Blow-up Analysis to a Quasilinear Chemotaxis System with Nonlocal Logistic Effect

Abstract

In this paper, we consider the following quasilinear chemotaxis system involving nonlocal effect

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\nabla \cdot (\varphi (u)\nabla u)-\nabla \cdot (u\nabla v)+\mu u \left( 1-\int _{\Omega }u^{\alpha }\text {d}x\right) ,\ {} &{}\ \ x\in \Omega , \ t>0,\\[2.5mm] 0=\Delta v-m(t)+u,\ m(t)=\frac{1}{|\Omega |}\int _{\Omega } u(x,t)\text {d}x,\ {} &{}\ \ x\in \Omega , \ t>0,\\[2.5mm] u(x,0)=u_{0}(x), \ {} &{}\ \ x\in \Omega , \end{array} \right. \end{aligned}$$

where $\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 3)$ with $R>0,$ the parameters $\mu , \alpha$ are positive constants and diffusion function $\varphi (u)\le C_{0}(1+u)^{-m}$ for all $u\ge 0$ with $C_{0}>0$ and $m> -1.$ It has been shown that if

$$\begin{aligned} 0<\alpha <\min \left\{ 2,\frac{n}{2},\frac{n(m+1)}{2}\right\} , \end{aligned}$$

then there exist suitable initial data $u_{0}$ such that the corresponding radially symmetric solution blows up in finite time. In this work, we extend the blow-up result established by previous researchers.

1 Introduction

During the 1970s, in order to describe the movement of biological organisms toward the gradient of certain chemical signal substance, Keller and Segel [14] first introduced a famous chemotaxis model (also called Keller–Segel model)

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\Delta u-\chi \nabla \cdot ( u\nabla v),\ {} &{} x\in \Omega , \ t>0,\\[2.5mm] \tau v_{t}=\Delta v-v+u,\ {} &{} x\in \Omega , \ t>0, \end{array} \right. \end{aligned}$$

(1.1)

where $\Omega \subset {\mathbb {R}}^{n} (n\ge 1)$ is a bounded smooth domain, $\tau \in \{0,1\},$ and the chemotaxis sensitivity coefficient $\chi >0.$ Here, the function u(x, t) is the density of cell and v(x, t) stands for the concentration of the chemical signal secreted by cell. Over the past few decades, considerable efforts have been done on the dynamical behavior (including global existence and boundedness, as well as the existence of blow-up solution) of the solutions to system (1.1) (see [12, 19, 20, 22, 24,25,26,27, 40], for instance).

It has been turned out that the system (1.1) involving nonlinear diffusions may lead to a wide array of interesting properties to the solutions. Sometimes, such nonuniform diffusion was also called volume-filling effect proposed by Hillen and Painter [13, 23]. The related model was given by

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\nabla \cdot (D(u)\nabla u)-\nabla \cdot (S(u)\nabla v),\ {} &{} x\in \Omega , \ t>0,\\[2.5mm] \tau v_{t}=\Delta v-v+u,\ {} &{} x\in \Omega , \ t>0, \end{array} \right. \end{aligned}$$

(1.2)

where D(u) and S(u) are positive functions which are used to characterize the strength of diffusion and chemoattractant, respectively. For the case of fully parabolic type of system (1.2), when D(u)and S(u) are nonlinear functions of u,  the existence of global classical solutions or blow-up solutions depends on the value of $\frac{S(u)}{D(u)}.$ Namely, Winkler [38] proved that if $\frac{S(u)}{D(u)}\ge cu^{\alpha }$ with $\alpha >\frac{2}{n}$ and some constant $c>0$ for all $u>1$, then for any $M>0$ there exist solutions that blow up in either finite or infinite time with mass $\int _{\Omega }u_{0}=M$. Later on, Tao and Winkler [30] showed that such result is optimal, i.e., if $\frac{S(u)}{D(u)}\le cu^{\alpha }$ with $\alpha <\frac{2}{n},n\ge 1$ and some constant $c>0$ for all $u>1$, then the system (1.2) possesses globally bounded classical solutions. For the case of parabolic-elliptic type of system (1.2) with the second equation replaced by $0=\Delta v-m(t)+u$ with $m(t)=\frac{1}{|\Omega |}\int _{\Omega }u,$ there have been available results [3, 8]. The critical result has been established by Winkler and Djie [43] by letting $D(u)=(u+1)^{-p}$ with $p\ge 0$ and $S(u)=u(u+1)^{q-1}$ with $q\in {\mathbb {R}}.$ It has been showed that if $p+q<\frac{2}{n}$, the system has globally bounded classical solutions, whereas if $p+q>\frac{2}{n}$, then the solutions exist and will blow up in finite time. Considering $S(u)=u$ and $D(u)=(1+u)^{-p}$ with $p\in {\mathbb {R}},$ Cieślak and Winkler [8] showed that there exists a critical exponent $\frac{2}{n}-1,$ which distinguishes between finite-time blow-up and global-in-time existence of uniformly bounded solutions. Furthermore, for the system (1.2) and its variants, the corresponding criteria for global boundedness and blow-up were also established, please refer to [3,4,5,6,7, 17, 18, 32, 33, 36, 42, 47] for instance.

When considering the growth and death of cells, we arrive at the following model with logistic source

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\Delta u-\chi \nabla \cdot (u\nabla v)+f(u),\ {} &{} x\in \Omega , \ t>0,\\[2.5mm] 0=\Delta v-v+u,\ {} &{} x\in \Omega , \ t>0. \end{array} \right. \end{aligned}$$

(1.3)

The studies showed that the source term may substantially influence the dynamics behavior of solutions in chemotaxis system. For instance, if $f(u)\le u(a-bu)$ with $a,b>0,$ it has been shown by Tello and Winkler [31] that the classical solution of (1.3) is globally bounded whenever $\frac{n-2}{n}\chi <b, n\ge 3.$ In [37], when $f(u)=au-bu^{\kappa }$ with $\kappa >1,$ $a\ge 0$ and $b>0,$ Winkler introduced a concept of very weak solutions and obtained the global existence of such solutions for any nonnegative initial data $u_{0}\in L^{1}(\Omega )$ under the condition that $\kappa >2-\frac{1}{n}.$ When $f(u)\le u(a-bu^{s})$ and the second equation turns into $0=\Delta v-v+u^{k}$ with $k,s>0,$ Xiang [44] showed that if either $s>k$ or $s=k$ with $\frac{kn-2}{kn}\chi <b,$ the system (1.2) has global classical solutions. In contrast, if second equation of (1.3) is replaced by $0=\Delta v-m(t)+u,\ m(t)=\frac{1}{|\Omega |}\int _{\Omega } u(x,t)\text {d}x$ in a ball, Winkler [39] constructed radial solutions and proved that the solutions blow up in finite-time with $f(u)=\lambda u -\mu u^{\kappa }$ when $\lambda \ge 0, n\ge 5$ and $\kappa \in (1,\frac{3}{2}+\frac{1}{2(n-1)}).$ Moreover, if $n=3,4$ and $\Omega$ is a ball, Winkler [41] also showed that if $\kappa \in (1,\frac{7}{6}),$ the radial solutions will blow up. Fuest [10] showed that the exponent $\kappa =2$ is actually critical in the four and higher dimensional setting. Moreover, the quasilinear systems and fully parabolic systems with logistic source also have been widely studied in [2, 11, 16, 34, 35, 45, 46]

The logistic source mentioned in the above model is only a local (pointwise) reaction term. However, in reality it is of great significance to take nonlocal interactions into account. As it has been shown that the proliferation of the population may relies on the total mass of the population in a neighborhood. In [28], a cancer invasion model with nonlocal reaction terms

$$\begin{aligned} \mu u\left( 1-\int _{\Omega }K_{1}(x,y)u(y,t)\text {d}y-\int _{\Omega }K_{2}(x,y)v(y,t)\text {d}y \right) \end{aligned}$$

has been studied, where these nonlocal reaction terms are used to describe the inhibition of cell proliferation caused by the density of surrounding cancer cells and tissue, respectively. Based on the maximum principle, Negreanu and Tello [21] investigated a system of partial differential equations under chemotactic effects with nonlocal reaction terms and derived the convergence of corresponding classical solutions. In addition, Bian et al. [1] analyzed the following system with nonlocal nonlinear source

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\Delta u-\chi \nabla \cdot (u\nabla v)+ u^{\alpha } \left( 1-\int _{\Omega }u^{\beta }\text {d}x\right) ,\ {} &{} x\in \Omega , \ t>0,\\[2.5mm] 0=\Delta v-v+u, \ {} &{} x\in \Omega , \ t>0 \end{array} \right. \end{aligned}$$

(1.4)

with $\alpha \ge 1$ and $\beta >1,$ and they proved that the system admits a global classical solution if $n\ge 3,$ either $2\le \alpha <1+\frac{2\beta }{n}$ or $\alpha <2$ and $\frac{(2+n)(2-\alpha )}{n}<1+\frac{2\beta }{n}-\alpha .$ Tao and Fang [29] considered the corresponding quasilinear system of (1.4) (i.e., the first equation replaced by $u_{t}= \text {d}\nabla \cdot ((1+u)^{m-1} \nabla u)-\chi \nabla \cdot (u(1+u)^{\sigma -2}\nabla v)+ u^{\alpha } \left( 1-\int _{\Omega }u^{\beta }\text {d}x\right)$ with $d,m,\chi >0$ and $\sigma \ge 1$ ) and proved that a global classical solution exists under the condition $\sigma +\frac{n}{2}(\sigma -m)-\beta<\alpha <m+\frac{2}{n}\beta .$ Nevertheless, there is a problem left in [29, Remark 1] that it is unclear whether the solution remains bounded or not in the case of $\beta \le \frac{n}{2}(\sigma -m).$ Recently, Du and Liu [9] studied the following chemotaxis system with nonlocal effect

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\Delta u-\nabla \cdot (u\nabla v)+\mu u \left( 1-\int _{\Omega }u^{\alpha }\text {d}x\right) ,\ {} &{} x\in \Omega , \ t>0,\\[2.5mm] 0=\Delta v-m(t)+u,\ m(t)=\frac{1}{|\Omega |}\int _{\Omega } u(x,t)\text {d}x,\ &{} x\in \Omega , \ t>0, \end{array} \right. \end{aligned}$$

(1.5)

and showed that the system has finite-time blow-up solutions in radial setting under the assumption that $0<\alpha <\min \big \{2,\frac{n}{2}\big \}.$

Inspired by the work mentioned above, we consider the quasilinear version of (1.5) as follows

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\nabla \cdot (\varphi (u)\nabla u)-\nabla \cdot (u\nabla v)+\mu u \left( 1-\int _{\Omega }u^{\alpha }\text {d}x\right) ,\ {} &{} x\in \Omega , \ t>0,\\[2.5mm] 0=\Delta v-m(t)+u,\ m(t)=\frac{1}{|\Omega |}\int _{\Omega } u(x,t)\text {d}x,\ {} &{} x\in \Omega , \ t>0,\\[2.5mm] \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0, \ {} &{} x\in \partial \Omega , \ t>0,\\[2.5mm] u(x,0)=u_{0}(x), \ {} &{} x\in \Omega , \end{array} \right. \end{aligned}$$

(1.6)

in a ball $\Omega =B_{R}(0)\subset {\mathbb {R}}^{n}(n\ge 3)$ with $R>0,$ where $\nu$ denotes the outward unit normal vector on $\partial \Omega$ and the parameters $\mu , \alpha >0.$

In this paper, we assume that nonlinear diffusion function $0<\varphi \le C_{0}(1+u)^{-m}$ with $C_{0}>0$ and $m>-1$ satisfies

$$\begin{aligned} \varphi \in C^{2}([0,\infty )), \end{aligned}$$

(1.7)

and initial data $u_{0}$ fulfill

$$\begin{aligned} S&(m_{0},m_{1},r_{1})=\bigg \{0\le u_{0}\in C^{1}(\overline{\Omega }) \ \text{ is } \text{ radially } \text{ symmetric } \text{ and } \text{ nonincreasing } \nonumber \\&\text{ with } \text{ respect } \text{ to } \ |x|,\ \text{ as } \text{ well } \text{ as }\ \frac{1}{|\Omega |}\int _{\Omega }u_{0}\text {d}x=m_{0} \ \text{ and } \ \frac{1}{|\Omega |}\int _{ B_{r_{1}(0)}}u_{0}\text {d}x\ge m_{1}\bigg \}, \end{aligned}$$

(1.8)

where constants $m_{0},m_{1},r_{1}>0.$

More precisely, we state our main result as follows.

Theorem 1.1

Let $\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 3)$ be a ball with $R>0$ and the parameters $\alpha ,\mu >0.$ Denote $m_{0}=\frac{1}{|\Omega |}\int _{\Omega }u_{0}\text {d}x >0.$ Assume that the diffusion function $\varphi \in C^{2}([0,\infty ))$ satisfies $\varphi (u)\le C_{0}(1+u)^{-m}$ with $C_{0}>0$ and $m> -1.$ If

$$\begin{aligned} 0<\alpha <\min \left\{ 2,\frac{n}{2},\frac{n(m+1)}{2}\right\} , \end{aligned}$$

(1.9)

then there exists $r_{1}\in (0,R)$ such that for any $u_{0}$ fulfilling the condition (1.8) with $m_{1}\in (0,m_{0}),$ the system (1.6) admits a classical solution, which blows up in finite time.

Remark 1.2

The blow-up result in this paper extends the one established in [9]. Especially, when $m\in (-1,0),$ we have $\frac{n}{2}>\frac{n(m+1)}{2}.$ Thus, for this case the blow-up interval for $\alpha$ somewhat can be reduced to a smaller one.

The outline of this paper is arranged as follows. In Sec. 2, we state a result on local existence of solutions and then transform the system (1.6) into a scalar problem. In Sec. 3, we get some properties to function w(s, t) defined in Sec. 2, which are crucial for blow-up analysis of solutions. In Sec. 4, we show that the solution of system (1.6) blows up in finite time for some suitable initial data $u_{0}.$

2 Local existence and transformation to a scalar problem

To begin with, we state a lemma involving the local existence of solutions to (1.6), which can be proved under the framework of standard fixed point argument (see [39, 43], for instance).

Lemma 2.1

Let $\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 1)$ be a ball with $R>0$ and the parameters $\alpha ,\mu >0.$ Then there exist $T_{\max }\in (0,+\infty ]$ and a pair of nonnegative functions (u, v) fulfilling

$$\begin{aligned} \left\{ \begin{array}{ll} u\in C^{0}(\overline{\Omega }\times {[0,T_{\max })})\cap C^{2,1}({\overline{\Omega }}\times (0,T_{\max })), \\[2.5mm] v\in \cap _{q>n}L^{\infty }((0,T_{\max });W^{1,q}(\Omega ))\cap C^{2,0}({\overline{\Omega }}\times (0,T_{\max })), \end{array} \right. \end{aligned}$$

(2.1)

such that (u, v) is a classical solution to the system (1.6), and that

$$\begin{aligned} \text{ either } \ T_{\max }=\infty , \ \text{ or } \ \limsup _{t\nearrow T_{\max }}\Vert u(\cdot ,t)\Vert _{L^{\infty }(\Omega )}=+\infty . \end{aligned}$$

(2.2)

Furthermore, $u\ge 0$ in $\Omega \times (0,T_{\max }),$ and v is radially symmetric with respect to |x|.

From now on, we will assume by contradiction that $T_{\max }=\infty$ and establish all kinds of estimates for $t\in [0,1]$ to get a contradiction. In the radial setting, we transform the system (1.6) into a scalar problem

$$\begin{aligned} \left\{ \begin{array}{ll} r^{n-1}u_t=(r^{n-1}\varphi (u)u_r)_r-(r^{n-1}uv_r)_r\\ +\mu r^{n-1}u\big (1-\int _{\Omega }u^{\alpha }\text {d}x\big ), \ {} &{} r\in (0,R), \ t\in (0,1],\\[2.5mm] 0=(r^{n-1}v_r)_r-r^{n-1}m(t)+r^{n-1}u, \ {} &{} r\in (0,R), \ t\in (0,1],\\[2.5mm] u_r=v_r=0, \ {} &{} r=R, \ t\in (0,1],\\[2.5mm] u(r,0)=u_0(r), \ {} &{} r\in (0,R). \end{array} \right. \end{aligned}$$

(2.3)

Let

$$\begin{aligned} w(s,t)=\int _0^{s^\frac{1}{n}}\rho ^{n-1}u(\rho ,t)\text {d}\rho , \ \ \ s=r^{n}\in [0,R^{n}], t\in [0,1]. \end{aligned}$$

(2.4)

Recalling (1.8), we know that $w\in C^{0}({\overline{\Omega }}\times [0,1])\cap C^{2,1}({\overline{\Omega }}\times (0,1]).$ By a simple calculation, it is easy to get

$$\begin{aligned} w_s(s,t)&=\frac{1}{n}u(s^{\frac{1}{n}},t) \ \text{ and } \ w_{ss}(s,t)=\frac{1}{n^2}s^{\frac{1}{n}-1}u_r(s^\frac{1}{n},t), \nonumber \\&\quad s\in (0,R^{n}), t\in (0,1]. \end{aligned}$$

(2.5)

Thus, it can be deduced from (2.3)–(2.5) that

$$\begin{aligned} w_t (s,t) =&\,\int _0^{s^\frac{1}{n}}\bigg [(\rho ^{n-1}\varphi (u)u_\rho )_\rho -(\rho ^{n-1}uv_\rho ) _\rho +\mu \rho ^{n-1} u\big (1-\int _{\Omega }u^{\alpha }\text {d}x\big )\bigg ]\text {d}\rho \nonumber \\ =&\,s^{1-\frac{1}{n}}\varphi (u)u_r(s^{\frac{1}{n}},t)-s^{1-\frac{1}{n}}u (s^{\frac{1}{n}},t) v_r(s^{\frac{1}{n}},t)+\int _0^{s^\frac{1}{n}}\mu \rho ^{n-1} u\big (1-\int _{\Omega }u^{\alpha }\text {d}x\big )\text {d}\rho \nonumber \\ =&\, s^{1-\frac{1}{n}}\varphi (u)n^2s^{1-\frac{1}{n}}w_{ss}(s,t)-s^{1-\frac{1}{n}}nw_s(s,t) v_r(s^{\frac{1}{n}},t) +\int _0^{s^\frac{1}{n}}\mu \rho ^{n-1} u\text {d}\rho \nonumber \\&\quad -\int _0^{s^\frac{1}{n}}\mu \rho ^{n-1} u\int _{\Omega }u^{\alpha }\text {d}x\text {d}\rho \nonumber \\ =&\,n^2s^{2-\frac{2}{n}}\varphi (nw_s)w_{ss}-ns^{1-\frac{1}{n}}w_s(s,t) v_r(s^\frac{1}{n},t)+\mu w \nonumber \\&\quad -\mu n^\alpha \beta (n)w\int _0^{R^{n}} w_s^{\alpha }(\sigma ,t)\text {d}\sigma , \end{aligned}$$

(2.6)

where $\beta (n)=|B_{1}(0)|.$ Besides, we integrate the second equation of system (2.3) over (0, r) to obtain

$$\begin{aligned} r^{n-1}v_r=m(t)\int _0^r\rho ^{n-1}\text {d}\rho -\int _0^r\rho ^{n-1}u\text {d}\rho . \end{aligned}$$

(2.7)

Let $r=s^{\frac{1}{n}}.$ It is easy to see that

$$\begin{aligned} v_r=\frac{m(t)}{n}s^{\frac{1}{n}}-s^{\frac{1}{n}-1}w. \end{aligned}$$

(2.8)

Substituting (2.8) into (2.6), we get

$$\begin{aligned} w_t(s,t)=&\,n^2s^{2-\frac{2}{n}}\varphi (nw_s)w_{ss}-ns^{1-\frac{1}{n}}w_s(s,t) \bigg (\frac{m(t)}{n}s^{\frac{1}{n}} -s^{\frac{1}{n}-1}w\bigg ) +\mu w(s,t)\nonumber \\&-\mu n^\alpha \beta (n)w\int _0^{R^{n}}w_s^{\alpha }(\sigma ,t)\text {d}\sigma \nonumber \\ =&\,n^2s^{2-\frac{2}{n}}\varphi (nw_s)w_{ss}+nww_s-m(t)sw_s +\mu w \nonumber \\&-\mu n^\alpha \beta (n)w\int _0^{R^{n}} w_s^{\alpha }(\sigma ,t)\text {d}\sigma \end{aligned}$$

(2.9)

for all $s\in (0, R^n)$ and $t\in (0,1],$ with the properties that

$$\begin{aligned} 0=w(0,t)\le w(s,t)\le w(R^n,t)=\frac{m(t)R^{n}}{n}, \end{aligned}$$

(2.10)

as well as

$$\begin{aligned} w_s(s,t)\ge 0. \end{aligned}$$

(2.11)

3 Some properties for w

In this section, we shall prove the concavity of $w(\cdot ,t)$ throughout evolution. As a preparation for this, we first give an upper bound estimate for m(t).

Lemma 3.1

Let $\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 3)$ be a ball with $R>0$ and the parameters $\alpha ,\mu >0.$ Then m(t) can be estimated as

$$\begin{aligned} m(t)\le m_{0}e^{\mu }, \ t\in [0,1], \end{aligned}$$

(3.1)

where $m_{0}=\frac{1}{|\Omega |}\int _{\Omega }u_{0}\text {d}x$ is defined as in (1.8).

Proof

Integrating the first equation of (1.6) over $\Omega ,$ due to $\mu >0,$ we can get

$$\begin{aligned} \frac{d}{dt}\int _{\Omega }u(\cdot ,t)\text {d}x =&\,\mu \int _{\Omega }u\text {d}x\bigg (1-\int _{\Omega }u^{\alpha }\text {d}x\bigg )\nonumber \\ \le&\,\mu \int _{\Omega } u \text {d}x. \end{aligned}$$

(3.2)

By a simple calculation, it is not difficult to see that

$$\begin{aligned} \int _{\Omega } u \text {d}x\le \int _{\Omega } u_{0}\text {d}x\cdot e^{\mu t}\le \int _{\Omega } u_{0}\text {d}x\cdot e^{\mu }, \ t\in [0,1]. \end{aligned}$$

(3.3)

Thus, from the definition of m(t),  we can directly obtain the desired result (3.1). $\square$

Lemma 3.2

Let $\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 3)$ be a ball with $R>0$ and the parameters $\alpha ,\mu >0.$ Assume that the diffusion function $\varphi$ and initial data $u_{0}$ satisfy conditions (1.7) and (1.8), respectively. Then, there hold

$$\begin{aligned} u_{r}(r,t)\le 0 \ \text{ for } \text{ all } \ (r,t)\in (0,R)\times (0,1], \end{aligned}$$

(3.4)

as well as

$$\begin{aligned} w_{ss}(s,t)\le 0 \ \text{ for } \text{ all } \ {(s,t)\in (0,R^{n})\times (0,1]}, \end{aligned}$$

(3.5)

where w is defined by (2.4).

Proof

We first prove (3.4) with the additional assumption $u_{0}\in C^{2}([0,\infty ))$ except for condition (1.8). Then, according to the well-known theory on higher regularity in scalar parabolic equations [15], we derive

$$\begin{aligned} u,u_{r}\in C^{0}([0,R]\times [0,1])\cap C^{2,1}((0,R)\times (0,1]). \end{aligned}$$

(3.6)

The second equation of system (2.3) enables us to obtain

$$\begin{aligned} v_{rr}+\frac{n-1}{r}v_{r}=m(t)-u. \end{aligned}$$

(3.7)

We substitute (3.7) into the first equation of system (2.3) to get

$$\begin{aligned} u_{t}=\varphi (u)u_{rr}+\frac{n-1}{r}\varphi (u)u_{r}+\varphi '(u)u_{r}^{2} -u_{r}v_{r}-um(t)+u^{2}+\mu u \bigg (1-\int _{\Omega }u^{\alpha }\text {d}x\bigg ). \end{aligned}$$

(3.8)

By differentiating Eq. (3.8) with respect to r,  it is easy to deduce

$$\begin{aligned} u_{rt}=&\varphi (u)u_{rrr}+\varphi '(u)u_{r}u_{rr}-\frac{n-1}{r^{2}}\varphi (u)u_{r}+\frac{n-1}{r}\varphi '(u)u_{r}^{2}+\frac{n-1}{r}\varphi (u)u_{rr} -u_{rr}v_{r}\nonumber \\&-v_{rr}u_{r}+\varphi ''(u)u_{r}^{3}+2\varphi '(u)u_{r}u_{rr} -m(t)u_{r}+2uu_{r}+\mu \bigg (1-\int _{\Omega }u^{\alpha }\text {d}x\bigg )u_{r}. \end{aligned}$$

(3.9)

Since $u\in C^{0}([0,R]\times [0,1]),$ we may choose a positive constant $\lambda$ large enough such that

$$\begin{aligned} \lambda \ge 6\Vert u\Vert _{L^{\infty }((0,R)\times (0,1))}+2\mu . \end{aligned}$$

(3.10)

For any $\varepsilon >0,$ we thereupon define

$$\begin{aligned} h(r,t)=u_{r}(r,t)-\varepsilon e^{\lambda t} \ \text{ for } \text{ all } \ (r,t)\in [0,R] \times [0,1]. \end{aligned}$$

(3.11)

Thus, h belongs to $C^{0}([0,R]\times [0,1])$ and satisfies

$$\begin{aligned} h_{t}=&\,\varphi (u)h_{rr}+\bigg (3\varphi '(u)h+{3\varphi '(u)\varepsilon e^{\lambda t }}+\frac{n-1}{r}\varphi (u)-v_{r}\bigg )h_{r}+\varphi ''(u)(h+\varepsilon e^{\lambda t})^{3}\nonumber \\&+\frac{n-1}{r}\varphi '(u)(h+\varepsilon e^{\lambda t})^{2}+b(r,t)(h+\varepsilon e^{\lambda t})-\varepsilon \lambda e^{\lambda t} \end{aligned}$$

(3.12)

for all $(r,t)\in (0,R] \times (0,1],$ where

$$\begin{aligned} b(r,t)=-\frac{n-1}{r^{2}}\varphi (u)-v_{rr}-m(t)+2u+\mu \bigg (1-\int _{\Omega }u^{\alpha }\text {d}x\bigg ) \end{aligned}$$

(3.13)

and

$$\begin{aligned} h(r,0)=u_{0r}(r,0)-\varepsilon <0 \ \text{ for } \text{ all } \ r\in [0,R], \end{aligned}$$

due to (1.8), as well as also

$$\begin{aligned} \ h(0,t)=h(R,t)=- \varepsilon e^{\lambda t}<0 \ \text{ for } \text{ all } \ t\in [0,1]. \end{aligned}$$

Nextly, we are going to prove that actually $h(r,t)<0$ for all $(r,t)\in [0,R]\times \in [0,1]$ by contradiction. Suppose that this is false. The standard maximum principle enables us to find some $r_{0}\in (0,R)$ and $t_{0}\in (0,1]$ such that

$$\begin{aligned} h(r_{0},t_{0})=0, h_{r}(r_{0},t_{0})=0,h_{rr}(r_{0},t_{0})\le 0 \ \ \text{ and } \ \ h_{t}(r_{0},t_{0})\ge 0. \end{aligned}$$

(3.14)

Using (3.12) and (3.14), we can conclude from the positivity of $\varphi (u)$ that

$$\begin{aligned} 0\le h_{t}(r_{0},t_{0})\le&\varphi ''(u)\varepsilon ^{3} e^{3\lambda t_{0}} +\frac{n-1}{r}\varphi '(u)\varepsilon ^{2}e^{2\lambda t_{0}}+(b(r_{0},t_{0})-\lambda )\varepsilon e^{\lambda t_{0}}. \end{aligned}$$

(3.15)

Substituting (3.7) into (3.13), it is easy to get

$$\begin{aligned} b(r,t)=-\frac{n-1}{r^{2}}\varphi (u)+\frac{n-1}{r}v_{r}-2m(t)+3u+\mu - \mu \int _{\Omega }u^{\alpha }\text {d}x \end{aligned}$$

(3.16)

for all $(r,t)\in (0,R)\times (0,1).$ Due to (2.7), we can easily obtain

$$\begin{aligned} v_r=\frac{m(t)}{n}r-r^{1-n}\int _0^r\rho ^{n-1}u\text {d}\rho . \end{aligned}$$

(3.17)

Combining (3.16) with (3.17), we have

$$\begin{aligned} b(r,t)=&\,-\frac{n-1}{r^{2}}\varphi (u)+\frac{n-1}{r}\bigg (\frac{m(t)}{n}r- r^{1-n}\int _0^r\rho ^{n-1}u\text {d}\rho \bigg )\nonumber \\&-2m(t)+3u+\mu - \mu \int _{\Omega }u^{\alpha }\text {d}x \nonumber \\ =&\, -\frac{n-1}{r^{2}}\varphi (u)+\frac{n-1}{n}m(t)-2m(t)- \frac{n-1}{r}r^{1-n}\int _0^r\rho ^{n-1}u\text {d}\rho \nonumber \\&+3u+\mu - \mu \int _{\Omega }u^{\alpha }\text {d}x\nonumber \\ \le&\, 3u+\mu \le \frac{\lambda }{2} \ \text{ for } \text{ all } \ (r,t)\in [0,R]\times \in [0,1]. \end{aligned}$$

(3.18)

Thus, based on (3.15), we derive

$$\begin{aligned} 0\le h_{t}(r_{0},t_{0})\le&\varphi ''(u)\varepsilon ^{3} e^{3\lambda t_{0}} +\frac{n-1}{r}\varphi '(u)\varepsilon ^{2}e^{2\lambda t_{0}}- \frac{\varepsilon \lambda }{2}e^{\lambda t_{0}}. \end{aligned}$$

(3.19)

Choosing $\varepsilon > 0$ sufficiently small, the inequality (3.19) means that

$$\begin{aligned} 0\le h_{t}(r_{0},t_{0})\le - \frac{\varepsilon \lambda }{4}e^{\lambda t_{0}}<0, \end{aligned}$$

(3.20)

which is contradictory. Hence, there holds $h(r,t)<0$ for all $(r,t)\in [0,R]\times [0,1].$ The desired result (3.4) can be deduced from (3.11) by letting $\varepsilon \searrow 0.$ Thus $w_{ss}(s,t)=\frac{1}{n^2}s^{\frac{1}{n}-1}u_r(s^\frac{1}{n},t)\le 0.$ For arbitrary $u_{0}$ merely satisfying (1.8), we can use the same method as in [42] to obtain the desired conclusions, here we omit it. $\square$

Lemma 3.3

Suppose that the initial data $u_{0}$ satisfy condition (1.8). For any $s\in [0,R^{n}]$ and $t\in [0,1],$ the following inequality holds

$$\begin{aligned} w_{s}(s,t)\le \frac{w(s,t)}{s}\le w_{s}(0,t). \end{aligned}$$

(3.21)

Proof

For fixed $t\in [0,1]$ and $s\in [0,R^{n}],$ we can get from mean value theorem that

$$\begin{aligned} w(s,t)=sw_{s}(\xi ,t), \end{aligned}$$

where $\xi \in (0,s).$ Due to $w_{ss}(s,t)=\frac{1}{n^2}s^{\frac{1}{n}-1}u_r(s^\frac{1}{n},t)$ and Lemma 3.2, it is easy to see that $w_{s}$ is decreasing and thus

$$\begin{aligned} w_s(s,t)\le w_s(\xi ,t)\le w_s(0,t). \end{aligned}$$

Hence, we can easily derive (3.21). $\square$

4 The finite-time blow-up analysis

In this section, we introduce a generalized moment-like functional $\phi (t)$ as defined in [41, 42]

$$\begin{aligned} \phi (t):=\int _0^{s_{0}}s^{-\gamma }(s_0-s)w(s,t)ds,\ \ t\in [0,1], \end{aligned}$$

(4.1)

where $\gamma \in (-\infty ,1)$ and $s_0\in (0,R^n),$ and w is defined in (2.4). Since u and $u_{t}$ are continuous in ${\overline{\Omega }}\times [0,1]$ and in $\overline{\Omega }\times (0,1),$ respectively, it can be verified that for any such $\gamma$ and $s_{0}$ the mapping $(0,s_{0})\ni s\mapsto s^{-\gamma }(s-s_{0})$ is integrable, the function $\phi (t)$ is well-defined and belongs to $C^{0}([0,1))\cap C^{1}((0,1)).$

In the following, in order to establish a superlinear differential inequality for $\phi (t),$ we give some restrictions on the corresponding parameters. For $m> -1$ and $0<\alpha <\min \left\{ 2,\frac{n}{2},\frac{n(m+1)}{2}\right\} ,$ let

$$\begin{aligned} 2-\frac{2}{\alpha }<\gamma <\min \left\{ 1, 2-\frac{4}{n}, 2-\frac{4}{n(m+1)}\right\} , \end{aligned}$$

(4.2)

and define the parameter $\tau$ as

$$\begin{aligned} 0<\tau <\min \left\{ 1, 2-\alpha , 2-\frac{4}{n}, 2-\frac{4}{n(m+1)}\right\} . \end{aligned}$$

(4.3)

For $\theta \in (0,1)$ and $\eta >\theta -(1-\theta )(1-\gamma ),$ we define W used later as follows

$$\begin{aligned} W(\eta ,\theta ,\gamma )=\bigg (\frac{\eta -\theta }{1-\theta }+1-\gamma \bigg )^{\theta -1}, \end{aligned}$$

(4.4)

where $\gamma$ is as in (4.2).

Lemma 4.1

Suppose that the diffusion function $\varphi (u)\le C_{0}(1+u)^{-m}$ with $C_{0}>0$ and $m>-1.$ Then, for any $s_{0}\in (0,R^{n})$ and $\gamma$ satisfying (4.2), there holds

$$\begin{aligned} \phi '(t)\ge&I_{1}+n\int _0^s s^{-\gamma }(s_0-s)w w_s\text {d}s -m(t) \int _0^{s_0}s^{1-\gamma }(s_0-s)w_s\text {d}s\nonumber \\&-\mu n^{\alpha }\beta (n)\int _0^{s_0}\int _0^{R^{n}}s^{-\gamma }(s_0-s)w(s,t) w_s^{\alpha }(\sigma ,t)\text {d}\sigma \text {d}s \ \ \text{ for } \ {t\in (0,1]}, \end{aligned}$$

(4.5)

with $\phi (t)$ defined in (4.1), where

$$\begin{aligned} I_{1}:=\left\{ \begin{array}{ll} -\frac{nC_0}{m-1}(2-\frac{2}{n}-\gamma )\int _0^{s_0}s^{1-\frac{2}{n}-\gamma }(s_0-s)ds,\ {} &{} \text{ if } \,\, m\in (1,\infty ), \\ -nC_0(2-\frac{2}{n}-\gamma )\int _0^{s_0}s^{1-\frac{2}{n}-\gamma }(s_0-s)\ln (1+nw_s)ds, \ {} &{} \text{ if } \,\, m=1,\\ -\frac{nC_0}{1-m}(2-\frac{2}{n}-\gamma )\int _0^{s_0}s^{1-\frac{2}{n}-\gamma } (s_0-s)(1+nw_s)^{1-m}ds, \ {} &{} \text{ if } \ m\in (-1,1). \end{array} \right. \end{aligned}$$

(4.6)

Proof

Recalling (2.9), we remove the positive term $\mu w$ to obtain

$$\begin{aligned} \phi '(t) =&\,\int _0^{s_0}s^{-\gamma }(s_0-s)w_t(s,t)ds {=}\int _0^{s_{0}}s^{-\gamma }(s_0-s)\bigg [n^2s^{2-\frac{2}{n}} \varphi (nw_s)w_{ss}+nww_s \nonumber \\&-m(t)sw_s(s,t) +\mu w(s,t)-\mu n^\alpha \beta (n)w\int _0^{R^{n}} w_s^{\alpha }(\sigma ,t)\text {d}\sigma \bigg ]\text {d}s\nonumber \\ \ge&\,n^2\int _0^{s_0}s^{2-\frac{2}{n}-\gamma }(s_0-s)\varphi (nw_s)w_{ss}\text {d}s +n\int _0^{s_0}s^{-\gamma }(s_0-s)ww_s\text {d}s\nonumber \\&-m(t)\int _0^{s_0}s^{1-\gamma }(s_0-s)w_sds-\mu n^{\alpha }\beta (n)\int _0^{R^{n}}w_s^{\alpha }(\sigma ,t)\text {d}\sigma \int _0^{s_0}s^{-\gamma }(s_0-s)w\text {d}s\nonumber \\ =&\,I_1+nI_2+I_3+I_4. \end{aligned}$$

(4.7)

For $\varphi \in C^2([0,\infty ])$, define

$$\begin{aligned} \Psi (\xi )=\int _0^\xi \varphi (\sigma )\text {d}\sigma . \end{aligned}$$

(4.8)

Since $\varphi (u)\le C_0(1+u)^{-m},$

$$\begin{aligned} 0<\Psi (\xi )=\int _0^\xi \varphi (\sigma )\text {d}\sigma \le \ \left\{ \begin{array}{ll} \frac{C_0}{m-1},\ {} &{} \text{ if }\quad m\in (1,\infty ),\\ C_0\ln (1+\xi ),\ {} &{} \text{ if } \quad m=1,\\ \frac{C_0}{1-m}(1+\xi )^{1-m}, \ {} &{} \text{ if } \quad m\in (-1,1).\\ \end{array} \right. \end{aligned}$$

(4.9)

Due to $\gamma <2-\frac{2}{n}$ as defined in (4.2), we use integration by parts to get

$$\begin{aligned} \begin{aligned} I_{1}=&n^2\int _0^{s_0}s^{2-\frac{2}{n}-\gamma }(s_0-s)\varphi (nw_s)w_{ss}\text {d}s =n\int _0^{s_0}s^{2-\frac{2}{n}-\gamma }(s_0-s)\text {d}\Psi (nw_s)\\ =&ns^{2-\frac{2}{n}-\gamma }{(s_{0}-s)}\Psi (nw_s)|_0^{s_{0}}+n\int _0^{s_0}s^{2-\frac{2}{n}-\gamma }\Psi (nw_s)\text {d}s\\&-n(2-\frac{2}{n}-\gamma )\int _0^{s_0}s^{1-\frac{2}{n}-\gamma }(s_0-s)\Psi (nw_s)\text {d}s\\ \ge&\left\{ \begin{array}{lll} -\frac{nC_0}{m-1}(2-\frac{2}{n}-\gamma )\int _0^{s_0}s^{1-\frac{2}{n}-\gamma }(s_0-s)\text {d}s, \ {} &{} \text{ if } \ m\in (1,\infty ),\\[2.5mm] \ -nC_0(2-\frac{2}{n}-\gamma )\int _0^{s_0}s^{1-\frac{2}{n}-\gamma }(s_0-s)\ln (1+nw_s)\text {d}s,\ {} &{} \text{ if } \ m=1,\\[2.5mm] -\frac{nC_0}{1-m}(2-\frac{2}{n}-\gamma )\int _0^{s_0}s^{1-\frac{2}{n}-\gamma } (s_0-s)(1+nw_s)^{1-m}\text {d}s,\ {} &{} \text{ if } \ m\in (-1,1).\\[2.5mm] \end{array} \right. \end{aligned} \end{aligned}$$

(4.10)

Thus, we complete the proof of Lemma 4.1. $\square$

Lemma 4.2

Let $\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 3)$ be a ball with $R>0$ and the parameters $\alpha ,\mu >0.$ If $2-\frac{2}{\alpha }<\gamma <\min \left\{ 1, 2-\frac{4}{n}, 2-\frac{4}{n(m+1)}\right\} ,$ then there exist $\theta \in (0,1)$ and $\eta >\theta -(1-\theta )(1-\gamma )$ such that

$$\begin{aligned} \int _0^{s_0}s^{\eta -\gamma }(s_{0}-s)w_{s}^{2\theta }\text {d}s\le W(\eta ,\theta ,\gamma )s_{0}^{\eta -\theta +(2-\gamma )(1-\theta )}I_{2}^{\theta }, \end{aligned}$$

(4.11)

where W is as in (4.4) and $I_{2}=\int _0^{s_0}s^{-\gamma }(s_0-s)ww_sds.$

Proof

Combining (3.21) and Hölder’s inequality, we know that

$$\begin{aligned} \int _0^{s_0}s^{\eta -\gamma }(s_{0}-s)w_{s}^{2\theta }ds \le&\,\int _0^{s_0}s^{\eta -\gamma }(s_{0}-s)\bigg (\frac{w}{s}\bigg ) ^{\theta }w_{s}^{\theta }\text {d}s\nonumber \\ \le&\,\bigg (\int _0^{s_0}s^{-\gamma }(s_{0}-s)ww_{s}\text {d}s\bigg )^{\theta } \bigg (\int _0^{s_0}s^{\frac{\eta -\theta }{1-\theta }-\gamma } (s_{0}-s)\text {d}s\bigg )^{1-\theta }\nonumber \\ \le&\,\bigg (\int _0^{s_0}s^{-\gamma }(s_{0}-s)ww_{s}\text {d}s\bigg )^{\theta } \bigg (s_{0}\int _0^{s_0}s^{\frac{\eta -\theta }{1-\theta }-\gamma }\text {d}s\bigg ) ^{1-\theta }\nonumber \\ \le&\,\bigg (\frac{\eta -\theta }{1-\theta }+1-\gamma \bigg )^{\theta -1} s_{0}^{\eta -\theta +(2-\gamma )(1-\theta )}I_{2}^{\theta }. \end{aligned}$$

(4.12)

$\square$

In the following, in order to deduce the contradiction, let us assume that $s_{0}\in (0,1).$ And we shall estimate each term of $I_{i},$ $i=1,2,3,4.$

Lemma 4.3

Let $\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 3)$ be a ball with $R>0$ and the parameters $\alpha ,\mu >0.$ Suppose that the diffusion function $\varphi (u)\le C_{0}(1+u)^{-m}$ with $C_{0}>0$ and $m>-1.$ If $2-\frac{2}{\alpha }<\gamma <\min \left\{ 1, 2-\frac{4}{n}, 2-\frac{4}{n(m+1)}\right\} ,$ then for any $s_{0}\in (0,1),$ there exists $C>0$ such that

$$\begin{aligned} \begin{aligned} I_{1}\ge \left\{ \begin{array}{lll} -\frac{nC_0}{m-1}s_{0}^{1-\gamma +\tau },\ {} &{} \text{ if } \ m\in (1,\infty ),\\[2.5mm] -\frac{1}{2}I_{2}-Cs_{0}^{1-\gamma +\tau },\ {} &{} \text{ if } \ m\in (-1,1], \end{array} \right. \end{aligned} \end{aligned}$$

(4.13)

where $\tau$ is defined in (4.3) and $I_{2}=\int _0^{s_0}s^{-\gamma }(s_0-s)ww_sds.$

Proof

According to (4.10), we know that if $m>1,$ there holds

$$\begin{aligned} I_{1} {\ge }&-\frac{nC_0}{m-1}\left( 2-\frac{2}{n}-\gamma \right) \int _0^{s_0} s^{1-\frac{2}{n}-\gamma }(s_0-s)\text {d}s\nonumber \\ \ge&-\frac{nC_0}{m-1}\left( 2-\frac{2}{n}-\gamma \right) s_{0}\int _0^{s_0} s^{1-\frac{2}{n}-\gamma }\text {d}s\nonumber \\ {=}&-\frac{nC_0}{m-1}s_{0}^{3-\frac{2}{n}-\gamma } \ge -\frac{nC_0}{m-1}s_{0}^{1-\gamma +\tau }, \end{aligned}$$

(4.14)

for $s_{0}\in (0,1).$ The last inequality in (4.14) can be ensured by $\tau<2-\frac{4}{n}<2-\frac{2}{n}$ defined in (4.3). If $m=1,$ using the fact $\frac{\ln (1+x)}{x}<1$ for any $x>0,$ we can estimate $I_{1}$ as

$$\begin{aligned} I_{1}{\ge }&-nC_0\left( 2-\frac{2}{n}-\gamma \right) \int _0^{s_0}s^{1-\frac{2}{n}-\gamma } (s_0-s)\ln (1+nw_s)\text {d}s\nonumber \\ \ge&-n^{2}C_0\left( 2-\frac{2}{n}-\gamma \right) \int _0^{s_0}s^{1-\frac{2}{n}-\gamma } (s_0-s)w_s\text {d}s\nonumber \\ \ge&-n^{2}C_0\left( 2-\frac{2}{n}-\gamma \right) W\left( 1-\frac{2}{n},\frac{1}{2},\gamma \right) s_{0}^{\frac{3-\gamma }{2}-\frac{2}{n}}I_{2}^{\frac{1}{2}}\nonumber \\ \ge&-\frac{1}{2}I_{2}-\frac{1}{2}n^{4}C_{0}^{2}\left( 2-\frac{2}{n}-\gamma \right) ^{2} W^{2}\left( 1-\frac{2}{n},\frac{1}{2},\gamma \right) s_{0}^{3-\frac{4}{n}-\gamma }\nonumber \\ {=}&-\frac{1}{2}I_{2}-C_{1}s_{0}^{3-\frac{4}{n}-\gamma }, \end{aligned}$$

(4.15)

with $C_{1}=\frac{1}{2}n^{4}C_{0}^{2}\left( 2-\frac{2}{n}-\gamma \right) ^{2} W^{2}\left( 1-\frac{2}{n},\frac{1}{2},\gamma \right) >0$ due to $\gamma <2-\frac{4}{n},$ where we have used Lemma 4.2 by choosing $\eta =1-\frac{2}{n}$ and $\theta =\frac{1}{2}.$ Since $\tau <2-\frac{4}{n}$ in (4.3), we have

$$\begin{aligned} I_{1}\ge -\frac{1}{2}I_{2}-C_{1}s_{0}^{3-\frac{4}{n}-\gamma }\ge -\frac{1}{2}I_{2} -C_{1}s_{0}^{1-\gamma +\tau } \end{aligned}$$

(4.16)

for $s_{0}\in (0,1).$ If $-1<m<1,$ we can get

$$\begin{aligned} I_{1}{\ge }&-\frac{nC_0}{1-m}\left( 2-\frac{2}{n}-\gamma \right) \int _0^{s_0}s^{1-\frac{2}{n}-\gamma } (s_0-s)(1+nw_s)^{1-m}\text {d}s\nonumber \\ \ge&-2^{1-m}\frac{nC_0}{1-m}\left( 2-\frac{2}{n}-\gamma \right) \int _0^{s_0} s^{1-\frac{2}{n}-\gamma } (s_0-s)\text {d}s\nonumber \\&-2^{1-m}\frac{n^{2-m}C_0}{1-m}\left( 2-\frac{2}{n}-\gamma \right) \int _0^{s_0}s^{1-\frac{2}{n}-\gamma }(s_0-s)w_{s}^{1-m}\text {d}s\nonumber \\ \ge&-2^{1-m}\frac{n^{2-m}C_0}{1-m}\left( 2-\frac{2}{n}-\gamma \right) W\left( 1-\frac{2}{n}, \frac{1-m}{2},\gamma \right) s_{0}^{\frac{(3-\gamma )(1+m)}{2}-\frac{2}{n}} I_{2}^{\frac{1-m}{2}}\nonumber \\&-2^{1-m}\frac{nC_0}{1-m}s_{0}^{3-\frac{2}{n}-\gamma }\nonumber \\ \ge&-\frac{1}{2}I_{2}-C_{2}s_{0}^{3-\frac{4}{n(1+m)}-\gamma } -2^{1-m}\frac{nC_0}{1-m}s_{0}^{3-\frac{2}{n}-\gamma } \end{aligned}$$

(4.17)

with $C_{2}=\frac{1}{2}\bigg (2^{1-m}\frac{n^{2-m}C_0}{1-m}\bigg )^{\frac{2}{1+m}} \left( 2-\frac{2}{n}-\gamma \right) ^{\frac{2}{1+m}}W^{\frac{2}{1+m}}\left( 1-\frac{2}{n}, \frac{1-m}{2},\gamma \right) >0$ due to $\gamma <2-{\frac{4}{n(1+m)}},$ where we have used Hölder’s inequality and Lemma 4.2 by setting $\eta =1-\frac{2}{n}$ and $\theta =\frac{1-m}{2}>0$ with $m>-1.$ From (4.3), we know that $3-\frac{4}{n(1+m)}-\gamma >1-\gamma +\tau$ and $3-\frac{2}{n}-\gamma >1-\gamma +\tau ,$ thus

$$\begin{aligned} I_{1}\ge -\frac{1}{2}I_{2}-C_{3}s_{0}^{1-\gamma +\tau }, \end{aligned}$$

(4.18)

where $C_{3}=C_{2}+2^{1-m}\frac{nC_0}{1-m}>0.$ By taking $C=\max \{C_{1},C_{3}\},$ thus we complete the proof of Lemma 4.3. $\square$

Lemma 4.4

Let $\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 3)$ be a ball with $R>0$ and the parameters $\alpha ,\mu >0.$ Suppose that the diffusion function $\varphi (u)\le C_{0}(1+u)^{-m}$ with $C_{0}>0$ and $m>-1.$ If $2-\frac{2}{\alpha }<\gamma <\min \left\{ 1, 2-\frac{4}{n}, 2-\frac{4}{n(m+1)}\right\} ,$ then there exists $C_{4}=C_{4}(\gamma )$ such that

$$\begin{aligned} I_{2}\ge C_{4}s_{0}^{-(3-\gamma )}\phi ^{2}(s_{0},t) \end{aligned}$$

(4.19)

for $s_{0}\in (0,1)$ and $t\in (0,1],$ where $\phi$ is defined as (4.1) and $I_{2}=\int _0^{s_0}s^{-\gamma }(s_0-s)ww_sds.$

Proof

As demonstrated in [41], we have

$$\begin{aligned} \phi (s_{0},t)\le C_{4}^{\frac{1}{2}}s_{0}^{\frac{3-\gamma }{2}}I_{2}^{\frac{1}{2}} \end{aligned}$$

(4.20)

for $s_{0}\in (0,1)$ and $t\in (0,1],$ where $C_{4}=2\bigg (\int _{0}^{1}x^{-\frac{\gamma }{2}}(1-x)^{-\frac{1}{2}}\text {d}x\bigg )^{2}$ is finite due to $\gamma <2.$ $\square$

Lemma 4.5

Let $\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 3)$ be a ball with $R>0$ and the parameters $\alpha ,\mu >0.$ Suppose that the diffusion function $\varphi (u)\le C_{0}(1+u)^{-m}$ with $C_{0}>0$ and $m>-1.$ If $2-\frac{2}{\alpha }<\gamma <\min \left\{ 1, 2-\frac{4}{n}, 2-\frac{4}{n(m+1)}\right\} ,$ there holds

$$\begin{aligned} I_{3}\ge -\frac{1}{2}I_{2}-C_{5}s_{0}^{1-\gamma +\tau } \end{aligned}$$

(4.21)

for $s_{0}\in {(0,1)}$ and $t\in (0,1],$ where $C_{5}=\frac{1}{2}m_{0}^{2}e^{2\mu }W^{2}(1,\frac{1}{2},\gamma )>0$ with W defined as in (4.4) and $I_{3}=-m(t)\int _0^{s_0}s^{1-\gamma }(s_0-s)w_sds.$

Proof

According to Lemma 3.1, we conclude from (4.11) with $\eta =1, \theta =\frac{1}{2}$ that

$$\begin{aligned} I_{3}&=-m(t)\int _0^{s_0}s^{1-\gamma }(s_0-s)w_s\text {d}s \ge -m_{0}e^{\mu }W(1,\frac{1}{2},\gamma )s_{0}^{\frac{3}{2}-\frac{\gamma }{2}} I_{2}^{\frac{1}{2}}\nonumber \\&\ge -\frac{1}{2}I_{2}-\frac{1}{2}m_{0}^{2}e^{2\mu }W^{2} (1,\frac{1}{2},\gamma )s_{0}^{3-\gamma } \ge -\frac{1}{2}I_{2}-C_{5}s_{0}^{3-\gamma }. \end{aligned}$$

(4.22)

From (4.3), it is easy to get $3-\gamma >1-\gamma +\tau .$ Thus, we have

$$\begin{aligned} I_{3}\ge -\frac{1}{2}I_{2}-C_{5}s_{0}^{1-\gamma +\tau } \end{aligned}$$

(4.23)

for $s_{0}\in (0,R^{n})$ and $t\in (0,1],$ where with $C_{5}=\frac{1}{2}m_{0}^{2}e^{2\mu }W^{2}(1,\frac{1}{2},\gamma )>0.$ $\square$

Lemma 4.6

Let $\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 3)$ be a ball with $R>0$ and the parameters $\alpha ,\mu>0, m>-1.$ If $2-\frac{2}{\alpha }<\gamma <\min \left\{ 1, 2-\frac{4}{n}, 2-\frac{4}{n(m+1)}\right\} ,$ then we have

$$\begin{aligned} I_{4}\ge -I_{2}-(C_{6}+C_{7})s_{0}^{1-\gamma +\tau } \end{aligned}$$

(4.24)

for $s_{0}\in {(0,1)}$ and $t\in (0,1],$ with $C_{6}=\frac{2-\alpha }{2}\bigg [\frac{\mu n^{\alpha }\beta (n)m_{0}e^{\mu }R^{n}}{(1-\gamma )n}W(\gamma ,\frac{\alpha }{2}, \gamma )\bigg ]^{\frac{2}{2-\alpha }}>0$ and $C_{7}=\frac{\mu R^{n(\alpha +1)}e^{\mu (\alpha +1)}m_{0}^{\alpha +1}\beta (n)}{n(1-\gamma ) (2-\gamma -\alpha )}+\frac{\mu R^{n(3-\tau )}\beta (n)e^{\mu (\alpha +1)}m_{0}^{\alpha +1}}{n(1-\gamma ) (2-\alpha -\tau )}>0,$ where $I_{4}=-\mu n^{\alpha }\beta (n)\int _0^{R^{n}}w_s^{\alpha }(\sigma ,t)\text {d}\sigma \int _0^{s_0}s^{-\gamma }(s_0-s)wds.$

Proof

The term $I_{4}$ can be rewritten as follows

$$\begin{aligned} I_{4}=&\,-\mu n^{\alpha }\beta (n)\int _0^{s_0}\int _0^{R^{n}}s^{-\gamma }(s_0-s) w(s,t)w_s^{\alpha }(\sigma ,t)\text {d}\sigma \text {d}s\nonumber \\ =&\,-\mu n^{\alpha }\beta (n)\int _0^{s_0}\int _0^{s}s^{-\gamma }(s_0-s) w(s,t)w_s^{\alpha }(\sigma ,t)\text {d}\sigma \text {d}s\nonumber \\&-\mu n^{\alpha }\beta (n)\int _0^{s_0}\int _s^{R^{n}}s^{-\gamma }(s_0-s) w(s,t)w_s^{\alpha }(\sigma ,t)\text {d}\sigma \text {d}s\nonumber \\ =:&\,J_{1}+J_{2}. \end{aligned}$$

(4.25)

Using (2.10), it is easy to get from Lemma 3.1 that

$$\begin{aligned} w(s,t)\le \frac{m(t)R^{n}}{n}\le \frac{m_{0}e^{\mu }R^{n}}{n}. \end{aligned}$$

(4.26)

From Fubini’s theorem and (4.26), using (4.11) with $\eta =\gamma , \theta =\frac{\alpha }{2}$ and Young’s inequality, we get

$$\begin{aligned} J_{1} =&-\mu n^{\alpha }\beta (n)\int _0^{s_0}\int _\sigma ^{s_{0}}s^{-\gamma }(s_0-s) w(s,t)w_s^{\alpha }(\sigma ,t)\text {d}s\text {d}\sigma \nonumber \\ \ge&-\mu n^{\alpha }\beta (n)\int _0^{s_0}\bigg (\int _\sigma ^{s_{0}}s^{-\gamma }\text {d}s\bigg ) (s_0-\sigma )\frac{m_{0}e^{\mu }R^{n}}{n}w_s^{\alpha }(\sigma ,t)\text {d}\sigma \nonumber \\ \ge&-\frac{\mu n^{\alpha }\beta (n)m_{0}e^{\mu }R^{n}s_{0}^{1-\gamma }}{(1-\gamma )n}\int _0^{s_0} (s_0-\sigma )w_s^{\alpha }(\sigma ,t)\text {d}\sigma \nonumber \\ \ge&-\frac{\mu n^{\alpha }\beta (n)m_{0}e^{\mu }R^{n}s_{0}^{1-\gamma }}{(1-\gamma )n}W(\gamma , \frac{\alpha }{2},\gamma )s_{0}^{\gamma -\frac{\alpha }{2} +\left( 1-\frac{\alpha }{2}\right) (2-\gamma )}I_{2}^{\frac{\alpha }{2}}\nonumber \\ \ge&-\frac{\mu n^{\alpha }\beta (n)m_{0}e^{\mu }R^{n}}{(1-\gamma )n}W(\gamma , \frac{\alpha }{2},\gamma )s_{0}^{\left( 1-\frac{\alpha }{2}\right) (3-\gamma )} I_{2}^{\frac{\alpha }{2}}\nonumber \\ \ge&-\frac{\alpha }{2}I_{2}-\frac{2-\alpha }{2}\bigg [ \frac{\mu n^{\alpha }\beta (n)m_{0}e^{\mu }R^{n}}{(1-\gamma )n}W\bigg (\gamma , \frac{\alpha }{2},\gamma \bigg )\bigg ]^{\frac{2}{2-\alpha }}s_{0}^{3-\gamma }\nonumber \\ \ge&-I_{2}-C_{6}s_{0}^{3-\gamma } \end{aligned}$$

(4.27)

for $s_{0}\in {(0,1)}$ and $t\in (0,1],$ where $C_{6}=\frac{2-\alpha }{2}\bigg [\frac{\mu n^{\alpha }\beta (n)m_{0}e^{\mu }R^{n}}{(1-\gamma )n}W(\gamma , \frac{\alpha }{2},\gamma )\bigg ]^{\frac{2}{2-\alpha }}>0$ due to $\gamma >2-\frac{2}{\alpha }$ and $\alpha <2.$ From inequality (4.3), we can deduce that $3-\gamma >1-\gamma +\tau .$ Thus we have

$$\begin{aligned} J_{1}\ge -I_{2}-C_{6}s_{0}^{1-\gamma +\tau }. \end{aligned}$$

(4.28)

Using the formula to interchange order of integration, we obtain

$$\begin{aligned} J_{2} \ge&-\mu n^{\alpha }\beta (n)\int _0^{s_0}\int _0^{\sigma }s^{-\gamma }(s_0-s) w(s,t)w_s^{\alpha }(\sigma ,t)\text {d}s\text {d}\sigma \nonumber \\&-\mu n^{\alpha }\beta (n)\int _{s_{0}}^{R^{n}}\int _0^{s_{0}}s^{-\gamma }(s_0-s) w(s,t)w_s^{\alpha }(\sigma ,t)\text {d}s\text {d}\sigma \nonumber \\ =&:K_{1}+K_{2}. \end{aligned}$$

(4.29)

Recalling $0<\alpha <\min \left\{ 2,\frac{n}{2},\frac{n(m+1)}{2}\right\}$ and $0<\tau <\min \left\{ 2-\alpha , 1, 2-\frac{4}{n}, 2-\frac{4}{n(m+1)}\right\}$, we can get from Lemma 3.3 that

$$\begin{aligned} K_{1} \ge&- \mu n^{\alpha }\beta (n)s_{0}\int _0^{s_0}\int _0^{\sigma }s^{-\gamma }ds w(\sigma ,t)w_s^{\alpha }(\sigma ,t)\text {d}\sigma \nonumber \\ \ge&-\mu n^{\alpha }\beta (n)s_{0}\int _0^{s_0}\int _0^{\sigma }s^{-\gamma }ds \bigg (\frac{w(\sigma ,t)}{\sigma }\bigg )^{\alpha }w(\sigma ,t)\text {d}\sigma \nonumber \\ \ge&-\frac{\mu R^{n(\alpha +1)}e^{\mu (\alpha +1)}m_{0}^{\alpha +1}\beta (n)}{n(1-\gamma )} s_{0}\int _0^{s_0}\sigma ^{1-\gamma -\alpha }\text {d}\sigma \nonumber \\ =&-\frac{\mu R^{n(\alpha +1)}e^{\mu (\alpha +1)}m_{0}^{\alpha +1}\beta (n)}{n(1-\gamma ) (2-\gamma -\alpha )}s_{0}^{3-\gamma -\alpha }\nonumber \\ \ge&-\frac{\mu R^{n(\alpha +1)}e^{\mu (\alpha +1)}m_{0}^{\alpha +1}\beta (n)}{n(1-\gamma ) (2-\gamma -\alpha )}s_{0}^{1-\gamma +\tau }. \end{aligned}$$

(4.30)

for $s_{0}\in {(0,1)}$ and $t\in (0,1].$ Using Lemmas 3.1, 3.3 and (4.26), we get

$$\begin{aligned} K_{2} \ge&-\mu n^{\alpha }\beta (n)s_{0}\int _{s_{0}}^{R^{n}}\int _0^{s_{0}}s^{-\gamma }ds w(\sigma ,t)w_s^{\alpha }(\sigma ,t)\text {d}\sigma \nonumber \\ \ge&-\frac{\mu n^{\alpha }\beta (n)}{1-\gamma }s_{0}^{2-\gamma }\int _{s_{0}}^{R^{n}}\sigma ^{-\alpha } w^{\alpha +1}\text {d}\sigma \nonumber \\ \ge&-\frac{\mu R^{n(\alpha +1)}e^{\mu (\alpha +1)}m_{0}^{\alpha +1}\beta (n)}{n(1-\gamma )} s_{0}^{2-\gamma }\int _{s_{0}} ^{R^{n}}\sigma ^{-\alpha }\text {d}\sigma \nonumber \\ \ge&-\frac{\mu R^{n(\alpha +1)}e^{\mu (\alpha +1)}m_{0}^{\alpha +1}\beta (n)}{n(1-\gamma )} s_{0}^{1-\gamma +\tau }\int _{s_{0}}^{R^{n}}\sigma ^{1-\tau }\sigma ^{-\alpha } \text {d}\sigma \nonumber \\ \ge&-\frac{\mu R^{n(3-\tau )}\beta (n)e^{\mu (\alpha +1)}m_{0}^{\alpha +1}}{n(1-\gamma ) (2-\alpha -\tau )}s_{0}^{1-\gamma +\tau }. \end{aligned}$$

(4.31)

Combining (4.27)–(4.31), we can directly infer (4.24). $\square$

Now, we are in a position to prove Theorem 1.1.

Proof of Theorem 1.1

To begin with, we recall the restrictions on the system (1.6). Let $\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 3)$ be a ball with $R>0$ and the parameters $\alpha ,\mu >0.$ The diffusion function $\varphi \in C^{2}([0,\infty ))$ and $\varphi (u)\le C_{0}(1+u)^{-m}$ for all $u\ge 0$ with $C_{0}>0$ and $m> -1.$ Assume that the initial data $u_{0}$ satisfies the condition (1.8). For $m>-1,$ there exists $0<\alpha <\min \left\{ 2,\frac{n}{2},\frac{n(m+1)}{2}\right\}$ such that

$$\begin{aligned} 2-\frac{2}{\alpha }<\gamma <\min \left\{ 1, 2-\frac{4}{n}, 2-\frac{4}{n(m+1)}\right\} \end{aligned}$$

(4.32)

and

$$\begin{aligned} 0<\tau <\min \left\{ 1, 2-\alpha , 2-\frac{4}{n}, 2-\frac{4}{n(m+1)}\right\} . \end{aligned}$$

(4.33)

Therefore, we collect Lemmas 4.3–4.6 to get

$$\begin{aligned} \begin{aligned} \phi '(t)(s_{0},t)\ge&\left\{ \begin{array}{lll} (n-\frac{3}{2})I_{2} -C_{8}s_{0}^{1-\gamma +\tau },\ {} &{} \text{ if } \ m\in (1,\infty ),\\[2.5mm] (n-2)I_{2}-C_{9}s_{0}^{1-\gamma +\tau },\ {} &{} \text{ if } \ m\in (-1,1], \end{array} \right. \end{aligned} \end{aligned}$$

(4.34)

where $C_{8}=\frac{nC_{0}}{m-1}+C_{5}+C_{6}+C_{7}>0$ and $C_{9}=C+C_{5}+C_{6}+C_{7}>0.$

Thus, for $m>-1,$ we can easily get from $n\ge 3$ that

$$\begin{aligned} \phi '(t)(s_{0},t)\ge I_{2}-C_{4}C_{10}s_{0}^{1-\gamma +\tau } \ge C_{4}\bigg [s_{0}^{-(3-\gamma )}\phi ^{2}(s_{0},t) -C_{10}s_{0}^{1-\gamma +\tau }\bigg ] \end{aligned}$$

(4.35)

for $s_{0}\in (0,1)$ and $t\in (0,1],$ where $C_{10}=\frac{\max \{C_{8},C_{9}\}}{C_{4}}.$

Let $r_{1}=\big (\frac{s_{0}}{4}\big )^{\frac{1}{n}}$ and $u_{0}$ satisfy the condition (1.8). We can derive from (2.11) that

$$\begin{aligned} w(s,0)\ge w(\frac{s_{0}}{4},0)\ge \frac{m_{1}}{n\omega _{n}} \end{aligned}$$

(4.36)

for $s\in (\frac{s_{0}}{4},R^{n}).$ Substituting (4.36) into the expression of $\phi ,$ we can get the following estimate

$$\begin{aligned} \phi (s_{0},0)\ge&\int _{\frac{s_{0}}{4}}^{\frac{s_{0}}{2}}s^{-\gamma }(s_{0}-s) w(s,0)\text {d}s\nonumber \\ \ge&\int _{\frac{s_{0}}{4}}^{\frac{s_{0}}{2}}(\frac{s_{0}}{2})^{-\gamma } \frac{s_{0}}{2}\frac{m_{1}}{n\omega _{n}}\text {d}s\nonumber \\ \ge&C_{11}s_{0}^{2-\gamma }, \end{aligned}$$

(4.37)

where $C_{11}=\frac{2^{\gamma -3}m_{1}}{n\omega _{n}}.$ We define $s_{0}>0$ small enough to satisfy

$$\begin{aligned} s_{0}^{\tau }<\frac{C_{11}^{2}}{2C_{10}} \end{aligned}$$

(4.38)

and

$$\begin{aligned} s_{0}<\frac{C_{4}C_{11}}{4}. \end{aligned}$$

(4.39)

According to (4.35), (4.37) and (4.38), we derive that

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{lll} \phi '(t)(s_{0},t)\ge C_{4}\bigg (s_{0}^{-(3-\gamma )}\phi ^{2}(s_{0},t) -C_{10}s_{0}^{1-\gamma +\tau }\bigg ),\ \ t\in (0,1],\\[2.5mm] \frac{1}{2}s_{0}^{-(3-\gamma )}\phi ^{2}(s_{0},0)>C_{10}s_{0}^{1-\gamma +\tau }. \end{array} \right. \end{aligned} \end{aligned}$$

(4.40)

It can be deduced from the comparison principle that $\frac{1}{2}s_{0}^{-(3-\gamma )}\phi ^{2}(s_{0},t)>C_{12}s_{0}^{1-\gamma +\tau }.$ Thus, one may obtain

$$\begin{aligned} \phi '(t)(s_{0},t)\ge \frac{C_{4}}{2}s_{0}^{-(3-\gamma )}\phi ^{2}(s_{0},t),\ \ t\in (0,1]. \end{aligned}$$

(4.41)

Integrating (4.41) from 0 to 1,  we obtain from (4.39) and (4.42) that

$$\begin{aligned} 1\le&\int _{0}^{1}\frac{2s_{0}^{3-\gamma }\phi '(t)}{C_{4}\phi ^{2}(t)} dt\le \frac{2s_{0}^{3-\gamma }}{C_{4}}\int _{\phi (s_{0},0)}^{\phi (s_{0},1)} \sigma ^{-2}\text {d}\sigma \nonumber \\ \le&\frac{2s_{0}^{3-\gamma }}{C_{4}}\phi ^{-1}(t)(s_{0},0) \le \frac{2s_{0}^{3-\gamma }}{C_{4}}\big (C_{13}s_{0}^{2-\gamma }\big )^{-1} =\frac{2s_{0}}{C_{4}C_{11}}\le \frac{1}{2}, \end{aligned}$$

(4.42)

which leads to an absurd conclusion. Thus, we complete the proof. $\square$