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

Boundary Value Problems20152015:194

https://doi.org/10.1186/s13661-015-0456-8

  • Received: 27 August 2015
  • Accepted: 9 October 2015
  • Published:

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.

Keywords

  • dynamic boundary condition
  • blow up
  • p-Laplacian
  • reaction-diffusion

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 [57], 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 \(T^{*}\le T_{0}\). □

Declarations

Acknowledgements

The first author was supported in part by the Scientific Research Foundation of Graduate School of South China Normal University (No. 2013kyjj022). The second author and the third author were supported in part by NNSFC (No. 11071099). The third author was supported in part by CPSF (No. 2015M572301) and the Fundamental Research Funds for the Central Universities.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China
(2)
Department of Mathematics, South China University of Technology, Guangzhou, 510640, China

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