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.

1 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 [5–12]). Some authors have started to consider the blow-up of these problems under Robin boundary conditions (see [13–17]). In papers [18–21], 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 [22–24]. 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 [25–29]). 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 $s>\max \{\frac{2n}{2n-1},p+q+1-m\}$.

2 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.

3 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. □