Global existence of solutions of initial-boundary value problem for nonlocal parabolic equation with nonlocal boundary condition

TL;DR

This paper investigates the conditions under which solutions to a nonlinear nonlocal parabolic equation with nonlocal boundary conditions exist globally or blow up, depending on variable coefficients over time.

Contribution

It establishes criteria for global existence and blow-up of solutions for a class of nonlinear nonlocal parabolic equations with nonlocal boundary conditions.

Findings

Conditions for global existence of solutions

Criteria for solution blow-up

Dependence on variable coefficients over time

Abstract

We prove global existence and blow-up of solutions of initial-boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for large values of time.

Equations

$$u_t=\Delta u+a(x,t)u^r{\int }_{\Omega }{u}^p(y,t)dy-b(x,t)u^q,x\in \Omega ,t>0,$$

$$u(x,t)={\int }_{\Omega }k(x,y,t)u^l(y,t)dy,x\in \partial \Omega ,t>0,$$

$$u(x,0)=u_0(x),x\in \Omega ,$$

$$a(x,t),b(x,t)\in C_{loc}^{\alpha }(\overline{\Omega }\times [0,\infty )),0<\alpha <1,a(x,t)\ge 0,b(x,t)\ge 0;$$

$$k(x,y,t)\in C(\partial \Omega \times \overline{\Omega }\times [0,\infty )),k(x,y,t)\ge 0;$$

$$u_0(x)\in C(\overline{\Omega }),u_0(x)\ge 0,x\in \overline{\Omega },u_0(x)={\int }_{\Omega }k(x,y,0)u_0^l(y)dy,x\in \partial \Omega .$$

$$u_t\ge \Delta u+a(x,t)u^r{\int }_{\Omega }{u}^p(y,t)dy-b(x,t)u^q,(x,t)\in Q_T,$$

$$u(x,t)\ge {\int }_{\Omega }k(x,y,t)u^l(y,t)dy,(x,t)\in S_T,$$

$$u(x,0)\ge u_0(x),x\in \Omega ,$$

$$b(x,t)>0\text{ for }x\in \overline{\Omega }\text{ and }t\ge 0.$$

$$M=\max \left(\underset{Q_T}{\sup }a(x,t),\underset{\partial \Omega \times Q_T}{\sup }k(x,y,t)\right).$$

$$\Delta \phi (x)+\lambda \phi (x)=0,x\in \Omega ,\phi (x)=0,x\in \partial \Omega ,$$

$$M{\int }_{\Omega }\frac{dy}{(\phi (y)+\epsilon {)}^l}\le 1.$$

$$v(x,t)=\frac{\eta \exp (\mu t)}{\phi (x)+\epsilon }$$

$$\eta \ge \max \left(\underset{\Omega }{\sup }{u}_0(x)\underset{\Omega }{\sup }\left(\phi (x)+\epsilon \right),1\right),$$

$$\mu \ge {\lambda }_1+\underset{\Omega }{\sup }\frac{2|\nabla \phi (x){|}^2}{{\left(\phi (x)+\epsilon \right)}^2}+M\underset{\Omega }{\sup }(\phi (x)+\epsilon {)}^{1-r}{\int }_{\Omega }\frac{dy}{{\left(\phi (y)+\epsilon \right)}^p}.$$

$$\eta \ge \max \left(\underset{\Omega }{\sup }{u}_0(x)\underset{\Omega }{\sup }\left(\phi (x)+\epsilon \right),{\left(\frac{M}{\underline{b}}\underset{\Omega }{\sup }(\phi (x)+\epsilon {)}^{q-r}{\int }_{\Omega }\frac{dy}{{\left(\phi (y)+\epsilon \right)}^p}\right)}^{\frac{1}{q-r-p}},1\right),$$

$$\mu \ge {\lambda }_1+\underset{\Omega }{\sup }\frac{2|\nabla \phi (x){|}^2}{{\left(\phi (x)+\epsilon \right)}^2}.$$

$$\Delta g(\overline{x},s)=\frac{{\partial }^{2}g}{\partial s^2}(\overline{x},s)-\sum _{j=1}^{n-1}\frac{H_j(\overline{x})}{1-s{H}_j(\overline{x})}\frac{\partial g}{\partial s}(\overline{x},s),$$

$$\left|\sum _{j=1}^{n-1}\frac{H_j(\overline{x})}{1-s{H}_j(\overline{x})}\right|\le \overline{c}.$$

$$v(x,t)=v((\overline{x},s),t)={\left[(s+\epsilon {)}^{-\gamma }-{\omega }^{-\gamma }\right]}_{+}^{\frac{\beta }{\gamma }}+A,$$

$$\left|\frac{\partial v}{\partial s}\right|\le \beta \min \left({\left[D(s)\right]}^{\frac{\gamma +1}{\gamma }}{\left[(s+\epsilon {)}^{-\gamma }-{\omega }^{-\gamma }\right]}_{+}^{\frac{\beta +1}{\gamma }},(s+\epsilon {)}^{-(\beta +1)}\right),$$

$$\left|\frac{{\partial }^{2}v}{\partial s^2}\right|\le \beta (\beta +1)\min \left({\left[D(s)\right]}^{\frac{2(\gamma +1)}{\gamma }}{\left[(s+\epsilon {)}^{-\gamma }-{\omega }^{-\gamma }\right]}_{+}^{\frac{\beta +2}{\gamma }},(s+\epsilon {)}^{-(\beta +2)}\right),$$

$$D(s)=\frac{(s+\epsilon {)}^{-\gamma }}{(s+\epsilon {)}^{-\gamma }-{\omega }^{-\gamma }}.$$

$$1\le D(s)\le 1+\overline{\epsilon },0<s\le \overline{s},$$

$$Lv\equiv v_t-\Delta v-a(x,t)v^r{\int }_{\Omega }{v}^p(y,t)dy+b(x,t)v^q$$

$$\overline{J}=\underset{0<s<\delta }{\sup }{\int }_{\partial \Omega }|J(\overline{y},s)|d\overline{y},$$

$${\int }_{\Omega }{v}^p(y,t)dy$$

$$Lv\ge \underline{b}{\left({\left[(s+\epsilon {)}^{-\gamma }-{\omega }^{-\gamma }\right]}_{+}^{\frac{\beta }{\gamma }}+A\right)}^q-\beta (\beta +1){\left[D(s)\right]}^{\frac{2(\gamma +1)}{\gamma }}{\left[(s+\epsilon {)}^{-\gamma }-{\omega }^{-\gamma }\right]}_{+}^{\frac{\beta +2}{\gamma }}$$

$$-\beta \overline{c}{\left[D(s)\right]}^{\frac{\gamma +1}{\gamma }}{\left[(s+\epsilon {)}^{-\gamma }-{\omega }^{-\gamma }\right]}_{+}^{\frac{\beta +1}{\gamma }}$$

Full text

1. Introduction

We consider nonlinear nonlocal parabolic equation

[TABLE]

with nonlinear nonlocal boundary condition

[TABLE]

and initial datum

[TABLE]

where $r,\,p,\,q,\,l$ are positive constants, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq 1$ with smooth boundary $\partial\Omega.$

Throughout this paper we suppose that $a(x,t),\;b(x,t),\;k(x,y,t)$ and $u_{0}(x)$ satisfy the following conditions:

[TABLE]

[TABLE]

[TABLE]

For global existence and blow-up of solutions for parabolic equations with nonlocal boundary conditions we refer to [1, 2, 6], [11]–[18], [21, 23, 24, 30, 32, 33] and the references therein. Initial-boundary value problems for nonlocal parabolic equations with nonlocal boundary conditions were considered in many papers also (see, for example, [4, 7, 9, 10, 26, 27, 34]). In particular, blow-up problem for nonlocal parabolic equations with boundary condition (1.2) was investigated in [5, 8, 25, 28, 29, 31, 35, 36]. So, for example, the authors of [5] studied (1.1)–(1.3) with $b(x,t)\equiv 0,\,$ $a(x,t)\equiv a(x)\,$ and $k(x,y,t)\equiv k(x,y),$ and problem (1.1)–(1.3) with $r=0,\,$ $a(x,t)\equiv 1,\,$ $b(x,t)\equiv b>0\,$ and $k(x,y,t)\equiv k(x,y)$ was considered in [31]. The authors of [15] studied (1.1)–(1.3) with $a(x,t)\equiv 0.$

The existence of classical local solutions and comparison principle for (1.1)–(1.3) were proved in [19] and [20].

In this paper we prove global existence and blow-up of solutions of (1.1)–(1.3). Obtained results depend on the behavior of variable coefficients $a(x,t),\,$ $b(x,t)\,$ and $k(x,y,t)$ as $t\to\infty.$

This paper is organized as follows. Global existence of solutions for any initial data and blow-up in finite time of solutions for large initial data are proved in section 2. In section 3 we present finite time blow-up of all nontrivial solutions as well as the existence of global solutions for small initial data.

2. Global existence and blow-up of solutions

Let $Q_{T}=\Omega\times(0,T),\;S_{T}=\partial\Omega\times(0,T),$ $\Gamma_{T}=S_{T}\cup\overline{\Omega}\times\{0\},$ $T>0.$

Definition 2.1**.**

We say that a nonnegative function $u(x,t)\in C^{2,1}(Q_{T})\cap C(Q_{T}\cup\Gamma_{T})$ is a supersolution of (1.1)–(1.3) in $Q_{T}$ if

[TABLE]

[TABLE]

[TABLE]

and $u(x,t)\in C^{2,1}(Q_{T})\cap C(Q_{T}\cup\Gamma_{T})$ is a subsolution of (1.1)–(1.3) in $Q_{T}$ if $u\geq 0$ and it satisfies (2.1)–(2.3) in the reverse order. We say that $u(x,t)$ is a solution of (1.1)–(1.3) in $Q_{T}$ if $u(x,t)$ is both a subsolution and a supersolution of (1.1)–(1.3) in $Q_{T}.$

We will repeatedly use the following comparison principle (see [19], [20]).

Theorem 2.2**.**

Let $\underline{u}(x,t)$ and $\overline{u}(x,t)$ be a subsolution and a supersolution of problem (1.1)–(1.3) in $Q_{T},$ respectively. Suppose that $\underline{u}(x,t)>0$ or $\overline{u}(x,t)>0$ in $Q_{T}\cup\Gamma_{T}$ if $\min(r,p,l)<1.$ Then $\overline{u}(x,t)\geq\underline{u}(x,t)$ in $Q_{T}\cup\Gamma_{T}.$

To prove global existence of solutions of (1.1)–(1.3) we suppose that

[TABLE]

Theorem 2.3**.**

Let $\max{(r+p,l)}\leq 1$ or (2.4) hold and either $l\leq 1,\,1<r+p<q$ or $1<l<(q+1)/2,\,\max(r+p,2p+1)<q.$ Then problem (1.1)–(1.3) has global solutions for any initial data.

Proof.

Let $T$ be any positive constant and

[TABLE]

In order to prove global existence of solutions we construct a suitable explicit supersolution of (1.1)–(1.3) in $Q_{T}.$

Suppose at first that $\max(r+p,l)\leq 1.$ Let $\lambda_{1}$ be the first eigenvalue of the following problem

[TABLE]

and $\varphi(x)$ be the corresponding eigenfunction which is chosen to satisfy that for some $0<\varepsilon<1$

[TABLE]

Then it is easy to check that

[TABLE]

is a supersolution of (1.1)–(1.3) in $Q_{T}$ if

[TABLE]

[TABLE]

By Theorem 2.2 problem (1.1)–(1.3) has global solutions for any initial data.

From (2.4) we conclude that $\underline{b}=\inf\limits_{Q_{T}}{b(x,t)}>0.$ Let $l\leq 1,\;1<r+p<q.$ Then $v(x,t)$ in (2.7) is a supersolution of (1.1)–(1.3) in $Q_{T}$ if

[TABLE]

[TABLE]

Let $1<l<(q+1)/2,\,\max(r+p,2p+1)<q.$ To construct a supersolution we use the change of variables in a neighborhood of $\partial\Omega$ as in [3]. Let $\overline{x}\in\partial\Omega$ and $\widehat{n}(\overline{x})$ be the inner unit normal to $\partial\Omega$ at the point $\overline{x}.$ Since $\partial\Omega$ is smooth it is well known that there exists $\delta>0$ such that the mapping $\psi:\partial\Omega\times[0,\delta]\to\mathbb{R}^{n}$ given by $\psi(\overline{x},s)=\overline{x}+s\widehat{n}(\overline{x})$ defines new coordinates ($\overline{x},s)$ in a neighborhood of $\partial\Omega$ in $\overline{\Omega}.$ A straightforward computation shows that, in these coordinates, $\Delta$ applied to a function $g(\overline{x},s)=g(s),$ which is independent of the variable $\overline{x},$ evaluated at a point $(\overline{x},s)$ is given by

[TABLE]

where $H_{j}(\overline{x})$ for $j=1,...,n-1,$ denotes the principal curvatures of $\partial\Omega$ at $\overline{x}.$ For $0\leq s\leq\delta$ and small $\delta$ we have

[TABLE]

Let $0<\varepsilon<\omega<\min(\delta,1),$ $2/(q-1)<\beta<\min(1/p,1/(l-1)),$ $0<\gamma<\beta/2,$ $A\geq\sup_{\Omega}u_{0}(x).$ For points in $Q_{\delta,T}=\partial\Omega\times[0,\delta]\times[0,T]$ of coordinates $(\overline{x},s,t)$ define

[TABLE]

where $s_{+}=\max(s,0).$ For points in $\overline{Q_{T}}\setminus Q_{\delta,T}$ we set $v(x,t)=A.$ We prove that $v(x,t)$ is a supersolution of (1.1)–(1.3) in $Q_{T}.$ It is not difficult to check that

[TABLE]

[TABLE]

where

[TABLE]

Then $D^{\prime}(s)>0$ and for any $\overline{\varepsilon}>0$

[TABLE]

where ${\overline{s}}=[\overline{\varepsilon}/(1+\overline{\varepsilon})]^{1/\gamma}\omega-\varepsilon,$ $\varepsilon<[\overline{\varepsilon}/(1+\overline{\varepsilon})]^{1/\gamma}\omega.$ We denote

[TABLE]

and

[TABLE]

where $J(\overline{y},s)$ is Jacobian of the change of variables in neighborhood of $\partial\Omega.$ We use the inequality $(a+b)^{p}\leq 2^{p}(a^{p}+b^{p}),\,a\geq 0,b\geq 0,p>0$ to estimate the integral in (2.14)

[TABLE]

Here $|\Omega|$ is Lebesque measure of $\Omega.$ By (2.8)–(2.14), (2.16) we can choose $\overline{\varepsilon}$ small and $A$ large so that in $Q_{{\overline{s}},T}$

[TABLE]

[TABLE]

[TABLE]

Let $s\in[{\overline{s}},\delta].$ From (2.8)–(2.12) we have

[TABLE]

and by (2.16) $Lv\geq 0$ for large $A.$ Obviously, in $\overline{Q_{T}}\setminus Q_{\delta,T}$

[TABLE]

for large $A.$

Now we prove the following inequality

[TABLE]

for a suitable choice of $\varepsilon.$ To do this we use the change of variables in neighborhood of $\partial\Omega.$ Estimating the integral $I$ in the right hand side of (2.17), we get

[TABLE]

where

[TABLE]

On the other hand, we have

[TABLE]

Hence, (2.17) holds for small values of $\varepsilon$ and by Theorem 2.2 $u(x,t)\leq v(x,t)$ in $\overline{Q_{T}}.$ ∎

To prove finite time blow-up result we need lower bound for solutions of (1.1)–(1.3) with large initial data.

Lemma 2.4**.**

Let $u(x,t)$ be a solution of (1.1)–(1.3) in $\overline{Q_{T}}.$ For any $\Omega_{0}\subset\subset\Omega_{1}\subset\subset\Omega$ and any positive constant $C$ there exists positive constant $c_{1}$ such that if $u_{0}(x)\geq c_{1}$ in $\Omega_{1},$ then

[TABLE]

Proof.

Let $y(x,t)$ be a solution of the following problem

[TABLE]

where $\chi(x)\in C_{0}^{\infty}(\Omega_{1}),$ $\chi(x)=1$ in $\Omega_{0}$ and $0\leq\chi(x)\leq 1.$ By strong maximum principle

[TABLE]

Suppose that $q\geq 1.$ We put $m=\max\left(\sup_{Q_{T}}{u(x,t)},\sup_{Q_{T}}{b(x,t)}\right)$ and define function $v(x,t)=\exp{(\rho t)}u(x,t).$ For $\rho\geq m^{q}$ we have in $Q_{T}$

[TABLE]

We assume $u_{0}(x)\geq c_{1}\chi(x)$ in $\Omega_{1},$ where constant $c_{1}$ will be chosen below. Then by comparison principle for (2.19) we get $v(x,t)\geq c_{1}y(x,t)$ in $\overline{\Omega_{1}}\times[0,T].$ Taking into account (2.20), we have (2.18) if $c_{1}=C\exp{(\rho T)}\left(\inf_{\Omega_{0}\times(0,T)}{y(x,t)}\right)^{-1}.$

Let $q<1.$ We set $w(x,t)=\exp{(mt)}(u(x,t)+1).$ Since $u^{q}\leq u+1,$ we conclude that

[TABLE]

in $Q_{T}.$ Arguing as in previous case, we obtain

[TABLE]

Choosing $c_{1}=(C+1)\exp{(mT)}\left(\inf_{\Omega_{0}\times(0,T)}{y(x,t)}\right)^{-1},$ we have (2.18). ∎

Now we prove that problem (1.1)–(1.3) has finite time blow-up solutions if either $l>\max{(1,(q+1)/2)}$ and

[TABLE]

for some positive constants $k_{0}$ and $t_{0}$ or $r+p>\max(q,1)$ and

[TABLE]

for some positive constants $a_{0}$ and $t_{1}.$

Theorem 2.5**.**

There exist finite time blow-up solutions of (1.1)–(1.3) if either $l>\max{(1,(q+1)/2)}$ and (2.21) holds or $r+p>\max(q,1)$ and (2.22) holds.

Proof.

We suppose at first that $l>\max{(1,(q+1)/2)}$ and (2.21) holds. Let us consider the following problem

[TABLE]

As it is proved in [15], problem (2.26) has positive finite time blow-up solutions. We note that any solution of (2.26) is a subsolution of (1.1)–(1.3). Applying Theorem 2.2, we prove the theorem.

Now we assume that $r+p>\max(q,1)$ and (2.22) holds. We put $\overline{b}=\sup\limits_{Q_{t_{1}}}{b(x,t)}.$

Let $r\geq q>1.$ We denote

[TABLE]

where $\varphi(x)$ is the solution of (2.6) satisfying

[TABLE]

Then using (1.1), (2.6), Green’s identity and the inequality

[TABLE]

where $\nu$ is unit outward normal on $\partial\Omega,$ we get for $t<t_{1}$

[TABLE]

By Lemma 2.4

[TABLE]

for $t<t_{1}$ and large initial data. Taking into account (2.30), (2.31), H$\ddot{o}$lder’s and Jensen’s inequalities, we have for $t<t_{1}$

[TABLE]

[TABLE]

Hence, $J(t)$ blows up in finite time $T\;(T<t_{1})$ for large initial data.

Let $r>1\geq q.$ By Lemma 2.4

[TABLE]

for $t<t_{1}$ and large initial data. Then using (2.30), (2.32), $u^{q}\leq u+1$ and Jensen’s inequalities, we obtain for $t<t_{1}$

[TABLE]

and again $J(t)$ blows up in finite time $T\;(T<t_{1})$ for large initial data.

Let $r<q.$ Without loss of generality, we may assume that $\Omega$ contains the origin. We introduce designations

[TABLE]

and consider the auxiliary problem

[TABLE]

where $A>0$ will be determined below and $T<\min\{1,t_{1}\}$ we choose in such a way that points $x$ with $|x|\leq A\sqrt{T}$ belong to $\Omega.$ Let $u(x,t)$ be a positive in $\overline{G}$ solution of (1.1)–(1.3) such that $u_{0}(x)\geq\left(A^{2}-|x|^{2}/T\right)_{+}.$ Obviously, $u(x,t)$ is a supersolution of (2.36). We construct a subsolution of (2.36) in the following form

[TABLE]

where $V(\xi)=\left(A^{2}-\xi^{2}\right)_{+},$ $\xi=|x|/\sqrt{T-t}$ and $\gamma>0$ will be chosen below. It is easy to see $\underline{u}(0,t)\to\infty$ as $t\to T_{1}$ and $T_{1}\to T.$ We show that

[TABLE]

in $G,$ where

[TABLE]

Note that

[TABLE]

By (2.39) we obtain

[TABLE]

for points of $G.$ Further we distinguish the two zones $0\leq\xi<\theta A$ and $\theta A\leq\xi<A,$ where $\theta\in(0,1)$ will be chosen below.

For $\theta A\leq\xi<A$ we have

[TABLE]

From (2.40), (2.41) it follows that

[TABLE]

We put

[TABLE]

and estimate the first term on the right hand side of (2.42)

[TABLE]

By (2.41)

[TABLE]

for small values of $T$ and $\gamma>n/[2(r+p-q)].$ From (2.42), (2.44), (2.45) it follows (2.38) for $\xi\in[\theta A,A).$

For $0\leq\xi<\theta A$ we have

[TABLE]

Then by (2.40) inequality (2.38) still holds for $0\leq\xi<\theta A$ if $T$ is small and

[TABLE]

Applying comparison principle for (2.36), we obtain $u(x,t)\geq\underline{u}(x,t)$ in $G.$ Hence, $u(x,t)$ blows up in finite time.

In the case $q\leq r\leq 1$ we have $\gamma q<\gamma+1.$ Then the function in (2.37) satisfies (2.38) for $0\leq\xi<A.$ Indeed, by virtue of (2.42)–(2.44) we have

[TABLE]

for $\theta A\leq\xi<A$ and small values of $T.$ For $0\leq\xi<\theta A$ inequality (2.38) holds if $\gamma>(n+2)/[2(r+p-1)]$ and $T$ is small. Arguing as in the previous case, we complete the proof. ∎

3. Blow-up of all nontrivial solutions and global existence of solutions for small initial data

In this section we find conditions which guarantee blow-up in finite time of all nontrivial solutions and prove global existence of solutions for small initial data.

First we show that for $q<\min(r+p,1)$ under some conditions problem (1.1)–(1.3) has nontrivial global solutions for any $a(x,t)$ and $k(x,y,t).$ Suppose that

[TABLE]

Theorem 3.1**.**

Let (3.1) hold and either $q<\min(r+p,1),\;l>1$ or $r\geq q,$ $(q+1)/2<l\leq 1.$ Then problem (1.1)–(1.3) has global solutions for small initial data.

Proof.

We put $b_{0}=\inf\limits_{Q_{T}}{b(x,t)}$ and choose $T$ so that $b_{0}>0.$

Suppose that $q<\min(r+p,1),\;l>1.$ A straightforward computation shows that for small $\beta,$ $\varepsilon$ and $u_{0}(x)$

[TABLE]

is a supersolution of (1.1)–(1.3) in $Q_{T}.$ Applying Theorem 2.2, we have $u(x,t)\leq g(t)$ in $Q_{T}$. Passing to the limit as $\varepsilon\rightarrow 0,$ we obtain

[TABLE]

Now we put $u(x,t)\equiv 0$ for $t\geq T.$

For $r\geq q,$ $(q+1)/2<l\leq 1$ to construct a supersolution we use the change of variables as in Theorem 2.3. For points of $Q_{\delta,t_{0}}$ define

[TABLE]

where $\delta>0,$ $\varepsilon>0,$ $t_{0}<\min(\delta,T),$ $2/(1-q)<\gamma<1/(1-l)$ for $l<1$ and $2/(1-q)<\gamma$ for $l=1.$ In $\overline{Q_{t_{0}}}\setminus Q_{\delta,t_{0}}$ we put $v(x,t)=\varepsilon.$ Then $v(x,t)$ is a supersolution of (1.1)–(1.3) in $Q_{t_{0}}$ if $u_{0}(x)\leq(\delta-s)^{\gamma}_{+}.$ Indeed, by (2.5), (2.8), (2.9), (2.14)–(2.16) for small $\delta$ and $\varepsilon$ we have $Lv\geq 0$ in $Q_{t_{0}}\setminus Q_{\delta,t_{0}}$ and

[TABLE]

[TABLE]

Estimating the integral $I$ in right hand side of (2.17), we obtain

[TABLE]

On the other hand, we have $v((\overline{x},0),t)=(\delta-t)^{\gamma}_{+}+\varepsilon$ and (2.17) holds for

[TABLE]

By Theorem 2.2 $u(x,t)\leq v(x,t)$ in $\overline{Q_{t_{0}}}$ and passing to the limit as $t_{0}\to\delta$ and $\varepsilon\to 0,$ we deduce

[TABLE]

We put $u(x,t)\equiv 0$ for $t\geq\delta$ and complete the proof. ∎

Now suppose that $q=1.$ We set

[TABLE]

Problem (1.1)–(1.3) has global solutions for small initial data if $q=1,\,\min(r+p,l)>1$ and

[TABLE]

[TABLE]

and conversely (1.1)–(1.3) has no global nontrivial solutions if either $q=1,$ $\min(r,p)\geq 1$ and

[TABLE]

or $q=1,\,l>1$ and

[TABLE]

Theorem 3.2**.**

Let $q=1,\,\min(r+p,l)>1$ and (3.3), (3.4) hold. Then there exist global solutions of (1.1)–(1.3) for small initial data. If either $q=1,\,\min(r,p)\geq 1$ and (3.5) holds or $q=1,\,l>1$ and (3.6) holds, then any nontrivial solution of (1.1)–(1.3) blows up in finite time.

Proof.

Assume that $T$ is any positive constant, $q=1,\,\min(r+p,l)>1$ and (3.3), (3.4) hold. We choose $\widetilde{\lambda}$ in such a way that

[TABLE]

Let $\widetilde{\Omega}$ be bounded domain in $\mathbb{R}^{n}$ with smooth boundary such that $\Omega\subset\subset\widetilde{\Omega}$ and $\widetilde{\lambda}$ be the first eigenvalue of (2.6) in $\widetilde{\Omega}.$ Then correspondent eigenfunction $\widetilde{\varphi}(x)$ satisfies

[TABLE]

for some $d>0.$ Choosing

[TABLE]

we have $\inf\limits_{\partial\Omega}{\widetilde{\varphi}(x)}>\varepsilon.$ We put $N=\sup\limits_{\Omega}{\widetilde{\varphi}^{r-1}(x)\int_{\Omega}{\widetilde{\varphi}^{p}(y)}\,dy}$ and

[TABLE]

where

[TABLE]

It is easy to check that

[TABLE]

is a supersolution of (1.1)–(1.3) in $Q_{T}$ for $u_{0}(x)\leq B^{-\frac{1}{r+p-1}}\widetilde{\varphi}(x).$ By Theorem 2.2 there exist global solutions of (1.1)–(1.3).

Now suppose that $q=1,\,\min(r,p)\geq 1$ and (3.5) holds. Multiplying (1.1) by $\varphi(x)\exp(\lambda_{1}t),$ where $\varphi(x)$ is defined in (2.6) and (2.28), and integrating the obtained equation over $\Omega,$ from (2.27), (2.29), Green’s identity and Jensen’s inequality, we obtain

[TABLE]

Now (3.5) guarantees blow-up of $J(t)$ in finite time. The case $q=1,\,l>1$ is treated similarly. ∎

Remark 3.3*.*

The conclusion of Theorem 3.2 is not true if $\sigma>\lambda_{1}$ in (3.3), or $\gamma=\lambda_{1}$ in (3.4), or $\lambda_{1}$ is replaced by a smaller value in (3.5) or (3.6).

Further we consider the case $q>1.$ To prove blow-up of all nontrivial solutions we need universal lower bound for solutions of (1.1)–(1.3). Assume that

[TABLE]

where

[TABLE]

Lemma 3.4**.**

Let $u(x,t)$ be a solution of (1.1)–(1.3) in $Q_{T},$ $q>1,$ $u_{0}(x)\not\equiv 0$ and (3.7), (3.8) hold. Then for any $t_{0}\in(0,T)$ there exists $d>0,$ which does not depend on $T,$ such that

[TABLE]

where $\varphi(x)$ is defined in (2.6) and (2.28).

Proof.

For $T_{0}\in(0,T)$ denote $m_{0}=\max\left(\sup\limits_{Q_{T_{0}}}{u(x,t)},\sup\limits_{Q_{T_{0}}}{b(x,t)}\right).$ Let $v(x,t)$ be a solution of the problem

[TABLE]

where $v_{0}(x)\in C_{0}^{\infty}(\Omega),\,$ $0\leq v_{0}(x)\leq u_{0}(x)\,$ and $v_{0}(x)\not\equiv 0.$ Obviously, $u(x,t)$ is a supersolution of (3.13). By comparison principle for (3.13) we obtain

[TABLE]

We put $h(x,t)=\exp(\mu t)v(x,t),$ where $\mu\geq m_{0}^{q}.$ Then in $Q_{T_{0}}$

[TABLE]

Since $h(x,0)=v_{0}(x)$ and $v_{0}(x)$ is nontrivial nonnegative function in $\Omega,$ by strong maximum principle

[TABLE]

By virtue of Theorem 3.6 in [22]

[TABLE]

where $t_{0}\in(0,T_{0}).$ From (3.15) and (3.16) it follows that

[TABLE]

Then there exists positive constant $d_{0}$ such that

[TABLE]

By (3.14) and (3.17)

[TABLE]

A straightforward computation shows that for large $f_{0}$

[TABLE]

is a subsolution of (3.13) in $Q_{T_{0}}\setminus\overline{Q_{t_{0}}}$ with initial datum $v(x,t_{0})=u(x,t_{0}).$ Application of comparison principle for (3.13) completes the proof. ∎

Next we assume that

[TABLE]

and

[TABLE]

or $b(x,t)$ satisfies (3.7), (3.8), where

[TABLE]

and

[TABLE]

for large values of $t.$

Theorem 3.5**.**

If $l>1,\;1<q<r+p$ and (3.18), (3.19) hold, then there exist global solutions of (1.1)–(1.3) for small initial data. If $l\geq q>1$ and (3.7), (3.8), (3.20), (3.21) hold, then any nontrivial solution of (1.1)–(1.3) blows up in finite time.

Proof.

Let $l>1,\;1<q<r+p$ and (3.18), (3.19) hold. We choose $\widetilde{\lambda}_{1}$ in the following way

[TABLE]

Let $\widetilde{\Omega}$ be bounded domain in $\mathbb{R}^{n}$ with smooth boundary such that $\Omega\subset\subset\widetilde{\Omega}$ and $\widetilde{\lambda}_{1}$ be the first eigenvalue of (2.6) in $\widetilde{\Omega}.$ The correspondent eigenfunction $\widetilde{\varphi}(x)$ is chosen to satisfy that $\sup_{\widetilde{\Omega}}\widetilde{\varphi}(x)=1.$ Obviously, $\inf_{\Omega}\widetilde{\varphi}(x)>d$ for some $d>0.$ Then $v(x,t)=\beta\exp{(-\widetilde{\lambda}_{1}t)}\widetilde{\varphi}(x)$ is a supersolution of (1.1)–(1.3) in $Q_{T}$ for any $T>0$ if

[TABLE]

By Theorem 2.2 there exist global solutions of (1.1)–(1.3).

Let $u(x,t)$ be nontrivial global solution of (1.1)–(1.3), $l\geq q>1$ and (3.7), (3.8), (3.20), (3.21) hold. Then by (1.2), (3.9) and (3.21) there exist positive constants $t_{1}$ and $d_{1}$ such that

[TABLE]

Let us consider auxiliary problem

[TABLE]

where $t_{2}\geq t_{1}.$ Using (3.7), (3.20), (3.22), we check that $\underline{u}(x,t)=d_{2}\exp(-\lambda_{1}t)$ is a subsolution of (3.26) under suitable choice of $t_{2}$ and $d_{2}>0.$ Comparison principle for (3.26) gives

[TABLE]

Let $\varphi(x)$ satify (2.6) and (2.28). Multiplying (1.1) by $\varphi(x)\exp(\lambda_{1}t),$ integrating over $\Omega$ and using

[TABLE]

Green’s identity, Jensen’s inequality and (2.27), (2.29), (3.7), (3.20)–(3.22), (3.27), we obtain

[TABLE]

for some $d_{3}>0$ and large values of $t>0.$ Integrating differential inequality, we prove the theorem. ∎

Remark 3.6*.*

Theorem 3.5 does not hold if $\sigma=\lambda_{1}(l-1)$ in (3.18) or $\lambda_{1}$ is replaced by a smaller value in (3.21). Furthermore, we can not $\varepsilon(t)$ replace by any positive constant in (3.7). Indeed, let $a(x,t)\equiv 0,\,$ $b(x,t)=b\exp[\lambda_{1}(q-1)t],\,$ $k(x,y,t)=k\exp{[\lambda_{1}(l-1)t]},$ where $b$ and $k$ are positive constants. Then

[TABLE]

is a supersolution of (1.1)–(1.3) if $\min(q,l)>1,\,$

[TABLE]

By Theorem 2.2 there exist global solutions of (1.1)–(1.3).

To prove blow-up of all nontrivial solutions of (1.1)–(1.3) for $\max(r,p)\geq q>1$ we assume that

[TABLE]

[TABLE]

where $\lim_{t\to\infty}\gamma(t)=\infty,$ $\underline{a}(t)$ and $\overline{b}(t)$ are defined in (3.2).

Theorem 3.7**.**

Let $\max(r,p)\geq q>1$ and (3.7), (3.8), (3.28), (3.29) hold. Then any nontrivial solution of (1.1)–(1.3) blows up in finite time.

Proof.

Denote

[TABLE]

where $\varphi(x)$ is defined in (2.6) and (2.28). Suppose at first that $p\geq q.$ By (3.9), (3.28), (3.30) and H$\ddot{o}$lder’s inequality it follows

[TABLE]

[TABLE]

[TABLE]

for large values of $t.$ Using H$\ddot{o}$lder’s inequality again and (2.30), (3.9), (3.31), we obtain

[TABLE]

[TABLE]

[TABLE]

for large values of $t.$ By (3.29), (3.32) $J(t)$ blows up in finite time.

Suppose that $r\geq q.$ Arguing as in previous case, we obtain

[TABLE]

for large values of $t$ and $J(t)$ blows up in finite time again. ∎

Remark 3.8*.*

Theorem 3.7 does not hold if $\gamma(t)$ is a bounded function in (3.28). Indeed, suppose that $p\geq 1,\,$ $r\geq q>1,\,$ $a(x,t)\equiv\underline{a}(t),\,$ $b(x,t)\equiv\overline{b}(t)\,$ and $k(x,y,t)\equiv 0.$ Let $\varphi(x)$ be defined in (2.6) and (2.28). Then $\overline{u}(x,t)=\beta\varphi(x)\exp(-\lambda_{1}t)$ is a supersolution of (1.1)–(1.3) for $u_{0}(x)\leq\beta\varphi(x),\;x\in\Omega$ and small $\beta>0.$ By Theorem 2.2 there exist global solutions of (1.1)–(1.3).

Remark 3.9*.*

We note here an importance of divergence of the integral in (3.29) for blow-up of all nontrivial solutions of (1.1)–(1.3). Suppose that $a(x,t)\equiv\underline{a}(t)\,$ and $k(x,y,t)\equiv 0.$ Let $\varphi(x)$ be the solution of (2.6) with $\sup_{\Omega}{\varphi(x)}=1$ and $f(t)$ be a solution of the following differential equation

[TABLE]

Since the integral in (3.29) converges, $f(t)$ exists for any $t\geq 0$ if $f(0)$ is small enough. Then $\overline{u}(x,t)=\varphi(x)f(t)$ is a supersolution of (1.1)–(1.3) with $u_{0}(x)\leq\varphi(x)f(0).$ By Theorem 2.2 there exist global solutions of (1.1)–(1.3).

Remark 3.10*.*

Theorem 3.7 is not true if $\lambda_{1}$ is replaced by a larger value in (3.7). Indeed, let $\sigma>0.$ We put

[TABLE]

[TABLE]

[TABLE]

It is easy to see that

[TABLE]

is the solution of (1.1)–(1.3) with $u_{0}(x)\equiv 1,\;x\in\Omega.$

Remark 3.11*.*

From Theorem 3.7 it follows that Theorem 3.5 does not hold for $\omega>\lambda_{1}(r+p-q)$ in (3.19).