\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla U\cdot \nu \ge0, \\& U_{t}-F_{n,M}[U]=U_{t}-\operatorname {div}\biggl(\biggl(\frac{1}{n}+\vert \nabla U\vert ^{2} \biggr)^{\frac{p-2}{2}}\nabla U\biggr) +\overset{\rightarrow}{g}(U)\cdot \nabla U-f(U) \\& \hphantom{U_{t}-F_{n,M}[U]} \ge g_{j}(U) (\alpha U)^{\frac{1}{p-1}}-f(U)\ge0,\quad x \in\Omega, t>0, \end{aligned}$$

and

$$U(x,0)=\frac{1}{\alpha}\bigl(Ke^{\eta(0)}+x_{j}-\overline{x}_{j}\bigr)^{\alpha}\ge \sup_{\Omega}u_{0} \ge u_{0,n}(x), \quad x\in\overline{\Omega}. $$

Lemma 2.2 implies that $U(x,t)$ is an upper solution of problem (2.1)-(2.3). □

Now we give some conditions on the functions f, g, and σ, which ensure the global existence of local solutions.

Theorem 3.1

Suppose $p>2$, $(\inf_{x\in\partial\Omega}\sigma(x,\cdot ))^{-1}\in L_{\mathrm{loc}}^{1}([0,+\infty))$, there exist an integer $1\le j\le N$ and a constant $M>1$ such that

$$g_{j}(s)s^{\frac{1}{p-1}}\ge f(s), \quad s\ge M. $$

Then the strong solution of problem (1.1)-(1.3) is a global solution.

Proof

Take $K=\max\{1,(\alpha\sup_{\Omega}u_{0})^{\frac{1}{\alpha}}, (\alpha M)^{\frac{1}{\alpha}}\}+\overline{x}_{j}-\underline{x}_{j}$, and define

$$\eta(t)=2^{p}\int_{0}^{t} \Bigl(\inf _{x\in\partial\Omega}\sigma(x,\tau ) \Bigr)^{-1}\,d\tau+ \alpha2^{p}t, $$

where α, $\overline{x}_{j}$, $\underline{x}_{j}$ are the constants defined in Lemma 3.1. Thus, $U(x,t)=\frac{1}{\alpha}(Ke^{\eta(t)}+x_{j}-\overline{x}_{j})^{\alpha}$ is an upper solution to the regularized problem (2.1)-(2.3) for any $n\in\mathbb{Z}^{+}$ and $M>0$. That is, $u_{n,M}(x,t)\le U(x,t)$ for $x\in\overline{\Omega}$ and $t\ge0$. According to the definition of strong solution, we have $u(x,t)\le U(x,t)$. Hence u does not blow up in finite time. □

4 Blow-up

In this section, we investigate the blow-up phenomenon of problem (1.1)-(1.3). We need to construct a family of lower solutions of the regularized problem (2.1)-(2.3) whose supremum blows up in finite time.

Lemma 4.1

Suppose that $p>2$, Ω is a convex domain, and there exist constants $C_{1}, C_{2}>0$ such that

$$f(s)\ge C_{1}s^{p-1}, \qquad \bigl\vert \overset{\rightarrow}{g}(s) \bigr\vert \le C_{2}s^{p-2}, \quad s\ge0. $$

Choose $x_{0}\in\Omega$ with $B_{r}(x_{0})\subset\Omega$, $r>0$. For $A,B>0$ and $\varphi_{M}\in C^{1}([0,+\infty))$, define

$$v_{M}(x,t)=\bigl(A-B\vert x-x_{0}\vert ^{2} \bigr)\varphi_{M}(t), \quad x\in\overline{\Omega}, t\ge0. $$

Then the function $v_{M}$ is a lower solution of the regularized problem (2.1)-(2.3) provided

$$\begin{aligned}& A\ge2Bd^{2}, \qquad 2Bd\varphi_{M}(0)\ge1,\qquad A\varphi_{M}(0) \le\inf_{\Omega}u_{0}, \qquad C_{1}\bigl(A \varphi_{M}(t)\bigr)^{p-1}\le M, \\& \varphi_{M}'\ge0, \qquad \sigma A\varphi_{M}' \le(2Br)^{p-1}\delta\varphi_{M}^{p-1},\qquad A \varphi_{M}'\le K\varphi_{M}^{p-1}, \end{aligned}$$

where $d=\sup_{\Omega} \vert x-x_{0}\vert$, $\delta=\inf_{\partial\Omega}\frac{x-x_{0}}{\vert x-x_{0}\vert }\cdot\nu >0$ (by the convexity of Ω), and

$$K=\frac{1}{2}C_{1}\biggl(\frac{1}{2}A \biggr)^{p-1}-2^{p}(2B)^{p-1}d^{p-2}(N+p)- \biggl(\frac {1}{2}C_{1}\biggr)^{-(p-2)}C_{2}^{p-1}(2Bd)^{p-1}. $$

Proof

Let $\rho(x)=\vert x-x_{0}\vert$. A direct calculation shows

$$\begin{aligned}& \nabla v_{M}=-2B\varphi_{M}(x-x_{0}),\qquad \frac{\partial v_{M}}{\partial t}=\bigl(A-B\rho^{2}\bigr)\varphi_{M}'(t), \\& \biggl(\frac{1}{n}+\vert \nabla v_{M}\vert ^{2} \biggr)^{\frac{p-2}{2}}\nabla v_{M} =-\biggl( \frac{1}{n}+(2B)^{2}\rho^{2}\varphi_{M}^{2} \biggr)^{\frac{p-2}{2}}2B\varphi_{M}(x-x_{0}), \\& \operatorname {div}\biggl(\biggl(\frac{1}{n}+\vert \nabla v_{M}\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla v_{M} \biggr) =-\biggl(\frac{1}{n}+(2B)^{2} \rho^{2}\varphi_{M}^{2}\biggr)^{\frac{p-2}{2}}2NB\varphi_{M} \\& \hphantom{\operatorname {div}\biggl(\biggl(\frac{1}{n}+\vert \nabla v_{M}\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla v_{M} \biggr) =}-(p-2) \biggl(\frac{1}{n}+(2B)^{2}\rho^{2} \varphi_{M}^{2}\biggr)^{\frac {p-2}{2}-1}(2B)^{2}\rho ^{2}\varphi_{M}^{2}{2B}\varphi_{M} \\& \hphantom{\operatorname {div}\biggl(\biggl(\frac{1}{n}+\vert \nabla v_{M}\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla v_{M} \biggr) }=-\biggl(\frac{1}{n}+(2B)^{2}\rho^{2} \varphi_{M}^{2}\biggr)^{\frac{p-2}{2}-1}2B\varphi_{M} \\& \hphantom{\operatorname {div}\biggl(\biggl(\frac{1}{n}+\vert \nabla v_{M}\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla v_{M} \biggr)= }\cdot \biggl(\biggl(\frac{1}{n}+(2B)^{2} \rho^{2}\varphi_{M}^{2}\biggr)N+(p-2) (2B)^{2}\rho ^{2}\varphi_{M}^{2} \biggr). \end{aligned}$$

Thus, we have

$$\begin{aligned} \biggl\vert \operatorname {div}\biggl(\biggl(\frac{1}{n}+ \vert \nabla v_{M}\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla v_{M} \biggr)\biggr\vert &\le\biggl( \frac{1}{n}+(2B)^{2}\rho^{2}\varphi_{M}^{2} \biggr)^{\frac {p-2}{2}}(N+p-1)2B\varphi_{M} \\ &\le\bigl(1+(2Bd)^{2}\varphi_{M}^{2} \bigr)^{\frac{p-2}{2}}(N+p-1)2B\varphi_{M} \\ &\le2^{p}(2B)^{p-1}d^{p-2}(N+p) \varphi_{M}^{p-1},\quad x\in\overline{\Omega}, t\le0, \end{aligned}$$

and

$$\begin{aligned} \biggl(\frac{1}{n}+\vert \nabla v_{M}\vert ^{2}\biggr)^{\frac{p-2}{2}}\nabla v_{M}\cdot\nu &=- \biggl(\frac{1}{n}+(2B)^{2}\rho^{2}\varphi_{M}^{2} \biggr)^{\frac{p-2}{2}}2B\varphi _{M}\rho \frac{x-x_{0}}{\vert x-x_{0}\vert } \cdot\nu \\ &\le -\biggl(\frac{1}{n}+(2B)^{2}\rho^{2} \varphi_{M}^{2}\biggr)^{\frac{p-2}{2}}2B\varphi _{M}\rho \delta \\ &\le -(2Br)^{p-1}\delta\varphi_{M}^{p-1},\quad x\in\partial\Omega, t\ge0. \end{aligned}$$

Young’s inequality shows

$$\bigl\vert \overset{\rightarrow}{g}(v_{M})\cdot\nabla v_{M}\bigr\vert \le C_{2}v_{M}^{p-2} \vert \nabla v_{M}\vert \le\frac{1}{2}C_{1}v_{M}^{p-1}+ \biggl(\frac{1}{2}C_{1}\biggr)^{-(p-2)}C_{2}^{p-1} \vert \nabla v_{M}\vert ^{p-1}. $$

We obtain

$$f_{M}(v_{M})=\min\bigl\{ M,f(v_{M})\bigr\} \ge C_{1}v_{M}^{p-1}, $$

and

$$\begin{aligned} f_{M}(v_{M})-\overset{\rightarrow}{g}\cdot\nabla v_{M} &\ge\frac{1}{2}C_{1}v_{M}^{p-1}- \biggl(\frac {1}{2}C_{1}\biggr)^{-(p-2)}C_{2}^{p-1} \vert \nabla v_{M}\vert ^{p-1} \\ &\ge\frac{1}{2}C_{1}\biggl(\frac{1}{2}A \varphi_{M}\biggr)^{p-1}-\biggl(\frac {1}{2}C_{1} \biggr)^{-(p-2)}C_{2}^{p-1}(2Bd\varphi_{M})^{p-1}. \end{aligned}$$

Furthermore,

$$\begin{aligned}& \frac{\partial v_{M}}{\partial t}-F_{n,M}[v_{M}]\le A\varphi_{M}'-K \varphi _{M}^{p-1}\le0,\quad (x,t)\in\partial\Omega\times \mathbb{R}^{+}, \\& B_{n}[v_{M}]=\sigma\frac{\partial v_{M}}{\partial t}+\biggl( \frac{1}{n}+\vert \nabla v_{M}\vert ^{2} \biggr)^{\frac{p-2}{2}}\frac{\partial v_{M}}{\partial\nu}\\& \hphantom{B_{n}[v_{M}]} \le\sigma A\varphi_{M}'-(2Br)^{p-1} \delta\varphi_{M}^{p-1}\le0, \quad (x,t)\in\partial\Omega\times \mathbb{R}^{+}, \\& v_{M}(x,0)\le A\varphi_{M}(0)\le\inf_{\Omega}u_{0}\le u_{0,n}(x),\quad x\in \overline{\Omega}. \end{aligned}$$

Lemma 2.2 implies that $v_{M}$ is a lower solution of problem (2.1)-(2.3). □

Theorem 4.1

Suppose that $p>2$, Ω is a convex domain, $\sigma\in L^{\infty }(\partial\Omega\times\mathbb{R}^{+})$, and there exist constants $C_{1}, C_{2}>0$ such that

$$f(s)\ge C_{1}s^{p-1},\qquad \bigl\vert \overset{\rightarrow}{g}(s) \bigr\vert \le C_{2}s^{p-2},\quad s\ge0. $$

Then the strong solution of problem (1.1)-(1.3) blows up in finite time provided that $\inf_{\Omega}u_{0}$ is sufficiently large.

Proof

Let $x_{0}$, r, d, δ, K be as defined in Lemma 4.1. Since $K=K(A,B)$ converges to $K(A,0)=\frac{1}{2}C_{1}(\frac {1}{2}A)^{p-1}>0$ as B tends to 0⁺, we can choose $A,B>0$ such that

$$C_{1}A^{p-1}=1, \qquad 2Bd^{2}\le A, \qquad K=K(A,B)>0. $$

Then set

$$\varphi_{M}(0)=\frac{1}{2Bd}>0,\qquad \overline{\sigma}=\sup _{\partial \Omega \times\mathbb{R}^{+}}\sigma,\qquad \gamma=\min\biggl\{ \frac{K}{A}, \frac{(2Br)^{p-1}\delta}{\overline{\sigma}A}\biggr\} . $$

By Lemma 4.1, the function $v_{M}=(A-B\vert x-x_{0}\vert ^{2})\varphi_{M}(t)$ is a lower solution provided

$$\inf_{\Omega}u_{0}\ge\frac{A}{2Bd},\qquad \varphi_{M}(t)\le M^{\frac{1}{p-1}},\qquad 0\le\varphi_{M}'(t) \le\gamma\varphi_{M}^{p-1}(t). $$

Define

$$\varphi_{M}(t)=\min\bigl\{ \bigl(\varphi_{M}^{2-p}(0)-(p-2) \gamma t\bigr)_{+}^{-\frac {1}{p-2}},M^{\frac{1}{p-1}}\bigr\} ,\quad t\ge0. $$

Although $\varphi_{M}(t)$ is not $C^{1}$ continuous, we can change the partial derivative $\frac{\partial}{\partial t}$ to the leftward partial derivative $\frac{\partial}{\partial t^{-}}$ in the proof of Lemma 2.1, Lemma 2.2, and Lemma 4.1, then we conclude that $v_{M}$ is a lower solution of problem (2.1)-(2.3). Hence $u_{n,M}(x,t)\ge v_{M}(x,t)$ for $(x,t)\in\overline{\Omega}\times \mathbb{R}^{+}$. By the definition of strong solution, we have

$$u(x,t)\ge\sup_{M\in\mathbb{Z}^{+}}v_{M}(x,t),\quad x\in\overline{\Omega}, t\in(0,T_{0}), $$

where $T_{0}=\frac{(2Bd)^{p-2}}{(p-2)\gamma}$. Since $v_{M}$ blows up at finite time $T_{0}$, the strong solution u must blow up at time $T^{*}\le T_{0}$. □