Skip to main content

Lower bound for the blow-up time for a general nonlinear nonlocal porous medium equation under nonlinear boundary condition

Abstract

In this paper, we study the blow-up phenomenon for a general nonlinear nonlocal porous medium equation in a bounded convex domain \((\varOmega\in \mathbb{R}^{n}, n\geq 3)\) with smooth boundary. Using the technique of a differential inequality and a Sobolev inequality, we derive the lower bound for the blow-up time under the nonlinear boundary condition if blow-up does really occur.

Introduction

Liu in paper [1] studied the blow-up phenomena for the solution of the following problems:

$$\begin{aligned}& \frac{\partial u}{\partial t}={\triangle } u^{m}+u^{p} \int _{\varOmega }u^{q}\,dx, \quad (x,t)\in \varOmega \times \bigl(0,t^{*} \bigr), \end{aligned}$$
(1.1)
$$\begin{aligned}& u(x,0)=f(x)\geq 0 ,\quad x\in \varOmega , \end{aligned}$$
(1.2)

under the Robin boundary condition

$$ \frac{\partial u}{\partial \nu }+ku=0 ,\quad (x,t)\in \varOmega \times \bigl(0,t^{*} \bigr). $$
(1.3)

He obtained a lower bound for the blow-up time of the system when the solution blows up.

In paper [2], the authors also studied equations (1.1) and (1.2) subject to either homogeneous Dirichlet boundary condition or homogeneous Neumann boundary condition. The lower bounds for the blow-up time under the above two boundary conditions were obtained. Equation (1.1) is used in the study of population dynamics (see [3]). For other systems in porous medium, one could see [4]. There have been a lot of papers in the literature on studying the question of blow-up for the solution of parabolic problems under a homogeneous Dirichlet boundary condition and Neumann boundary condition(one can see [512]). Some authors have started to consider the blow-up of these problems under Robin boundary conditions (see [1317]). In papers [1821], the authors studied the blow-up phenomena for the heat equation under nonlinear boundary conditions. Some new results about the nonlinear evolution equations may be founded in [2224]. These papers have mainly focused on the bounded convex domain in \({\mathbb{R}^{3}}\). Recently, there have been some papers starting to study the blow-up problems in \({\mathbb{R}^{n}}\) (\(n\geq 3\)) (see [2529]). We continue the work of [2] for a more general equation. Until now, the authors have not found any paper dealing with lower bound for the blow-up time of a nonlinear nonlocal porous medium equation under nonlinear boundary condition in \({\mathbb{R}^{n}}\) (\(n\geq 3\)). In this sense, the result obtained in this paper is new and interesting. In this paper, we consider the blow-up phenomena of the solution for the following equation:

$$ \bigl(h(u) \bigr)_{t}={\triangle } u^{m}+k_{1}(t)u^{p} \int _{\varOmega }u^{q}\,dx,\quad (x,t) \in \varOmega \times \bigl(0,t^{*} \bigr), $$
(1.4)

with the following boundary initial conditions:

$$\begin{aligned}& u(x,0)=f(x)\geq 0 ,\quad x\in \varOmega , \end{aligned}$$
(1.5)
$$\begin{aligned}& \frac{\partial u}{\partial \nu }=k_{2}(t) \int _{\varOmega }g(u)\,dx, \quad (x,t)\in \partial \varOmega \times \bigl(0,t^{*} \bigr), \end{aligned}$$
(1.6)

where Ω is a bounded convex domain in \({\mathbb{R}^{n}}\), \(n\geq 3\), with sufficiently smooth boundary, is the Laplace operator, ∂Ω is the boundary of Ω, and \(t^{*}\) is the possible blow-up time, \(\frac{\partial u}{\partial \nu }\) is the outward normal derivative of u. We assume \(\frac{k_{1}^{\prime }(t)}{k_{1}(t)}\leq \alpha \) and \(\frac{dh(u)}{du}\geq M>0\).

The function \(g(\xi )\) satisfies

$$ 0\leq g(\xi )\leq \xi ^{s},\quad \forall \xi >0, $$
(1.7)

where \(s>\max \{\frac{2n}{2n-1},p+q+1-m\}\).

Some useful inequalities

We will use the following useful inequalities later in the proof.

Lemma 2.1

We suppose thatuis a nonnegative function andσ, mare positive constants, then we have the result as follows:

$$ \int _{\partial \varOmega } u^{\sigma +m-2}\,dA\leq \frac{n}{\rho _{0}} \int _{\varOmega } u^{\sigma +m-2}\,dx + \frac{(\sigma +m-2)d}{\rho _{0}} \int _{\varOmega } u^{\sigma +m-3} \vert \nabla u \vert \,dx, $$
(2.1)

where\(\rho _{0} :=\min_{\partial \varOmega } \vert x\cdot \vec{\nu } \vert \), ν⃗is the outward normal vector of∂Ωand\(d:=\max_{\partial \varOmega } \vert x \vert \).

Proof

Applying the divergence definition, we have

$$ \operatorname{div} \bigl(u^{\sigma +m-2}x \bigr)=nu^{\sigma +m-2}+(\sigma +m-2)u^{ \sigma +m-3}(x\cdot \bigtriangledown u). $$
(2.2)

Integrating (2.2), we deduce

$$ \int _{\varOmega } \operatorname{div} \bigl(u^{\sigma +m-2}x \bigr) \,dx \leq n \int _{\varOmega } u^{ \sigma +m-2}\,dx +(\sigma +m-2) \int _{\varOmega } u^{\sigma +m-3} \vert x\cdot \nabla u \vert \,dx. $$

Applying the divergence theorem, we obtain

$$ \int _{\partial \varOmega } u^{\sigma +m-2}x\cdot \vec{\nu }\,dA= n \int _{ \varOmega } u^{\sigma +m-2}\,dx +(\sigma +m-2) \int _{\varOmega } u^{\sigma +m-3} \vert x \cdot \nabla u \vert \,dx. $$

Because Ω is a convex domain, we have \(\rho _{0} :=\min_{\partial \varOmega } \vert x\cdot \vec{\nu } \vert >0\). Then we derive

$$ \int _{\partial \varOmega } u^{\sigma +m-2}\,dA\leq \frac{n}{\rho _{0}} \int _{\varOmega } u^{\sigma +m-2}\,dx +\frac{(\sigma +m-2)d}{\rho _{0}} \int _{\varOmega } u^{\sigma +m-3} \vert x\cdot \nabla u \vert \,dx. $$

 □

Lemma 2.2

Supposing that\(u\in W^{1,2}(\varOmega )\)and\(n\geq 3\), we have

$$ \int _{\varOmega } u^{\frac{(\sigma +m-1)n}{n-2}}\,dx\leq C^{ \frac{2n}{n-2}}2^{\frac{n}{n-2}-1} \biggl[ \biggl( \int _{\varOmega } u^{ \sigma +m-1}\,dx \biggr)^{\frac{n}{n-2}} + \biggl( \int _{\varOmega } \bigl\vert \nabla ^{ \frac{\sigma +m-1}{2}}{u} \bigr\vert ^{2}\,dx \biggr)^{\frac{n}{n-2}} \biggr], $$
(2.3)

where\(C=C(n,\varOmega )\)is a Sobolev embedding constant depending onnand Ω.

Proof

In paper [30], we have \(W^{1,2}(\varOmega )\hookrightarrow L^{\frac{2n}{n-2}(\varOmega )}\), \(n\geq 3\). Then we deduce the Sobolev inequality as follows:

$$ \biggl( \int _{\varOmega } w^{\frac{2n}{n-2}}\,dx \biggr)^{ \frac{n-2}{2n}}\leq C \biggl( \int _{\varOmega } w^{2}\,dx+ \int _{\varOmega } \vert \nabla w \vert ^{2}\,dx \biggr)^{\frac{1}{2}}, $$

that is,

$$ \biggl( \int _{\varOmega } \bigl(u^{\frac{\sigma +m-1}{2}} \bigr)^{ \frac{2n}{n-2}}\,dx \biggr)^{\frac{n-2}{2n}}\leq C \biggl( \int _{\varOmega } \bigl(u^{ \frac{\sigma +m-1}{2}} \bigr)^{2}\,dx+ \int _{\varOmega } \bigl\vert \nabla u^{ \frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx \biggr)^{\frac{1}{2}}. $$

We can get

$$ \begin{aligned} \int _{\varOmega } u^{\frac{(\sigma +m-1)n}{n-2}}&\leq C^{ \frac{2n}{n-2}} \biggl( \int _{\varOmega } \bigl(u^{\frac{\sigma +m-1}{2}} \bigr)^{2}\,dx+ \int _{\varOmega } \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx \biggr)^{ \frac{n}{n-2}} \\ &\leq C^{\frac{2n}{n-2}}2^{\frac{n}{n-2}-1} \biggl[ \biggl( \int _{ \varOmega } u^{\sigma +m-1}\,dx \biggr)^{\frac{n}{n-2}} + \biggl( \int _{ \varOmega } \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx \biggr)^{ \frac{n}{n-2}} \biggr]. \end{aligned} $$

 □

Remark 2.1

For any nonnegative function u, the following Hölder inequality holds:

$$ \int _{\varOmega }u^{n_{1}+n_{2}}\,dx\leq \biggl( \int _{\varOmega }u^{ \frac{n_{1}}{x_{1}}}\,dx \biggr)^{x_{1}} \biggl( \int _{\varOmega }u^{ \frac{n_{2}}{x_{2}}\,dx} \biggr)^{x_{2}}, $$
(2.4)

where \({n_{1}}\), \({n_{2}}\), \({x_{1}}\), \({x_{2}}\) are positive constants and \({x_{1}}\), \({x_{2}}\) satisfy \({x_{1}}+{x_{2}}=1\).

Remark 2.2

The fundamental inequality

$$ (a+b)^{l}\leq a^{l}+b^{l}, $$
(2.5)

where \(a,b\geq 0 \) and \(0< l\leq 1\), holds.

Lower bound for the blow-up time

In this section it is useful in the sequel to define an auxiliary function of the following form:

$$ \phi (t)=k_{1}^{n}(t) \int _{\varOmega } u^{2n(s-1)}\,dx=k_{1}^{n}(t) \int _{\varOmega } u^{\sigma }\,dx, \quad {0\leq t< t^{*}}. $$
(3.1)

We will derive a differential inequality for \(\phi (t)\). From the inequality, we can establish the following theorem.

Theorem 3.1

Let\(u(x,t)\)be the classical nonnegative solution of problem (1.4)(1.7) in a bounded convex domainΩ (\(\varOmega \in R^{n}\) (\(n\geq 3\))). We assume that\(m+s>p+q+1>2\), \(m>3\), \(p>0\), \(q>0\). Then the quantity\(\phi (t) \)defined in (3.1) satisfies the differential inequality

$$ \phi '(t)\phi ^{-5}(t)\leq a(t) \phi ^{-4}(t)+b(t), $$
(3.2)

from which it follows that the blow-up time\(t^{*}\)is bounded below. We have

$$ t^{*}\geq \varTheta ^{-1} \biggl( \frac{1}{4\phi ^{4}(0)} \biggr), $$
(3.3)

where\(\varTheta ^{-1} \)is the inverse function ofΘ, and\(a(t)\), \(b(t)\)are defined in (3.21), (3.22) respectively.

Proof

Now we prove Theorem 3.1. For convenience, we set \(\phi (t)=\phi \), \(k_{1}(t)=k_{1}\), \(k_{2}(t)=k_{2}\). First we compute

$$ \begin{aligned} \phi '(t)&=nk_{1}^{n-1}k_{1}^{\prime } \int _{\varOmega }u^{\sigma }\,dx+k_{1}^{n} \sigma \int _{\varOmega }u^{\sigma -1}u_{t}\,dx \\ &=nk_{1}^{n-1}k_{1}^{\prime } \int _{\varOmega }u^{\sigma }\,dx+k_{1}^{n} \sigma \int _{\varOmega }u^{\sigma -1}\frac{1}{h^{\prime }(u)} \biggl[{\triangle } u^{m}+k_{1}u^{p} \int _{\varOmega }u^{q}\,dx \biggr]\,dx \\ &\leq n\alpha \phi +\frac{k_{1}^{n}\sigma }{M} \int _{\varOmega }u^{\sigma -1} \biggl[{\triangle } u^{m}+k_{1}u^{p} \int _{\varOmega }u^{q}\,dx \biggr]\,dx. \end{aligned} $$

Integrating by parts, we have

$$ \begin{aligned} \phi '(t)&\leq n\alpha \phi + \frac{k_{1}^{n}\sigma }{M} \biggl[m \int _{\partial \varOmega } u^{\sigma +m-2} \frac{\partial u}{\partial \nu }\,dA-m(\sigma -1) \int _{\varOmega }u^{ \sigma +m-3} \vert \nabla u \vert ^{2}\,dx \biggr] \\ &\quad{} + \frac{k_{1}^{n+1}\sigma \vert \varOmega \vert }{M} \int _{\varOmega }u^{\sigma +p+q-1}\,dx \\ &\leq n\alpha \phi +\frac{\sigma mk_{1}^{n}k_{2}}{M} \int _{\partial \varOmega } u^{\sigma +m-2}\,dA \int _{\varOmega }u^{s}\,dx- \frac{\sigma m(\sigma -1)k_{1}^{n}}{M} \int _{\varOmega }u^{\sigma +m-3} \vert \nabla u \vert ^{2}\,dx \\ &\quad {} +\frac{k_{1}^{n+1}\sigma \vert \varOmega \vert }{M} \int _{\varOmega }u^{\sigma +p+q-1}\,dx. \end{aligned} $$

Using the result of Lemma 2.1, we obtain

$$\begin{aligned} \begin{aligned}[b] \phi '(t)&\leq n \alpha \phi + \frac{\sigma mk_{1}^{n}k_{2}}{M}\frac{n}{\rho _{0}} \int _{\varOmega }u^{ \sigma +m-2}\,dx \int _{\varOmega }u^{s}\,dx \\ &\quad{} +\frac{\sigma mk_{1}^{n}k_{2}}{M} \frac{(\sigma +m-2)d}{\rho _{0}} \int _{\varOmega }u^{\sigma +m-3} \vert \nabla u \vert \,dx \int _{\varOmega }u^{s}\,dx \\ &\quad {}-\frac{\sigma m(\sigma -1)k_{1}^{n}}{M} \frac{4}{(\sigma +m-1)^{2}} \int _{\varOmega } \bigl\vert \nabla u^{ \frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx+\frac{k_{1}^{n+1}\sigma \vert \varOmega \vert }{M} \int _{\varOmega }u^{\sigma +p+q-1}\,dx \\ &\leq n\alpha \phi +r_{1}k_{1}^{n}k_{2} \int _{\varOmega }u^{\sigma +m+s-2}\,dx+r_{2}k_{1}^{n}k_{2} \int _{\varOmega }u^{\sigma +m-3} \vert \nabla u \vert \,dx \int _{\varOmega }u^{s}\,dx \\ &\quad {}-r_{3}k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx+r_{4}k_{1}^{n+1} \int _{\varOmega }u^{\sigma +p+q-1}\,dx, \end{aligned} \end{aligned}$$
(3.4)

where \(r_{1}=\frac{\sigma m}{M}\frac{n \vert \varOmega \vert }{\rho _{0}}\), \(r_{2}= \frac{\sigma m}{M}\frac{(\sigma +m-2)d}{\rho _{0}}\), \(r_{3}= \frac{\sigma m(\sigma -1)}{M}\frac{4}{(\sigma +m-1)^{2}}\), \(r_{4}= \frac{\sigma \vert \varOmega \vert }{M} \).

Now we estimate the third term of the right-hand side of (3.4). Using Hölder’s inequality, we have

$$ \int _{\varOmega }u^{s}\,dx\leq \biggl( \int _{\varOmega }u^{\sigma }\,dx \biggr)^{ \frac{s}{\sigma }} \vert \varOmega \vert ^{\frac{\sigma -s}{\sigma }} =k_{1}^{- \frac{ns}{\sigma }}\phi ^{\frac{s}{\sigma }} \vert \varOmega \vert ^{ \frac{\sigma -s}{\sigma }}. $$

Then we obtain

$$\begin{aligned}& k_{1}^{n} \int _{\varOmega }u^{\sigma +m-3} \vert \nabla u \vert \,dx \int _{\varOmega }u^{s}\,dx \\& \quad \leq k_{1}^{n} \int _{\varOmega }u^{\sigma +m-3} \vert \nabla u \vert dxk_{1}^{- \frac{ns}{\sigma }}\phi ^{\frac{s}{\sigma }} \vert \varOmega \vert ^{ \frac{\sigma -s}{\sigma }} \\& \quad =k_{1}^{-\frac{ns}{\sigma }} \vert \varOmega \vert ^{\frac{\sigma -s}{\sigma }} \frac{2}{\sigma +m-1}\phi ^{\frac{s}{\sigma }}k_{1}^{n} \int _{\varOmega }u^{ \frac{\sigma +m-3}{2}} \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert \,dx \\& \quad \leq \biggl(\varepsilon _{1}^{-1}r_{5}k_{1}^{n} \phi ^{ \frac{2s}{\sigma }} \int _{\varOmega } \bigl(u^{\frac{\sigma +m-3}{2}} \bigr)^{2}\,dx \biggr)^{\frac{1}{2}} \biggl(\varepsilon _{1}k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx \biggr)^{\frac{1}{2}} \\& \quad \leq \frac{1}{2}\varepsilon _{1}^{-1}r_{5}k_{1}^{n} \phi ^{ \frac{2s}{\sigma }} \int _{\varOmega }u^{\sigma +m-3}\,dx+\frac{1}{2} \varepsilon _{1}k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{ \frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx, \end{aligned}$$

where \(r_{5}=(k_{1}^{-\frac{ns}{\sigma }} \vert \varOmega \vert ^{ \frac{\sigma -s}{\sigma }}\frac{2}{\sigma +m-1})^{2}\), \(\varepsilon _{1}\) is a positive constant which will be defined later.

From the above deductions, we get

$$ \begin{aligned}[b] &r_{2}k_{2}k_{1}^{n} \int _{\varOmega }u^{\sigma +m-3} \vert \nabla u \vert \,dx \int _{\varOmega }u^{s}\,dx \\ &\leq \frac{1}{2}r_{2}k_{2}\varepsilon _{1}^{-1}r_{5}k_{1}^{n} \phi ^{ \frac{2s}{\sigma }} \int _{\varOmega }u^{\sigma +m-3}\,dx+\frac{1}{2}r_{2}k_{2} \varepsilon _{1}k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{ \frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx. \end{aligned} $$
(3.5)

Combining (3.4) and (3.5), we obtain

$$ \begin{aligned}[b] \phi '(t)&\leq n \alpha \phi +r_{1}k_{1}^{n}k_{2} \int _{\varOmega }u^{\sigma +m+s-2}\,dx+\frac{1}{2}r_{2}k_{2} \varepsilon _{1}^{-1}r_{5}k_{1}^{n} \phi ^{\frac{2s}{\sigma }} \int _{\varOmega }u^{\sigma +m-3}\,dx \\ &\quad {}+r_{4}k_{1}^{n+1} \int _{\varOmega }u^{\sigma +p+q-1}\,dx+ \biggl( \frac{1}{2}r_{2}k_{2} \varepsilon _{1}-r_{3} \biggr)k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx. \end{aligned} $$
(3.6)

Using (2.3), (2.4), and (2.5), we obtain

$$ \begin{aligned}[b] \int _{\varOmega }u^{\sigma +m+s-2}\,dx&\leq \biggl( \int _{ \varOmega } u^{\frac{(\sigma +m-1)n}{n-2}}\,dx \biggr)^{x_{1}} \biggl( \int _{ \varOmega } u^{\sigma }\,dx \biggr)^{x_{2}} \\ &\leq \bigl(C^{\frac{2n}{n-2}}2^{\frac{n}{n-2}-1} \bigr)^{x_{1}} \biggl[ \biggl( \int _{\varOmega }u^{\sigma +m-1}\,dx \biggr)^{\frac{x_{1}n}{n-2}} \\ &\quad{} + \biggl( \int _{\varOmega } \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx \biggr)^{ \frac{x_{1}n}{n-2}} \biggr] \biggl( \int _{\varOmega }u^{\sigma }\,dx \biggr)^{x_{2}} \\ &=r_{6} \biggl( \int _{\varOmega }u^{\sigma +m-1}\,dx \biggr)^{\frac{x_{1}n}{n-2}} \biggl( \int _{\varOmega }u^{\sigma }\,dx \biggr)^{x_{2}} \\ &\quad{}+r_{6} \biggl( \int _{\varOmega } \bigl\vert \nabla u ^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx \biggr)^{\frac{x_{1}n}{n-2}} \biggl( \int _{\varOmega }u^{\sigma }\,dx \biggr)^{x_{2}}, \end{aligned} $$
(3.7)

where

$$ \begin{aligned} &x_{1}=\frac{(m+s-2)(n-2)}{(m-1)n+2\sigma },\quad\quad x_{2}= \frac{(m-1)n+2\sigma +(2-m-s)(n-2)}{(m-1)n+2\sigma }, \\ &r_{6}= \bigl(C^{\frac{2n}{n-2}}2^{\frac{n}{n-2}-1} \bigr)^{x_{1}}. \end{aligned} $$

Using Hölder’s and Young’s inequalities, we have

$$ \begin{aligned}[b] & r_{6} \biggl( \int _{\varOmega }u^{\sigma +m-1}\,dx \biggr)^{ \frac{x_{1}n}{n-2}} \biggl( \int _{\varOmega }u^{\sigma }\,dx \biggr)^{x_{2}} \\ &\quad = \biggl(\frac{n-2}{x_{1}n} \int _{\varOmega }u^{\sigma +m-1}\,dx \biggr)^{ \frac{x_{1}n}{n-2}} \biggl\{ \biggl[ \biggl({\frac{n-2}{x_{1}n}} \biggr)^{- \frac{x_{1}n}{n-2}}r_{6} \biggl( \int _{\varOmega }u^{\sigma }\,dx \biggr)^{x_{2}} \biggr]^{\frac{n-2}{n-2-x_{1}n}} \biggr\} ^{\frac{n-2-x_{1}n}{n-2}} \\ &\quad \leq \int _{\varOmega }u^{\sigma +m-1}\,dx+r_{7} \biggl( \int _{\varOmega }u^{ \sigma } \,dx \biggr)^{\frac{x_{2}(n-2)}{n-2-x_{1}n}}, \end{aligned} $$
(3.8)

where \(r_{7}=\frac{n-2-x_{1}n}{n-2}({\frac{n-2}{x_{1}n}})^{- \frac{x_{1}n}{n-2-x_{1}n}}r_{6}^{\frac{n-2}{n-2-x_{1}n}}\).

By Hölder’s and Young’s inequalities, we get

$$ \begin{aligned} \int _{\varOmega }u^{\sigma +m-1}\,dx&\leq \biggl(\varepsilon _{2} \int _{\varOmega }u^{\sigma +m+s-2}\,dx \biggr)^{x_{10}} \biggl( \varepsilon _{2}^{- \frac{x_{10}}{x_{20}}} \int _{\varOmega }u^{\sigma }\,dx \biggr)^{x_{20}} \\ &\leq x_{10}\varepsilon _{2} \int _{\varOmega }u^{\sigma +m+s-2}\,dx+x_{20} \varepsilon _{2}^{-\frac{x_{10}}{x_{20}}} \int _{\varOmega }u^{\sigma }\,dx, \end{aligned} $$

where \(x_{10}=\frac{m-1}{m+s-2}\), \(n_{10}=\frac{(\sigma +m+s-2)(m-1)}{m+s-2}\), \(x_{20}= \frac{s-1}{m+s-2}\), \(n_{20}=\frac{(s-1)\sigma }{m+s-2}\).

If we choose \(\varepsilon _{2}\) such that \(x_{10}\varepsilon _{2}=\frac{1}{2}\), we have

$$ \int _{\varOmega }u^{\sigma +m-1}\,dx\leq \frac{1}{2} \int _{\varOmega }u^{\sigma +m+s-2}\,dx+x_{20}\varepsilon _{2}^{- \frac{x_{10}}{x_{20}}} \int _{\varOmega }u^{\sigma }\,dx. $$
(3.9)

Combining (3.7)–(3.9), we obtain

$$ \begin{aligned}[b] \int _{\varOmega }u^{\sigma +m+s-2}\,dx &\leq 2x_{20} \varepsilon _{2}^{-\frac{x_{10}}{x_{20}}} \int _{\varOmega }u^{\sigma }\,dx+2r_{7} \biggl( \int _{\varOmega }u^{\sigma }\,dx \biggr)^{\frac{x_{2}(n-2)}{n-2-x_{1}n}} \\ &\quad{} +2r_{6} \biggl( \int _{\varOmega } \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx \biggr)^{\frac{x_{1}n}{n-2}} \biggl( \int _{\varOmega }u^{\sigma }\,dx \biggr)^{x_{2}}. \end{aligned} $$
(3.10)

Then we can deduce

$$\begin{aligned}& k_{1}^{n} \int _{\varOmega }u^{\sigma +m+s-2}\,dx \\& \quad \leq 2x_{20} \varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}\phi +2r_{7}k_{1}^{n- \frac{x_{2}(n-2)n}{n-2-x_{1}n}} \biggl(k_{1}^{n} \int _{\varOmega }u^{\sigma }\,dx \biggr)^{\frac{x_{2}(n-2)}{n-2-x_{1}n}} \\& \quad \quad{} +2r_{6}k_{1}^{n-\frac{x_{1}n^{2}}{n-2}-nx_{2}} \biggl(k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx \biggr)^{ \frac{x_{1}n}{n-2}} \biggl(k_{1}^{n} \int _{\varOmega }u^{\sigma }\,dx \biggr)^{x_{2}} \\& \quad \leq 2x_{20}\varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}\phi +2r_{7}k_{1}^{n- \frac{x_{2}(n-2)n}{n-2-x_{1}n}}\phi ^{\frac{x_{2}(n-2)}{n-2-x_{1}n}} +2r_{6}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \biggl(k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{ \frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx \biggr)^{\frac{x_{1}n}{n-2}}\phi ^{x_{2}} \\& \quad \leq 2x_{20}\varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}\phi +2r_{7}k_{1}^{n- \frac{x_{2}(n-2)n}{n-2-x_{1}n}}\phi ^{\frac{x_{2}(n-2)}{n-2-x_{1}n}} +2r_{6}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}}\frac{x_{1}n}{n-2}\varepsilon _{3}k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx \\& \quad \quad {}+2r_{6}k_{1}^{nx_{1}-\frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2} \varepsilon _{3}^{-\frac{x_{1}n}{n-2-x_{1}n}} \phi ^{\frac{x_{2}(n-2)}{n-2-x_{1}n}} \\& \quad \leq 2x_{20}\varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}\phi + \biggl[2r_{7}k_{1}^{n- \frac{x_{2}(n-2)n}{n-2-x_{1}n}} +2r_{6}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{3}^{- \frac{x_{1}n}{n-2-x_{1}n}} \biggr]\phi ^{\frac{x_{2}(n-2)}{n-2-x_{1}n}} \\& \quad \quad {}+2r_{6}k_{1}^{nx_{1}-\frac{x_{1}n^{2}}{n-2}}\frac{x_{1}n}{n-2} \varepsilon _{3}k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{ \frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx, \end{aligned}$$
(3.11)

where \(\varepsilon _{3}\) is a positive constant which will be defined later.

If we choose \(x_{11}=\frac{m-3}{m+s-2}\), \(n_{11}=\frac{(\sigma +m+s-2)(m-3)}{m+s-2}\), \(x_{21}= \frac{s+1}{m+s-2}\), \(n_{21}=\frac{(s+1)\sigma }{m+s-2}\), using (2.4), we get

$$ \begin{aligned} \int _{\varOmega }u^{\sigma +m-3}\,dx&\leq \biggl( \int _{\varOmega }u^{\sigma +m+s-2}\,dx \biggr)^{x_{11}} \biggl( \int _{\varOmega }u^{\sigma }\,dx \biggr)^{x_{21}} \\ &\leq x_{11} \int _{\varOmega }u^{\sigma +m+s-2}\,dx+x_{21} \int _{\varOmega }u^{\sigma }\,dx. \end{aligned} $$

Then we obtain

$$ k_{1}^{n}\phi ^{\frac{2s}{\sigma }} \int _{\varOmega }u^{ \sigma +m-3}\,dx\leq x_{11}\phi ^{\frac{2s}{\sigma }}k_{1}^{n} \int _{ \varOmega } u^{\sigma +m+s-2}\,dx+x_{21}\phi ^{\frac{2s}{\sigma }+1}. $$
(3.12)

Combining (3.10) and (3.12), we have

$$\begin{aligned} k_{1}^{n}\phi ^{\frac{2s}{\sigma }} \int _{\varOmega }u^{ \sigma +m-3}\,dx&\leq \bigl(2x_{20} \varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}x_{11}+x_{21} \bigr) \phi ^{\frac{2s}{\sigma }+1} +x_{11}k_{1}^{n- \frac{x_{2}(n-2)n}{n-2-x_{1}n}}2r_{7} \phi ^{\frac{2s}{\sigma }+ \frac{x_{2}(n-2)}{n-2-x_{1}n}} \\ &\quad{} +2r_{6}x_{11}k_{1}^{nx_{1}-\frac{x_{1}n^{2}}{n-2}} \biggl(k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx \biggr)^{ \frac{x_{1}n}{n-2}}\phi ^{\frac{2s}{\sigma }+x_{2}} \\ &\leq \bigl(2x_{20}\varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}x_{11}+x_{21} \bigr) \phi ^{\frac{2s}{\sigma }+1} +x_{11}k_{1}^{n- \frac{x_{2}(n-2)n}{n-2-x_{1}n}}2r_{7} \phi ^{\frac{2s}{\sigma }+ \frac{x_{2}(n-2)}{n-2-x_{1}n}} \\ &\quad{} +2r_{6}x_{11}k_{1}^{nx_{1}-\frac{x_{1}n^{2}}{n-2}} \frac{x_{1}n}{n-2}\varepsilon _{4}k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{ \frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx \\ &\quad{} +2r_{6}x_{11}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{4}^{- \frac{x_{1}n}{n-2-x_{1}n}}\phi ^{ \frac{(2s+\sigma x_{2})(n-2)}{\sigma (n-2-x_{1}n)}}, \end{aligned}$$
(3.13)

where \(\varepsilon _{4}\) is a positive constant which will be defined later.

Similarly, if we choose \(x_{12}=\frac{p+q-1}{m+s-2}\), \(n_{12}= \frac{(\sigma +m+s-2)(p+q-1)}{m+s-2}\), \(x_{22}=\frac{m+s-(p+q+1)}{m+s-2}\), \(n_{22}= \frac{\sigma [m+s-(p+q+1)]}{m+s-2}\), using (2.4), we get

$$ \begin{aligned}[b] \int _{\varOmega }u^{\sigma +p+q-1}\,dx&\leq \biggl( \int _{\varOmega }u^{\sigma +m+s-2}\,dx \biggr)^{x_{12}} \biggl( \int _{\varOmega }u^{\sigma }\,dx \biggr)^{x_{22}} \\ &\leq x_{12} \int _{\varOmega }u^{\sigma +m+s-2}\,dx+x_{22} \int _{\varOmega }u^{\sigma }\,dx. \end{aligned} $$
(3.14)

Combining (3.10) and (3.14), we obtain

$$\begin{aligned}& k_{1}^{n+1} \int _{\varOmega }u^{\sigma +p+q-1}\,dx \\& \quad \leq x_{12}k_{1}^{n+1} \int _{\varOmega }u^{\sigma +m+s-2}\,dx+x_{22}k_{1}^{n+1} \int _{\varOmega }u^{\sigma }\,dx \\& \quad \leq \bigl(2x_{20}\varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}x_{12}k_{1}+x_{22}k_{1} \bigr) \phi \\& \quad \quad{} + \biggl(2r_{7}x_{12}k_{1}^{n+1-\frac{x_{2}(n-2)n}{n-2-x_{1}n}} +2r_{6}x_{12}k_{1}^{nx_{1}+1- \frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{5}^{- \frac{x_{1}n}{n-2-x_{1}n}} \biggr) \\& \quad \quad{} \cdot \phi ^{\frac{x_{2}(n-2)}{n-2-x_{1}n}} +2r_{6}x_{12}k_{1}^{nx_{1}+1- \frac{x_{1}n^{2}}{n-2}} \frac{x_{1}n}{n-2}\varepsilon _{5}k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx, \end{aligned}$$
(3.15)

where \(\varepsilon _{5}\) is a positive constant which will be defined later.

Combining (3.6), (3.11), (3.13), and (3.15), we have

$$\begin{aligned} \phi ^{\prime }(t)&\leq \bigl(n \alpha +2r_{1}k_{2}x_{20} \varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}+2r_{4}x_{20}\varepsilon _{2}^{- \frac{x_{10}}{x_{20}}}x_{12}k_{1} +r_{4}x_{22}k_{1} \bigr)\phi \\ & \quad{} + \biggl(2r_{1}k_{2}r_{7}k_{1}^{n-\frac{x_{2}(n-2)n}{n-2-x_{1}n}}+2r_{1}k_{2}r_{6}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{3}^{- \frac{x_{1}n}{n-2-x_{1}n}} \\ &\quad{} +2r_{4}r_{7}x_{12}k_{1}^{n+1- \frac{x_{2}(n-2)n}{n-2-x_{1}n}} +2r_{4}r_{6}x_{12}k_{1}^{nx_{1}+1-\frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{5}^{-\frac{x_{1}n}{n-2-x_{1}n}} \biggr) \phi ^{\frac{x_{2}(n-2)}{n-2-x_{1}n}} \\ &\quad{}+\frac{1}{2}r_{2}k_{2}\varepsilon _{1}^{-1}r_{5} \bigl(2x_{20} \varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}x_{11}+x_{21} \bigr)\phi ^{ \frac{2s}{\sigma }+1} \\ &\quad{} +\frac{1}{2}r_{2}k_{2} \varepsilon _{1}^{-1}r_{5}x_{11}k_{1}^{n- \frac{x_{2}(n-2)n}{n-2-x_{1}n}}2r_{7} \phi ^{\frac{2s}{\sigma }+ \frac{x_{2}(n-2)}{n-2-x_{1}n}} \\ &\quad{}+r_{2}k_{2}\varepsilon _{1}^{-1}r_{5}r_{6}x_{11}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{4}^{- \frac{x_{1}n}{n-2-x_{1}n}}\phi ^{ \frac{(2s+\sigma x_{2})(n-2)}{\sigma (n-2-x_{1}n)}} \\ &\quad{}+ \biggl(2r_{1}k_{2}r_{6}k_{1}^{nx_{1}-\frac{x_{1}n^{2}}{n-2}} \frac{x_{1}n}{n-2}\varepsilon _{3} +\frac{1}{2}r_{2}k_{2} \varepsilon _{1}+r_{2}k_{2} \varepsilon _{1}^{-1}r_{5}r_{6}x_{11}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \frac{x_{1}n}{n-2}\varepsilon _{4} \\ &\quad{}+ 2r_{4}r_{6}x_{12}k_{1}^{nx_{1}+1-\frac{x_{1}n^{2}}{n-2}} \frac{x_{1}n}{n-2}\varepsilon _{5}-r_{3} \biggr) k_{1}^{n} \int _{\varOmega } \bigl\vert \nabla u^{\frac{\sigma +m-1}{2}} \bigr\vert ^{2}\,dx. \end{aligned}$$
(3.16)

If we choose suitable \(\varepsilon _{1}\), \(\varepsilon _{3}\), \(\varepsilon _{4}\), \(\varepsilon _{5}\) such that

$$ \begin{aligned}[b] &2r_{1}k_{2}r_{6}k_{1}^{nx_{1}-\frac{x_{1}n^{2}}{n-2}} \frac{x_{1}n}{n-2}\varepsilon _{3} +\frac{1}{2}r_{2}k_{2} \varepsilon _{1}+r_{2}k_{2} \varepsilon _{1}^{-1}r_{5}r_{6}x_{11}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \frac{x_{1}n}{n-2}\varepsilon _{4} \\ &\quad{} +2r_{4}r_{6}x_{12}k_{1}^{nx_{1}+1-\frac{x_{1}n^{2}}{n-2}} \frac{x_{1}n}{n-2}\varepsilon _{5}-r_{3}=0. \end{aligned} $$
(3.17)

Substituting (3.17) into (3.16), we derive

$$\begin{aligned} \begin{aligned}[b]\phi ^{\prime }(t)&\leq \bigl(n \alpha +2r_{1}k_{2}x_{20} \varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}+2r_{4}x_{20}\varepsilon _{2}^{- \frac{x_{10}}{x_{20}}}x_{12}k_{1} +r_{4}x_{22}k_{1} \bigr)\phi \\ & \quad{} + \biggl(2r_{1}k_{2}r_{7}k_{1}^{n-\frac{x_{2}(n-2)n}{n-2-x_{1}n}}+2r_{1}k_{2}r_{6}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{3}^{- \frac{x_{1}n}{n-2-x_{1}n}} \\ &\quad{} +2r_{4}r_{7}x_{12}k_{1}^{n+1- \frac{x_{2}(n-2)n}{n-2-x_{1}n}} +2r_{4}r_{6}x_{12}k_{1}^{nx_{1}+1-\frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{5}^{-\frac{x_{1}n}{n-2-x_{1}n}} \biggr) \phi ^{1+\frac{2x_{1}}{n-2-x_{1}n}} \\ &\quad{} +\frac{1}{2}r_{2}k_{2}\varepsilon _{1}^{-1}r_{5} \bigl(2x_{20} \varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}x_{11}+x_{21} \bigr)\phi ^{1+ \frac{2s}{\sigma }} \\ &\quad{} +\frac{1}{2}r_{2}k_{2} \varepsilon _{1}^{-1}r_{5}x_{11}k_{1}^{n- \frac{x_{2}(n-2)n}{n-2-x_{1}n}}2r_{7} \phi ^{1+(\frac{2s}{\sigma }+ \frac{2x_{1}}{n-2-x_{1}n})} \\ &\quad{} +r_{2}k_{2}\varepsilon _{1}^{-1}r_{5}r_{6}x_{11}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{4}^{- \frac{x_{1}n}{n-2-x_{1}n}}\phi ^{1+ \frac{2s(n-2)+2x_{1}\sigma }{\sigma (n-2-x_{1}n)}}. \end{aligned} \end{aligned}$$
(3.18)

Using Hölder’s and Young’s inequalities, we have

$$ \phi ^{1+\gamma }\leq \biggl(1-\frac{\gamma }{4} \biggr)\phi + \frac{\gamma }{4}\phi ^{5}. $$
(3.19)

Applying (3.19) to \(\phi ^{1+\frac{2x_{1}}{n-2-x_{1}n}}\), \(\phi ^{1+\frac{2s}{\sigma }}\), \(\phi ^{1+(\frac{2s}{\sigma }+\frac{2x_{1}}{n-2-x_{1}n})}\), \(\phi ^{1+ \frac{2s(n-2)+2x_{1}\sigma }{\sigma (n-2-x_{1}n)}}\) in (3.18), respectively, we obtain

$$ \phi ^{\prime }(t)\leq a(t)\phi (t)+b(t)\phi ^{5}(t), $$
(3.20)

where

$$ \begin{aligned}[b] a(t)&= \bigl(n\alpha +2r_{1}k_{2}x_{20}\varepsilon _{2}^{- \frac{x_{10}}{x_{20}}}+2r_{4}x_{20}\varepsilon _{2}^{- \frac{x_{10}}{x_{20}}}x_{12}k_{1} +r_{4}x_{22}k_{1} \bigr) \\ &\quad{} + \biggl(2r_{1}k_{2}r_{7}k_{1}^{n-\frac{x_{2}(n-2)n}{n-2-x_{1}n}}+2r_{1}k_{2}r_{6}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{3}^{- \frac{x_{1}n}{n-2-x_{1}n}} \\ &\quad{} +2r_{4}r_{7}x_{12}k_{1}^{n+1- \frac{x_{2}(n-2)n}{n-2-x_{1}n}} \\ &\quad{}+2r_{4}r_{6}x_{12}k_{1}^{nx_{1}+1-\frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{5}^{-\frac{x_{1}n}{n-2-x_{1}n}} \biggr) \biggl[1- \frac{x_{1}}{2(n-2-x_{1}n)} \biggr] \\ &\quad{} +\frac{1}{2}r_{2}k_{2}\varepsilon _{1}^{-1}r_{5} \bigl(2x_{20} \varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}x_{11}+x_{21} \bigr) \biggl(1- \frac{s}{2\sigma } \biggr) \\ &\quad{}+\frac{1}{2}r_{2}k_{2}\varepsilon _{1}^{-1}r_{5}x_{11}k_{1}^{n- \frac{x_{2}(n-2)}{n-2-x_{1}n}}2r_{7} \biggl[1- \frac{s(x_{2}n-2)+x_{1}\sigma }{2\sigma (n-2-x_{1}n)} \biggr] \\ &\quad{}+r_{2}k_{2}\varepsilon _{1}^{-1}r_{5}r_{6}x_{11}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{4}^{- \frac{x_{1}n}{n-2-x_{1}n}} \biggl[1- \frac{s(n-2)+x_{1}\sigma }{2\sigma (n-2-x_{1}n)} \biggr] \end{aligned} $$
(3.21)

and

$$ \begin{aligned}[b] b(t)&= \biggl(2r_{1}k_{2}r_{7}k_{1}^{n- \frac{x_{2}(n-2)n}{n-2-x_{1}n}}+2r_{1}k_{2}r_{6}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{3}^{- \frac{x_{1}n}{n-2-x_{1}n}} \\ &\quad{} +2r_{4}r_{7}x_{12}k_{1}^{n+1- \frac{x_{2}(n-2)n}{n-2-x_{1}n}} \\ &\quad{} +2r_{4}r_{6}x_{12}k_{1}^{nx_{1}+1-\frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{5}^{-\frac{x_{1}n}{n-2-x_{1}n}} \biggr) \frac{x_{1}}{2(n-2-x_{1}n)} \\ &\quad{}+\frac{1}{2}r_{2}k_{2}\varepsilon _{1}^{-1}r_{5} \bigl(2x_{20} \varepsilon _{2}^{-\frac{x_{10}}{x_{20}}}x_{11}+x_{21} \bigr) \frac{s}{2\sigma } \\ &\quad{}+\frac{1}{2}r_{2}k_{2} \varepsilon _{1}^{-1}r_{5}x_{11}k_{1}^{n- \frac{x_{2}(n-2)}{n-2-x_{1}n}}2r_{7} \biggl[ \frac{s(x_{2}n-2)+x_{1}\sigma }{2\sigma (n-2-x_{1}n)} \biggr] \\ &\quad{} +r_{2}k_{2}\varepsilon _{1}^{-1}r_{5}r_{6}x_{11}k_{1}^{nx_{1}- \frac{x_{1}n^{2}}{n-2}} \frac{n-2-x_{1}n}{n-2}\varepsilon _{4}^{- \frac{x_{1}n}{n-2-x_{1}n}} \frac{s(n-2)+x_{1}\sigma }{2\sigma (n-2-x_{1}n)}. \end{aligned} $$
(3.22)

Multiplying both sides of (3.20) by \(\phi ^{-5}(t)\), we obtain

$$ \phi ^{\prime }(t)\phi ^{-5}(t)\leq a(t) \phi ^{-4}(t)+b(t). $$
(3.23)

That is,

$$ - \bigl(\phi ^{-4}(t) \bigr)^{\prime } \leq 4a(t)\phi ^{-4}(t)+4b(t). $$
(3.24)

Setting \(H(t)=\int _{0}^{t}a(\tau )\,d\tau \), (3.24) can be rewritten as

$$ \bigl(\phi ^{-4}(t)e^{4H(t)} \bigr)^{\prime }\geq -4b(t)e^{4H(t)}. $$
(3.25)

Integrating (3.25) from 0 to t, we have

$$ \phi ^{-4}(t)e^{4H(t)}-\phi ^{-4}(0)\geq -4 \int _{0}^{t}b( \tau )e^{4H(\tau )}\,d\tau . $$
(3.26)

That is to say,

$$ \frac{e^{4H(t)}}{\phi ^{4}(t)}-\frac{1}{\phi ^{4}(0)} \geq -4\varTheta (t), $$
(3.27)

where \(\varTheta (t)=\int _{0}^{t}b(\tau )e^{4H(\tau )}\,d\tau \).

Taking the limit to (3.27) as \(t\rightarrow t^{*}\), we get

$$ \varTheta \bigl(t^{*} \bigr)\geq \frac{1}{4\phi ^{4}(0)}. $$

From the definition of \(\varTheta (t)\), we have \(\frac{d\varTheta (t)}{dt}=b(t)e^{4H(t)}>0\). We get \(\varTheta (t)\) is a strictly increasing function. So we can get

$$ t^{*}\geq \varTheta ^{-1} \biggl(\frac{1}{4\phi ^{4}(0)} \biggr), $$

from which we complete the proof of Theorem 3.1. □

References

  1. 1.

    Liu, Y.: Blow up phenomena for the nonlinear nonlocal porous medium equation under Robin boundary condition. Comput. Math. Appl. 66, 2092–2095 (2013)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Liu, D.M., Mu, C.L., Qiao, X.: Lower bounds estimate for the blow up time of a nonlinear nonlocal porous medium equation. Acta Math. Sci. Ser. B Engl. Ed. 32, 1206–1212 (2012)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Galaktionov, V.A.: On asymptotic self-similar behavior for a quasilinear heat equation: single point blow-up. SIAM J. Math. Anal. 26, 675–693 (1995)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bögelein, V., Duzaar, F., Korte, R., Scheven, C.: The higher integrability of weak solutions of porous medium systems. Adv. Nonlinear Anal. 8(1), 1004–1034 (2019)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Payne, L.E., Philippin, G.A., Schaefer, P.W.: Blow-up phenomena for some nonlinear parabolic problems. Nonlinear Anal. 69, 3495–3502 (2008)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Payne, L.E., Philippin, G.A., Schaefer, P.W.: Bounds for blow-up time in nonlinear parabolic problems. J. Math. Anal. Appl. 338, 438–447 (2008)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Payne, L.E., Schaefer, P.W.: Lower bounds for blow-up time in parabolic problems under Dirichlet conditions. J. Math. Anal. Appl. 328, 1196–1205 (2007)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Payne, L.E., Schaefer, P.W.: Lower bounds for blow-up time in parabolic problems under Neumann conditions. Appl. Anal. 85, 1301–1311 (2006)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Payne, L.E., Song, J.C.: Lower bounds for blow-up time in a nonlinear parabolic problem. J. Math. Anal. Appl. 354, 394–396 (2009)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Payne, L.E., Song, J.C.: Lower bounds for the blow-up time in a temperature dependent Navier–Stokes flow. J. Math. Anal. Appl. 335, 371–376 (2007)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Schaefer, P.W.: Blow up phenomena in some porous medium problems. Preprint 394-396

  12. 12.

    Song, J.C.: Lower bounds for blow-up time in a non-local reaction-diffusion problem. Appl. Math. Lett. 5, 793–796 (2011)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Li, Y.F., Liu, Y., Lin, C.H.: Blow-up phenomena for some nonlinear parabolic problems under mixed boundary conditions. Nonlinear Anal., Real World Appl. 11, 3815–3823 (2010)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Payne, L.E., Schaefer, P.W.: Blow-up in parabolic problems under Robin boundary conditions. Appl. Anal. 87, 699–707 (2008)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Papageorgiou, N.S., Radulescu, V.D., Repovs, D.D.: Parametric nonlinear resonant Robin problems. Math. Nachr. 292(11), 2456–2480 (2019)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Pikula, M., Vladicic, V., Vojvodic, B.: Inverse spectral problems for Sturm–Liouville operators with a constant delay less than half the length of the interval and 2 Robin boundary conditions. Results Math. 74(1), Article ID UNSP 45 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Gala, S., Liu, Q., Ragusa, M.A.: A new regularity criterion for the nematic liquid crystal flows. Appl. Anal. 91(9), 1741–1747 (2012)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Liu, Y., Luo, S.G., Ye, Y.H.: Blow-up phenomena for a parabolic problem with a gradient nonlinearity under nonlinear boundary conditions. Comput. Math. Appl. 65, 1194–1199 (2013)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Payne, L.E., Philippin, G.A., Vernier Piro, S.: Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition. Z. Angew. Math. Phys. 61, 999–1007 (2010)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Payne, L.E., Schaefer, P.W.: Bounds for the blow-up time for the heat equation under nonlinear boundary conditions. Proc. R. Soc. Edinb. 139, 1289–1296 (2009)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Tang, G.S.: Blow-up phenomena for a parabolic system with gradient nonlinearity under nonlinear boundary conditions. Comput. Math. Appl. 74, 360–368 (2017)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Ghisi, M., Gobbino, M., Haraux, A.: Quantization of energy and weakly turbulent profiles of solutions to some damped second-order evolution equations. Adv. Nonlinear Anal. 8(1), 902–927 (2019)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Papageorgiou, N.S., Radulescu, V.D., Repovs, D.D.: Nonlinear second order evolution inclusions with noncoercive viscosity term. J. Differ. Equ. 264(7), 4749–4763 (2018)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Papageorgiou, N.S., Radulescu, V.D., Repovs, D.D.: Nonlinear Analysis—Theory and Methods. Springer Monographs in Mathematics. Springer, Cham (2019)

    Google Scholar 

  25. 25.

    Ding, J.T., Shen, X.H.: Blow-up analysis in quasilinear reaction–diffusion problems with weighted nonlocal source. Comput. Math. Appl. 75, 1288–1301 (2018)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Shen, X.H., Ding, J.T.: Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions. Comput. Math. Appl. 77, 3250–3263 (2019)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Ding, J.T., Kou, W.: Blow-up solutions for reaction diffusion equations with nonlocal boundary conditions. J. Math. Anal. Appl. 470, 1–15 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Ding, J.T., Hu, H.J.: Blow-up and global solutions for a class of nonlinear reaction diffusion equations under Dirichlet boundary conditions. J. Math. Anal. Appl. 433, 1718–1735 (2016)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Chen, W.H., Liu, Y.: Lower bound for the blow up time for some nonlinear parabolic equations. Bound. Value Probl. 2016, 161 (2016)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. In: Universitext. Springer, New York (2011)

    Google Scholar 

Download references

Acknowledgements

The authors express their heartfelt thanks to the editors and referees who have provided some important suggestions.

Availability of data and materials

This paper focuses on theoretical analysis, not involving experiments and data.

Funding

The work was supported by the National Natural Science Foundation of China (Grant ♯ 61907010), Natural Science in Higher Education of Guangdong, China (Grant ♯ 2018KZDXM048), the General Project of Science Research of Guangzhou (Grant ♯ 201707010126), and the Science Foundation of Huashang College Guangdong University of Finance & Economics (Grant ♯ 2019HSDS26).

Author information

Affiliations

Authors

Contributions

The authors have equal contributions to each part of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Yiwu Lin.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ouyang, B., Lin, Y., Liu, Y. et al. Lower bound for the blow-up time for a general nonlinear nonlocal porous medium equation under nonlinear boundary condition. Bound Value Probl 2020, 76 (2020). https://doi.org/10.1186/s13661-020-01372-x

Download citation

Keywords

  • Lower bound
  • Blow-up time
  • Robin boundary condition
  • Nonlocal porous medium equation