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Research article Special Issues

Blow-up and boundedness in quasilinear attraction-repulsion systems with nonlinear signal production


  • In this paper, we consider the quasilinear parabolic-elliptic-elliptic attraction-repulsion system

    {ut=(D(u)u)χ(uv)+ξ(uw),xΩ,t>0,0=Δvμ1(t)+f1(u),xΩ,t>0,0=Δwμ2(t)+f2(u),xΩ,t>0

    under homogeneous Neumann boundary conditions in a smooth bounded domain ΩRn, n2. The nonlinear diffusivity D and nonlinear signal productions f1,f2 are supposed to extend the prototypes

    D(s)=(1+s)m1, f1(s)=(1+s)γ1, f2(s)=(1+s)γ2, s0,γ1,γ2>0,mR.

    We proved that if γ1>γ2 and 1+γ1m>2n, then the solution with initial mass concentrating enough in a small ball centered at origin will blow up in finite time. However, the system admits a global bounded classical solution for suitable smooth initial datum when γ2<1+γ1<2n+m.

    Citation: Ruxi Cao, Zhongping Li. Blow-up and boundedness in quasilinear attraction-repulsion systems with nonlinear signal production[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 5243-5267. doi: 10.3934/mbe.2023243

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  • In this paper, we consider the quasilinear parabolic-elliptic-elliptic attraction-repulsion system

    {ut=(D(u)u)χ(uv)+ξ(uw),xΩ,t>0,0=Δvμ1(t)+f1(u),xΩ,t>0,0=Δwμ2(t)+f2(u),xΩ,t>0

    under homogeneous Neumann boundary conditions in a smooth bounded domain ΩRn, n2. The nonlinear diffusivity D and nonlinear signal productions f1,f2 are supposed to extend the prototypes

    D(s)=(1+s)m1, f1(s)=(1+s)γ1, f2(s)=(1+s)γ2, s0,γ1,γ2>0,mR.

    We proved that if γ1>γ2 and 1+γ1m>2n, then the solution with initial mass concentrating enough in a small ball centered at origin will blow up in finite time. However, the system admits a global bounded classical solution for suitable smooth initial datum when γ2<1+γ1<2n+m.



    Chemotaxis is the property of cells to move in an oriented manner in response to an increasing concentration of chemo-attractant or decreasing concentration of chemo-repellent, where the former is referred to as attractive chemotaxis and the later to repulsive chemotaxis. To begin with, it is important to study the quasilinear Keller-Segel system as follows

    {ut=(D(u)u)χ(ϕ(u)v),xΩ,t>0,τvt=Δvαv+βu,xΩ,t>0, (1.1)

    subject to homogeneous Neumann boundary conditions, where the functions D(u) and ϕ(u) denote the strength of diffusion and chemoattractant, respectively, and the function u=u(x,t) idealizes the density of cell, v=x(x,t) represents the concentration of the chemoattractant. Here the attractive (repulsive) chemotaxis corresponds to χ>0 (χ<0), and |χ|R{0} measures the strength of chemotactic response. The parameters τ{0,1}, and α,β>0 denote the production and degradation rates of the chemical. The above system describes the chemotactic interaction between cells and one chemical signal (either attractive or repulsive), and it has been investigated quite extensively on the existence of global bounded solutions or the occurrence of blow-up in finite time in the past four decades. In particular, the system (1.1) is the prototypical Keller-Segel model [1] when D(u)=1,ϕ(u)=u. In the case τ=1, there are many works to show that the solution is bounded [2,3,4,5], and blow-up in finite time [6,7,8,9,10,11]. If the cell's movement is much slower than the chemical signal diffusing, the second equation of (1.1) is reduced to 0=ΔvM+u, where M:=1|Ω|Ωu(x,t)dx and the simplified system has many significant results [12,13,14,15].

    For further information concerning nonlinear signal production, when the chemical signal function is denoted by g(u), authors derived for more general nonlinear diffusive system as follows

    {ut=(D(u)u)(ϕ(u)v),xΩ,t>0,0=ΔvM+g(u),xΩ,t>0, (1.2)

    where M:=1|Ω|Ωg(u(x,t))dx. Recently, when D(u)=up,ϕ(u)=u and g(u)=ul, it has been shown that all solutions are global and uniformly bounded if p+l<2n, whereas p+l>2n implies that the solution blows up in finite time [16]. What's more, there are many significant works [17,18,19] associated with this system.

    Subsequently, the attraction-repulsion system has been introduced in ([20,21]) as follows

    {ut=Δuχ(uv)+ξ(uw),xΩ,t>0,τ1vt=Δv+αuβv,xΩ,t>0,τ2wt=Δw+δuγw,xΩ,t>0, (1.3)

    subject to homogeneous Neumann boundary conditions, where χ,ξ,α,β,δ,γ>0 are constants, and the functions u(x,t),v(x,t) and w(x,t) denote the cell density, the concentration of the chemoattractant and chemorepellent, respectively. The above attraction-repulsion chemotaxis system has been studied actively in recent years, and there are many significant works to be shown as follows.

    For example, if τ1=τ2=0, Perthame [22] investigated a hyperbolic Keller-Segel system with attraction and repulsion when n=1. Subsequently, Tao and Wang [23] proved that the solution of (1.3) is globally bounded provided ξγχα>0 when n2, and the solution would blow up in finite time provided ξγχα<0,α=β when n=2. Then, there is a blow-up solution when χαξγ>0,δβ or χαδξγβ>0,δ<β for n=2 [24]. Moreover, Viglialoro [25] studied the explicit lower bound of blow-up time when n=2. In another hand, if τ1=1,τ2=0, Jin and Wang [26] showed that the solution is bounded when n=2 with ξγχα0, and Zhong et al. [27] obtained the global existence of weak solution when ξγχα0 for n=3. Furthermore, if τ1=τ2=1, Liu and Wang [28] obtained the global existence of solutions, and Jin et al. [29,30,31] also showed a uniform-in-time bound for solutions. In addition, there are plenty of available results of the attraction-repulsion system with logistic terms [32,33,34,35,36,37,38,39,40], and for further information concerning (1.3) based on the nonlinear signal production, it was used to model the aggregation patterns formed by some bacterial chemotaxis in [41,42,43].

    We turn our eyes into a multi-dimensional attraction-repulsion system

    {ut=Δuχ(ϕ(u)v)+ξ(ψ(u)w),xΩ,t>0,τ1vt=Δvμ1(t)+f(u),xΩ,t>0,τ2wt=Δwμ2(t)+g(u),xΩ,t>0, (1.4)

    where ΩRn(n2) is a bounded domain with smooth boundary, μ1(t)=1|Ω|Ωf(u)dx,μ2(t)=1|Ω|Ωg(u)dx and τ1,τ2{0,1}. Later on, the system (1.4) has attracted great attention of many mathematicians. In particular, when ϕ(u)=ψ(u)=u,f(u)=uk and g(u)=ul, Liu and Li [44] proved that all solutions are bounded if k<2n, while blow-up occurs for k>l and k>2n in the case τ1=τ2=0.

    Inspired by the above literature, we are devoted to deal with the quasilinear attraction-repulsion chemotaxis system

    {ut=(D(u)u)χ(uv)+ξ(uw),xΩ,t>0,0=Δvμ1(t)+f1(u),xΩ,t>0,0=Δwμ2(t)+f2(u),xΩ,t>0,uν=vν=wν=0,xΩ,t>0,u(x,0)=u0(x),xΩ, (1.5)

    in a bounded domain ΩRn,n2 with smooth boundary, where ν denotes outward normal derivatives on Ω. The function u(x,t) denotes the cell density, v(x,t) represents the concentration of an attractive signal (chemo-attractant), and w(x,t) is the concentration of a repulsive signal (chemo-repellent). The parameters satisfy χ,ξ0, which denote the strength of the attraction and repulsion, respectively. Here μ1(t)=1|Ω|Ωf1(u(x,t))dx, μ2(t)=1|Ω|Ωf2(u(x,t))dx, and f1,f2 are nonnegative Hölder continuous functions.

    In the end, we propose the following assumptions on D,f1,f2 and u0 for the system (1.5).

    (I1) The nonlinear diffusivity D is positive function satisfying

    DC2([0,)). (1.6)

    (I2) The function fi is nonnegative and nondecreasing and satisfies

    fiθ(0,1)Cθloc([0,))C1((0,)) (1.7)

    with i{1,2}.

    (I3) The initial datum

    u0θ(0,1)Cθ(¯Ω) is nonnegative and radially decreasing,u0ν=0 on Ω. (1.8)

    The goal of the article is twofold. On the one hand, we need to find out the mutual effect of the nonlinear diffusivity D(u) and the nonlinear signal production fi(u)(i=1,2). On the other hand, we need to make a substantial step towards the dynamic of blowing up in finite time. Hence, we draw our main results concerning (1.5) read as follows.

    Theorem 1.1. Let n2, R>0 and Ω=BR(0)Rn be a ball, and suppose that the function D satisfies (1.6) and f1,f2 are assumed to fulfill (1.7) as well as

    D(u)d(1+u)m1, f1(u)k1(1+u)γ1, f2(u)k2(1+u)γ2 for all u0,

    with mR, k1,k2,γ1,γ2,d>0 and

    γ1>γ2 and 1+γ1m>2n. (1.9)

    For any M>0 there exist ε=ε(γ1,M,R)(0,M) and r=r(γ1,M,R)(0,R) such that if u0 satisfies (1.8) with

    Ωu0=M and Br(0)u0Mε,

    then the corresponding solution of the system (1.5) blows up in finite time.

    Theorem 1.2. Let n2, ΩRn be a smooth bounded domain, and suppose that the function D satisfies (1.6) and f1,f2 are assumed to fulfill (1.7) as well as

    D(u)d(1+u)m1, f1(u)k1(1+u)γ1, f2(u)=k2(1+u)γ2 for all u0,

    with mR, k1,k2,γ1,γ2,d>0 and

    γ2<1+γ1<2n+m. (1.10)

    Then for each u0θ(0,1)Cθ(¯Ω), u00 with u0ν=0 on Ω, and the system (1.5) admits a unique global classical solution (u,v,w) with

    u,v,wC2,1(¯Ω×(0,))C0(¯Ω×[0,)).

    Furthermore, u,v and w are all non-negative and bounded.

    The structure of this paper reads as follows: In section 2, we will show the local-in-time existence of a classical solution to the system (1.5) and some lemmas that we will use later. In section 3, we will prove Theorem 1.1 by establishing a superlinear differential inequality. In section 4, we will solve the boundedness of u in L and prove Theorem 1.2.

    Firstly, we state one result concerning local-in-time existence of a classical solution to the system (1.5). Then, we denote some new variables to transfer the original equations in (1.5) to a new system according to the ideas in [19,20,21,22,23,24,25]. In addition, in order to prove the main result, we will state some lemmas which will be needed later.

    Lemma 2.1. Let ΩRn with n2 be a bounded domain with smooth boundary. Assume that D fulfills (1.6), f1,f2 satisfy (1.7) and u0θ(0,1)Cθ(¯Ω) with u0ν=0 on Ω as well as u00, then there exist Tmax(0,] and a classical solution (u,v,w) to (1.5) uniquely determined by

    {uC0(¯Ω×[0,Tmax))C2,1(¯Ω×(0,Tmax)),vq>nL((0,Tmax);W1,q(Ω))C2,0(¯Ω×(0,Tmax)),wq>nL((0,Tmax);W1,q(Ω))C2,0(¯Ω×(0,Tmax)).

    In addition, the function u0 in Ω×(0,Tmax) and if Tmax< then

    limtTmaxsupu(,t)L(Ω)=. (2.1)

    Moreover,

    Ωv(,t)=0,Ωw(,t)=0 for all t(0,Tmax). (2.2)

    Finally, the solution (u,v,w) is radially symmetric with respect to |x| if u0 satisfies (1.8).

    Proof. The proof of this lemma needs to be divided into four steps. Firstly, the method to solve the local time existence of the classical solution to the problem (1.5) is based on a standard fixed point theorem. Next, we will use the standard extension theorem to obtain (2.1). Then, we are going to use integration by parts to deduce (2.2). Finally, we would use the comparison principle to conclude that the solution is radially symmetric. For the details, we refer to [45,46,47,48].

    For the convenience of analysis and in order to prove Theorem 1.1, we set h=χvξw, then the system (1.5) is rewritten as

    {ut=(D(u)u)(uh),xΩ,t>0,0=Δhμ(t)+f(u),xΩ,t>0,uν=hν=0,xΩ,t>0,u(x,0)=u0(x),xΩ, (2.3)

    where μ(t)=χμ1(t)ξμ2(t)=1|Ω|Ωf(u(x,t))dx and f(u)=χf1(u)ξf2(u).

    For the same reason, we will convert the system (2.3) into a scalar equation. Let us assume Ω=BR(0) with some R>0 is a ball and the initial data u0=u0(r) with r=|x|[0,R] satisfies (1.8). In the radial framework, the system (2.3) can be transformed into the following form

    {rn1ut=(rn1D(u)ur)r(rn1uhr)r,r(0,R),t>0,0=(rn1hr)rrn1μ(t)+rn1f(u),r(0,R),t>0,ur=hr=0, r=R,t>0,u(r,0)=u0(r),r(0,R). (2.4)

    Lemma 2.2. Let us introduce the function

    U(s,t)=ns1n0ρn1u(ρ,t)dρ,s=rn[0,Rn], t(0,Tmax),

    then

    Us(t)=u(s1n,t), Uss(t)=1ns1n1ur(s1n,t), (2.5)

    and

    Ut(s,t)=n2s22nD(Us)Usssμ(t)Us+Uss0f(Us(σ,t))dσ. (2.6)

    Proof. Firstly, integrating the second equation of (2.4) over (0,r), we have

    rn1hr(r,t)=rnnμ(t)r0ρn1f(u(ρ,t))dρ,

    so

    s11nhr(s1n,t)=snμ(t)1ns0f(u(σ1n,t))dσ,s(0,Rn), t(0,Tmax).

    Then, a direct calculation yields

    Us(s,t)=u(s1n,t),s(0,Rn), t(0,Tmax),

    and

    Uss(s,t)=1ns1n1ur(s1n,t),s(0,Rn), t(0,Tmax),

    as well as

    Ut(s,t)=ns1n0ρn1ut(ρ,t)dρ=n2s22nD(Us)Ussns11nUshr=n2s22nD(Us)Usssμ(t)Us+Uss0f(Us(σ,t))dσ

    for all s(0,Rn) and t(0,Tmax).

    Furthermore, by a direct calculation and (1.7), we know that the functions U and f satisfy the following results

    {Us(s,t)=u(s1n,t)>0,s(0,Rn),t(0,Tmax),U(0,t)=0,U(Rn,t)=nωnΩu(,t)=nMωn,t[0,Tmax),|f(s)|,f1(s),f2(s)C0,0sA,C0=C0(A)>0, (2.7)

    where ωn=n|B1(0)| and A is a positive constant.

    Lemma 2.3. Suppose that (1.7), (1.8) and (2.7) hold, then we have

    hr(r,t)=1nμ(t)rr1nr0ρn1f(u(ρ,t))dρfor all r(0,R),t(0,Tmax).

    In particular,

    hr(r,t)1n(μ(t)+C0)r. (2.8)

    Proof. By integration the second equation in (2.4), we obtain that

    rn1hr=μ(t)r0ρn1dρr0ρn1f(u(ρ,t))dρ for all r(0,R),t(0,Tmax).

    According to (1.9), we can easily get that f(u)0 if uC=max{0,(k2ξk1χ)1γ1γ21}, and split

    \begin{equation} \nonumber \int^{r}_{0}\rho^{n-1}f(u(\rho,t))d\rho = \int^{r}_{0}\boldsymbol{\chi}_{\{u(\cdot,t)\geq C^{\ast}\}}(\rho)\cdot\rho^{n-1}f(u(\rho,t))d\rho+\int^{r}_{0}\boldsymbol{\chi}_{\{u(\cdot,t) < C^{\ast}\}}(\rho)\cdot\rho^{n-1}f(u(\rho,t))d\rho. \end{equation}

    Combining these we have

    \begin{align} h_{r}& = \frac{1}{n}\mu(t)r-r^{1-n}\int^{r}_{0}\boldsymbol{\chi}_{\{u(\cdot,t)\geq C^{\ast}\}}(\rho)\cdot\rho^{n-1}f(u(\rho,t))d\rho-r^{1-n}\\ &\int^{r}_{0} \boldsymbol{\chi}_{\{u(\cdot,t) < C^{\ast}\}}(\rho)\cdot\rho^{n-1}f(u(\rho,t))d\rho\\ &\leq\frac{1}{n}\mu(t)r-r^{1-n}\int^{r}_{0}\boldsymbol{\chi}_{\{u(\cdot,t) < C^{\ast}\}}(\rho)\cdot\rho^{n-1}f(u(\rho,t))d\rho\\ &\leq\frac{1}{n}\mu(t)r+C_{0}r^{1-n}\int^{r}_{0}\boldsymbol{\chi}_{\{u(\cdot,t) < C^{\ast}\}}(\rho)\cdot\rho^{n-1}d\rho\\ &\leq\frac{1}{n}\mu(t)r+C_{0}r^{1-n}\int^{r}_{0}\rho^{n-1}d\rho\\ &\leq\frac{1}{n}(\mu(t)+C_{0})r, \end{align}

    so we complete this proof.

    To show the existence of a finite-time blow-up solution of (2.4), we need to prove that U_{ss} is nonpositive by the following lemma. The proof follows the strategy in [48].

    Lemma 2.4. Suppose that D, f and u_{0} satisfy (I_{1}), (I_{2}) and (I_{3}) respectively. Then

    \begin{equation} u_{r}(r,t)\leq0 \ for \ all \ r\in(0,R), t\in(0,T_{max}). \end{equation} (2.9)

    Moreover,

    \begin{equation} U_{ss}(s,t)\leq0\ for \ all \ r\in(0,R), t\in(0,T_{max}). \end{equation} (2.10)

    Proof. Without loss of generality we may assume that \nonumber u_{0}\in C^{2} ([0, \infty)) and f\in C^{2}([0, \infty)) . Applying the regularity theory in ([49,50]), we all know that u and u_{r} belong to C^{0}([0, R]\times[0, T))\cap C^{2, 1}((0, R)\times(0, T)) and we fixed T\in(0, T_{max}) . From (2.4), we have for r\in(0, R) and t\in(0, T)

    \begin{equation} h_{rr}+\frac{n-1}{r}h_{r} = \mu(t)-f(u), \end{equation} (2.11)

    and from (2.4) we obtain

    \begin{align} u_{rt}& = \Big((D(u)u_{r})_{r}+\frac{n-1}{r}D(u)u_{r}+uf(u)-u\mu(t)-u_{r}h_{r}\Big)_{r}\\ & = (D(u)u_{r})_{rr}+a_{1}(D(u)u_{r})_{r}+a_{2}u_{rr}+bu_{r}, \end{align}

    for all r\in(0, R) and t\in(0, T) , where

    \begin{equation} \nonumber a_{1}(r,t) = \frac{n-1}{r}, \quad a_{2}(r,t) = -h_{r}, \quad b(r,t) = -\frac{n-1}{r^{2}}D(u)-\mu(t)-h_{rr}+f(u)+uf'(u), \end{equation}

    for all r\in(0, R) and t\in(0, T) . Moreover, we have h_{r}\leq\frac{r}{n}(\mu(t)+C_{0}) by (2.8) and from (2.11) such that

    \begin{equation} \nonumber-h_{rr} = \frac{n-1}{r}h_{r}-\mu(t)+f(u)\leq\frac{n-1}{n}\mu(t)+\frac{n-1}{n}C_{0}-\mu(t)+f(u)\leq f(u)+C_{0} \quad {\rm for\ all\ } r\in(0,R)\ {\rm and\ } t\in(0,T), \end{equation}

    then setting c_{1}: = \sup_{(r, t)\in(0, R)\times(0, T)}(2f(u)+uf'(u)+C_{0}) , we obtain

    \begin{equation} \nonumber b(r,t)\leq c_{1} \quad {\rm for\ all\ } r\in(0,R)\ {\rm and\ } t\in(0,T), \end{equation}

    and we introduce

    \begin{equation} \nonumber c_{2}: = \sup\limits_{(r,t)\in(0,R)\times(0,T)}((D(u))_{rr}+a_{1}(D(u))_{r}) < \infty, \end{equation}

    and set c_{3} = 2(c_{1}+c_{2}+1) . Since u_{r}(r, t) = 0 for r\in\{0, R\}, t\in(0, T) (because u is radially symmetric) and u_{0_{r}}\leq 0 , the function y:[0, R]\times[0, T]\rightarrow \mathbb{R} , (r, t)\mapsto u_{r}(r, t)-\varepsilon e^{c_{3}t} belongs to C^{0}([0, R]\times[0, T]) and fulfills

    \begin{equation} \left\{ \begin{split} y_{t}& = (D(u)(y+\varepsilon e^{c_{3}t}))_{rr}+a_{1}(D(u)(y+\varepsilon e^{c_{3}t}))_{r}+a_{2}y_{r}+b(y+\varepsilon e^{c_{3}t})-c_{3}\varepsilon e^{c_{3}t}\\ & = (D(u)y)_{rr}+a_{1}(D(u)y)_{r}+a_{2}y_{r}+by+\varepsilon e^{c_{3}t}((D(u))_{rr}+a_{1}(D(u))_{r}+b-c_{3})\\ &\leq(D(u)y)_{rr}+a_{1}(D(u)y)_{r}+a_{2}y_{r}+by+\varepsilon e^{c_{3}t}(c_{1}+c_{2}-c_{3}), \quad {\rm in\ } (0,R)\times(0,T),\\ y& < 0, \quad {\rm on\ } \{0,R\} \times(0,T),\\ y&(\cdot,0) < 0, \quad {\rm in\ } (0,R). \end{split} \right. \end{equation} (2.12)

    By the estimate for y(\cdot, 0) in (2.12) and continuity of y , the time t_{0}: = \sup\{t\in(0, T):y\leq 0\ { \rm in} \ [0, R]\times (0, T)\} \in (0, T] is defined. Suppose that t_{0} < T , then there exists r_{0}\in [0, R] such that y(r_{0}, t_{0}) = 0 and y(r, t)\leq 0 for all r\in[0, R] and t\in[0, t_{0}] ; hence, y_{t}(r_{0}, t_{0})\geq 0 . As D\geq 0 in [0, \infty) , not only y(\cdot, t_{0}) but also z:(0, R)\rightarrow \mathbb{R}, r\longmapsto D(u(r, t_{0}))y(r, t_{0}) attains its maximum 0 at r_{0} . Since the second equality in (2.12) asserts r_{0}\in(0, R) , we conclude z_{rr}(r_{0})\leq 0, z_{r}(r_{0}) = 0 and y_{r}(r_{0}, t_{0}) = 0 . Hence, we could obtain the contradiction

    \begin{align} 0&\leq y_{t}(r_{0},t_{0})\\ &\leq z_{rr}(r_{0})+a_{1}(r_{0},t_{0})z_{r}(r_{0})+a_{2}(r_{0},t_{0})y_{r}(r_{0},t_{0})+b(r_{0},t_{0})y(r_{0},t_{0})+\varepsilon e^{c_{3}t_{0}}\big(c_{1}+c_{2}-c_{3}\big)\\ &\leq-\frac{c_{3}}{2}\varepsilon e^{c_{3}t_{0}} < 0, \end{align}

    since we have

    \begin{equation} \nonumber c_{1}+c_{2}\leq \frac{c_{3}}{2}. \end{equation}

    So that t_{0} = T, implying y\leq 0 in [0, R]\times [0, T] and hence u_{r}\leq\varepsilon e^{c_{3}t} in [0, R]\times [0, T] . Letting first \varepsilon\searrow 0 and then T\nearrow T_{max} , this proves that u_{r}\leq 0 in [0, R]\times [0, T_{max}) , and we have U_{ss}\leq0 because of (2.5).

    In this section our aim is to establish a function and to select appropriate parameters such that the function satisfies ODI, which means finiteness of T_{max} by counter evidence. Firstly, we introduce a moment-like functional as follows

    \begin{equation} \phi(t): = \int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U(s,t)ds, \quad t\in[0,T_{max}), \end{equation} (3.1)

    with \gamma\in(-\infty, 1) and s_{0}\in(0, R^{n}) . As a preparation of the subsequent analysis of \phi , we denote

    \begin{equation} S_{\phi}: = \bigg\{t\in(0,T_{max})|\phi(t)\geq\frac{nM-s_{0}}{(1-\gamma)(2-\gamma)\omega_{n}}\cdot s_{0}^{2-\gamma}\bigg\}. \end{equation} (3.2)

    The following lemma provides a lower bound for U .

    Lemma 3.1. Let \gamma\in(-\infty, 1) and s_{0}\in(0, R^{n}) , then

    \begin{equation} U(\frac{s_{0}}{2},t)\geq\frac{1}{\omega_{n}}\cdot(nM-\frac{4s_{0}}{2^{\gamma}(3-\gamma)}). \end{equation} (3.3)

    Proof. If (3.3) was false for some t\in S_{\phi} such that U(\frac{s_{0}}{2}, t) < \frac{1}{\omega_{n}}\cdot(nM-\frac{4s_{0}}{2^{\gamma}(3-\gamma)}) , then necessarily \delta: = \frac{4s_{0}}{2^{\gamma}(3-\gamma)} < nM . By the monotonicity of U(\cdot, t) we would obtain that U(s, t) < \frac{nM-\delta}{\omega_{n}} for all s\in(0, \frac{s_{0}}{2}) . Since U(s, t) < \frac{nM}{\omega_{n}} for all s\in(0, R^{n}) , we have

    \begin{align} \phi(t)& < \frac{nM-\delta}{\omega_{n}}\cdot\int^{\frac{s_{0}}{2}}_{0}s^{-\gamma}(s_{0}-s)ds+\frac{nM}{\omega_{n}}\cdot\int^{s_{0}}_{\frac{s_{0}}{2}}s^{-\gamma}(s_{0}-s)ds\\ & = \frac{nM}{\omega_{n}}\cdot\int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)ds-\frac{\delta}{\omega_{n}}\cdot\int^{\frac{s_{0}}{2}}_{0}s^{-\gamma}(s_{0}-s)ds \end{align}
    \begin{equation} \nonumber = \frac{nM}{\omega_{n}}\cdot\frac{s_{0}^{2-\gamma}}{(1-\gamma)(2-\gamma)}-\frac{\delta}{\omega_{n}}\cdot\frac{2^{\gamma}(3-\gamma)s_{0}^{2-\gamma}}{4(1-\gamma)(2-\gamma)}. \quad \end{equation}

    In view of the definition of S_{\phi} , we find that nM-s_{0} < nM-\frac{2^{\gamma}(3-\gamma)\delta}{4} , which contradicts our definition of \delta .

    An upper bound for \mu is established by the following lemma.

    Lemma 3.2. Let \gamma\in(-\infty, 1) and s_{0} > 0 such that s_{0}\leq\frac{R^{n}}{6} . Then the function \mu(t) has property that

    \begin{equation} \mu(t)\leq C_{1}+\frac{1}{2s}\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma \quad \ for \ all\ s\in(0,s_{0}) \ and \ any\ t\in S_{\phi}, \end{equation} (3.4)

    where C_{1} = \frac{\frac{\chi}{2}C_{0}+C_{0}+C_{2}}{3}+C_{3} = \frac{1}{3}\bigg(\frac{\chi}{2}C_{0}+C_{0}+\frac{\chi k_{1}(\gamma_{1}-\gamma_{2})}{2\gamma_{2}}(\frac{2\xi k_{2}\gamma_{2}}{\chi k_{1}\gamma_{1}})^{\frac{\gamma_{1}}{\gamma_{1}-\gamma_{2}}}\bigg)+\chi f_{1}\bigg(\frac{2\delta}{\omega_{n}s_{0}}\bigg) .

    Proof. First for any fixed t\in S_{\phi} , we may invoke Lemma 3.1 to see that

    \begin{equation} \nonumber U(\frac{s_{0}}{2},t)\geq\frac{nM-\delta}{\omega_{n}}, \end{equation}

    and thus, as U\leq\frac{nM}{\omega_{n}} ,

    \begin{equation} \nonumber\frac{U(s_{0},t)-U(\frac{s_{0}}{2},t)}{\frac{s_{0}}{2}}\leq\frac{\frac{nM}{\omega_{n}}-\frac{nM-\delta}{\omega_{n}}}{\frac{s_{0}}{2}} = \frac{2\delta}{\omega_{n}s_{0}}. \end{equation}

    However, by concavity of U(\cdot, t) , as asserted by Lemma 2.4,

    \begin{equation} \nonumber\frac{U(s_{0},t)-U(\frac{s_{0}}{2},t)}{\frac{s_{0}}{2}}\geq U_{s}(s_{0},t)\geq U_{s}(s,t) \quad {\rm \ for \ all} \ s\in(s_{0},R^{n}). \end{equation}

    Then let s_{0}\in(0, R^{n}) , we know that

    \begin{align} \mu(t)& = \frac{1}{R^{n}}\int^{s_{0}}_{0}f(U_{s}(\sigma,t))d\sigma+\frac{1}{R^{n}}\int^{R^{n}}_{s_{0}}f(U_{s}(\sigma,t))d\sigma \\ & = \frac{1}{R^{n}}\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma+\frac{1}{R^{n}}\int^{s_{0}}_{s}f(U_{s}(\sigma,t))d\sigma+\frac{1}{R^{n}}\int^{R^{n}}_{s_{0}}f(U_{s}(\sigma,t))d\sigma, \forall t\in(0,T_{max}). \end{align} (3.5)

    Since \gamma_{1} > \gamma_{2} and Young's inequality such that \xi f_{2}(u)\leq\xi k_{2}(1+u)^{\gamma_{2}}\leq\frac{\chi k_{1}}{2}(1+u)^{\gamma_{1}}+C_{2}\leq\frac{\chi}{2}f_{1}(u)+C_{2} with C_{2} = \frac{\chi k_{1}(\gamma_{1}-\gamma_{2})}{2\gamma_{2}}(\frac{2\xi k_{2}\gamma_{2}}{\chi k_{1}\gamma_{1}})^{\frac{\gamma_{1}}{\gamma_{1}-\gamma_{2}}} for u\geq0 , then for all s\in(0, R^{n}) and t\in(0, T_{max}) we show that

    \begin{equation} \frac{\chi}{2}f_{1}(U_{s}(s,t))-C_{2}\leq f(U_{s}(s,t))\leq\chi f_{1}(U_{s}(s,t)). \end{equation} (3.6)

    Accordingly, by the monotonicity of U_{s}(\cdot, t) along with (1.7) and (3.6), we have

    \begin{align} \int^{s_{0}}_{s}f(U_{s}(\sigma,t))d\sigma&\leq\int^{s_{0}}_{s}\chi f_{1}(U_{s}(\sigma,t))d\sigma\\ &\leq\int^{s_{0}}_{s}\chi f_{1}(U_{s}(s,t))d\sigma\\ &\leq s_{0}\chi f_{1}(U_{s}(s,t)), \quad \forall s\in(0,s_{0}) ,\ t\in(0,T_{max}). \end{align}

    Since the condition of (2.7) implies that

    \begin{align} \int^{s}_{0}f(U_{s}(\sigma,t))d\sigma& = \int^{s}_{0}\boldsymbol{\chi}_{\{U_{s}(\cdot,t)\geq1\}}(\sigma)\cdot f(U_{s}(\sigma,t)d\sigma+\int^{s}_{0}\boldsymbol{\chi}_{\{U_{s}(\cdot,t) < 1\}}(\sigma)\cdot f(U_{s}(\sigma,t)d\sigma\\ &\geq\int^{s}_{0}\boldsymbol{\chi}_{\{U_{s}(\cdot,t)\geq1\}}(\sigma)\cdot\bigg(\frac{\chi}{2}f_{1}(U_{s}(\sigma,t))-C_{2}\bigg)d\sigma-C_{0}s\\ \quad &\geq\int^{s}_{0}\boldsymbol{\chi}_{\{U_{s}(\cdot,t)\geq1\}}(\sigma)\cdot\frac{\chi}{2}f_{1}(U_{s}(\sigma,t))d\sigma-(C_{0}+C_{2})s\\ & = \int^{s}_{0}\frac{\chi}{2}f_{1}(U_{s}(\sigma,t))d\sigma-\int^{s}_{0} \boldsymbol{\chi}_{\{U_{s}(\cdot,t) < 1\}}(\sigma)\cdot\frac{\chi}{2}f_{1}(U_{s}(\sigma,t))d\sigma-(C_{0}+C_{2})s\\ &\geq\int^{s}_{0}\frac{\chi}{2}f_{1}(U_{s}(s,t))d\sigma-\frac{\chi}{2}C_{0}s-(C_{0}+C_{2})s\\ &\geq\frac{s\chi}{2}f_{1}(U_{s}(s,t))-(\frac{\chi}{2}C_{0}+C_{0}+C_{2})s. \end{align}

    Therefore, we obtain

    \begin{equation} \nonumber\int^{s_{0}}_{s}f(U_{s}(\sigma,t))d\sigma\leq\frac{2s_{0}}{s}\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma+2(\frac{\chi}{2}C_{0}+C_{0}+C_{2})s_{0}. \end{equation}

    Since (3.5) we have for all s\in(0, s_{0})

    \begin{align} &\frac{1}{R^{n}}\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma+\frac{1}{R^{n}}\int^{s_{0}}_{s}f(U_{s}(\sigma,t))d\sigma\\ &\leq\frac{1}{R^{n}}\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma+\frac{2s_{0}}{R^{n}s}\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma+\frac{2(\frac{\chi}{2}C_{0}+C_{0}+C_{2})s_{0}}{R^{n}}, \end{align} (3.7)

    where s_{0}\leq\frac{R^{n}}{6} such that \frac{1}{R^{n}}\leq\frac{1}{6s_{0}}\leq\frac{1}{6s}, \frac{2s_{0}}{R^{n}s}\leq\frac{1}{3s} and \frac{s_{0}}{R^{n}}\leq\frac{1}{6} for all s\in(0, s_{0}) . Finally, we estimate the last summand of (3.5)

    \begin{equation} \frac{1}{R^{n}}\int^{R^{n}}_{s_{0}}f(U_{s}(\sigma,t))d\sigma\leq\frac{1}{R^{n}}\int^{R^{n}}_{s_{0}}\chi f_{1}(U_{s}(\sigma,t))d\sigma\leq\chi f_{1}\bigg(\frac{2\delta}{\omega_{n}s_{0}}\bigg) = C_{3}. \end{equation} (3.8)

    Together with (3.5), (3.7) and (3.8) imply (3.4).

    Lemma 3.3. Assume that \gamma\in(-\infty, 1) satisfying

    \begin{equation} \nonumber\gamma < 2-\frac{2}{n}, \end{equation}

    and s_{0}\in(0, \frac{R^{n}}{6}] . Then the function \phi:[0, T_{max})\rightarrow \mathbb{R} defined by (3.1) belongs to C^{0}([0, T_{max}))\cap C^{1}((0, T_{max})) and satisfies

    \begin{align} \phi'(t)&\geq n^{2}\int^{s_{0}}_{0}s^{2-\frac{2}{n}-\gamma}(s_{0}-s)U_{ss}D(U_{s})ds\\ & \quad +\frac{1}{2}\int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U_{s}\cdot\bigg\{\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma\bigg\}ds-C_{1}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}ds\\ & = :J_{1}(t)+J_{2}(t)+J_{3}(t), \end{align} (3.9)

    for all t\in[0, T_{max}) , where C_{1} is defined in Lemma 3.2.

    Proof. Combining (2.6) and (3.4) we have

    \begin{align} U_{t}(s,t)& = n^{2}s^{2-\frac{2}{n}}D(U_{s})U_{ss}-s\mu(t)U_{s}+U_{s}\cdot\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma\\ &\geq n^{2}s^{2-\frac{2}{n}}U_{ss}D(U_{s})+\frac{1}{2}U_{s}\cdot\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma-C_{1}sU_{s}. \end{align}

    Notice \phi(t) conforms \phi(t) = \int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U(s, t)ds . So (3.9) is a direct consequence.

    Lemma 3.4. Let s_{0}\in (0, \frac{R^{n}}{6}] , and \gamma\in(-\infty, 1) satisfying \gamma < 2-\frac{2}{n}. Then J_{1}(t) in (3.9) satisfies

    \begin{equation} J_{1}(t)\geq-I, \end{equation} (3.10)

    where

    \begin{equation} I: = \left\{ \begin{split} &-\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s), &\qquad &m < 0, \\ &n^{2}d(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)\ln(U_{s}+1),&\qquad &m = 0,\\ &\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)(U_{s}+1)^{m},&\qquad &m > 0, \end{split} \right. \end{equation} (3.11)

    for all t\in S_{\phi} .

    Proof. Since D\in C^{2}([0, \infty)) , suppose that

    \begin{equation} \nonumber G(\tau) = \int^{\tau}_{0}D(\delta)d\delta, \end{equation}

    then

    \begin{equation} 0 < G(\tau)\leq d\int^{\tau}_{0}(1+\delta)^{m-1}d\delta\leq\left\{ \begin{split} &\frac{-d}{m}, &\qquad m < 0, \\ &d\ln(\tau+1),&\qquad m = 0,\\ &\frac{d}{m}(\tau+1)^{m},&\qquad m > 0. \end{split} \right. \nonumber \end{equation}

    Here integrating by parts we obtain

    \begin{align} J_{1}(t)& = n^{2}\int^{s_{0}}_{0}s^{2-\frac{2}{n}-\gamma}(s_{0}-s)dG(U_{s})\\ & = n^{2}s^{2-\frac{2}{n}-\gamma}(s_{0}-s)G(U_{s})|^{s_{0}}_{0}+n^{2}\int^{s_{0}}_{0}s^{2-\frac{2}{n}-\gamma}G(U_{s})ds\\ & \quad -n^{2}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)G(U_{s})ds. \end{align}

    Hence a direct calculation yields

    \begin{equation} J_{1}(t)\geq\left\{ \begin{split} &\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s), &\qquad m < 0, \\ &-n^{2}d(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)\ln(U_{s}+1),&\qquad m = 0,\\ &-\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)(U_{s}+1)^{m},&\qquad m > 0,\\ \end{split} \right. \nonumber \end{equation}

    for all t\in S_{\phi} . We conclude (3.10).

    Lemma 3.5. Assume that \gamma\in(-\infty, 1) satisfying \gamma < 2-\frac{2}{n} and s_{0}\in(0, \frac{R^{n}}{6}] . Then we have

    \begin{equation} J_{2}(t)+J_{3}(t)\geq\frac{k_{1}\chi }{4}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}ds-C_{4}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}ds \end{equation} (3.12)

    for all t\in S_{\phi} , where C_{4} = C_{1}+\frac{(\frac{\chi}{2}C_{0}+C_{0}+C_{2})}{2} .

    Proof. Since Lemma 3.2 we have

    \begin{equation} \nonumber\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma\geq \frac{s}{2}\chi f_{1}(U_{s}(s,t))-(\frac{\chi}{2}C_{0}+C_{0}+C_{2})s \quad {\rm for\ all\ } s\in(0,s_{0})\ {\rm and\ } t\in(0,T_{max}). \end{equation}

    Therefore,

    \begin{align} J_{2}(t)& = \frac{1}{2}\int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U_{s}\cdot\bigg\{\int^{s}_{0}f(U_{s}(\sigma,t))d\sigma\bigg\}ds\\ &\geq\frac{\chi}{4}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}f_{1}(U_{s}(s,t))ds-\frac{(\frac{\chi}{2}C_{0}+C_{0}+C_{2})}{2}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}ds\\ &\geq\frac{k_{1}\chi }{4}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}ds-\frac{(\frac{\chi}{2}C_{0}+C_{0}+C_{2})}{2}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}ds, \end{align}

    where f_{1}(U_{s}(s, t))\geq k_{1}(1+U_{s})^{\gamma_{1}}\geq k_{1}(U_{s})^{\gamma_{1}} . Combining these inequalities we can deduce (3.12).

    Lemma 3.6. Let \gamma_{1} > \max\big\{0, m-1\big\} . For any \gamma\in(-\infty, 1) satisfying

    \begin{equation} \gamma\in\min\bigg\{2-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}, \ 2-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}\bigg\}, \end{equation} (3.13)

    and s_{0}\in(0, \frac{R^{n}}{6}], the function \phi:[0, T_{max})\rightarrow \mathbb{R} defined in (3.1) satisfies

    \begin{equation} \phi'(t)\geq\left\{ \begin{split} &C\psi(t)-Cs_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}, &m\leq1, \\ &C\psi(t)-Cs_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}},&m > 1, \ \end{split} \right. \end{equation} (3.14)

    with C > 0 for all t\in S_{\phi} , where \psi(t): = \int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}ds .

    Proof. From (3.10) and (3.12) we have

    \begin{equation} \phi'(t)\geq\frac{k_{1}\chi }{4}\psi(t)-I-C_{4}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}ds, \end{equation} (3.15)

    for all t\in S_{\phi} and I is given by (3.11). In the case m < 0 ,

    \begin{align} -\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)ds&\leq-\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)s_{0}\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}ds\\ & = -\frac{n^{2}d}{m}s_{0}^{3-\frac{2}{n}-\gamma}. \end{align}

    If m = 0 , we use the fact that \frac{\ln(1+x)}{x} < 1 for any x > 0 and Hölder's inequality to estimate

    \begin{align} &n^{2}d(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)\ln(U_{s}+1)ds\\ & = n^{2}d(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}[s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}]^{\frac{1}{1+\gamma_{1}}}\\ & \cdot s^{1-\frac{2}{n}-\gamma-\frac{1-\gamma}{1+\gamma_{1}}}(s_{0}-s)^{1-\frac{1}{1+\gamma_{1}}}\frac{\ln(1+U_{s})}{U_{s}}ds\\ &\leq n^{2}d(2-\frac{2}{n}-\gamma)\bigg\{\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}ds\bigg\}^{\frac{1}{1+\gamma_{1}}}\cdot \\ &\bigg\{\int^{s_{0}}_{0}\Big(s^{1-\frac{2}{n}-\gamma-\frac{1-\gamma}{1+\gamma_{1}}}(s_{0}-s)^{\frac{\gamma_{1}}{1+\gamma_{1}}}\Big)^{\frac{1+\gamma_{1}}{\gamma_{1}}}ds\bigg\}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\\ &\leq n^{2}d(2-\frac{2}{n}-\gamma)s_{0}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\bigg\{\int^{s_{0}}_{0}s^{\frac{(1-\frac{2}{n}-\gamma)\gamma_{1}-\frac{2}{n}}{\gamma_{1}}}ds\bigg\}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t)\\ & = C_{5}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\frac{2}{n}-\gamma)\gamma_{1}-\frac{2}{n}}{1+\gamma_{1}}}, \end{align}

    for all t\in S_{\phi} with C_{5}: = n^{2}d(2-\frac{2}{n}-\gamma)\cdot\Big(\frac{1}{2-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}\Big)^{\frac{\gamma_{1}}{1+\gamma_{1}}} > 0 by (3.13). In the case m > 0 , by using the elementary inequality (a+b)^{\alpha}\leq2^{\alpha}(a^{\alpha}+b^{\alpha}) for all a, b > 0 and every \alpha > 0 , we obtain

    \begin{align} &\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)(U_{s}+1)^{m}ds\\ &\leq2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)U_{s}^{m}ds+2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)ds, \end{align} (3.16)

    for all t\in S_{\phi} , and we first estimate the second term on the right of (3.16)

    \begin{equation} \nonumber2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)ds\leq2^{m}\frac{n^{2}d}{m}s_{0}^{3-\frac{2}{n}-\gamma}. \end{equation}

    Since \gamma_{1} > m-1 and by Hölder's inequality we deduce that

    \begin{align} 2^{m}\frac{n^{2}d}{m}&(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{1-\frac{2}{n}-\gamma}(s_{0}-s)U_{s}^{m}ds \quad \\ & = 2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\int^{s_{0}}_{0}s^{(1-\gamma)\cdot\frac{m}{1+\gamma_{1}}}(s_{0}-s)^{\frac{m}{1+\gamma_{1}}}U_{s}^{m}\cdot s^{1-\frac{2}{n}-\gamma-(1-\gamma)\cdot\frac{m}{1+\gamma_{1}}}(s_{0}-s)^{1-\frac{m}{1+\gamma_{1}}}ds\\ &\leq2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\bigg\{\int^{s_{0}}_{0}[s^{(1-\gamma)\cdot\frac{m}{1+\gamma_{1}}}(s_{0}-s)^{\frac{m}{1+\gamma_{1}}}U_{s}^{m}]^{\frac{1+\gamma_{1}}{m}}ds\bigg\}^{\frac{m}{1+\gamma_{1}}}\\ & \quad \times\bigg\{\int^{s_{0}}_{0}[s^{1-\frac{2}{n}-\gamma-(1-\gamma)\cdot\frac{m}{1+\gamma_{1}}}(s_{0}-s)^{1-\frac{m}{1+\gamma_{1}}}]^{\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}ds\bigg\}^{\frac{1+\gamma_{1}-m}{1+\gamma_{1}}}\\ &\leq2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\psi^{\frac{m}{1+\gamma_{1}}}(t)s_{0}^{\frac{1+\gamma_{1}-m}{1+\gamma_{1}}}\cdot\bigg\{\int^{s_{0}}_{0}s^{\frac{(1+\gamma_{1}-m)(1-\gamma)-\frac{2}{n}(1+\gamma_{1}) }{1+\gamma_{1}-m}}ds\bigg\}^{\frac{1+\gamma_{1}-m}{1+\gamma_{1}}}\\ &\leq C_{6}\psi^{\frac{m}{1+\gamma_{1}}}(t)s_{0}^{\frac{(1+\gamma_{1}-m)(3-\gamma)-\frac{2}{n}(1+\gamma_{1})}{1+\gamma_{1}}}, \end{align}

    for all t\in S_{\phi} with C_{6} = 2^{m}\frac{n^{2}d}{m}(2-\frac{2}{n}-\gamma)\Big(\frac{1}{2-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}\Big)^{\frac{1+\gamma_{1}-m}{1+\gamma_{1}}} > 0 where \gamma < 2-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m} from (3.13).

    Next, we can estimate the third expression on the right of (3.15) as follows

    \begin{align} &C_{4}\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}ds\\ & = C_{4}\int^{s_{0}}_{0}[s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}]^{\frac{1}{1+\gamma_{1}}}\cdot s^{1-\gamma-\frac{1-\gamma}{1+\gamma_{1}}}(s_{0}-s)^{1-\frac{1}{1+\gamma_{1}}}ds\\ &\leq C_{4}\bigg\{\int^{s_{0}}_{0}s^{1-\gamma}(s_{0}-s)U_{s}^{1+\gamma_{1}}ds\bigg\}^{\frac{1}{1+\gamma_{1}}}\cdot\bigg\{\int^{s_{0}}_{0}[s^{1-\gamma-\frac{1-\gamma}{1+\gamma_{1}}}(s_{0}-s)^{1-\frac{1}{1+\gamma_{1}}}]^{\frac{1+\gamma_{1}}{\gamma_{1}}}ds\bigg\}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\\ &\leq C_{4}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\bigg\{\int^{s_{0}}_{0}s^{1-\gamma}ds\bigg\}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\\ & = C_{7}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\gamma)\gamma_{1}}{1+\gamma_{1}}}, \end{align}

    where C_{7} = C_{4}\big(\frac{1}{2-\gamma}\big)^{\frac{\gamma_{1}}{1+\gamma_{1}}} for all t\in S_{\phi} . By (3.15) and collecting the estimates above we have

    \begin{equation} \phi'(t)\geq\left\{ \begin{split} &\frac{k_{1}\chi }{4}\psi(t)+\frac{n^{2}d}{m}s_{0}^{3-\frac{2}{n}-\gamma}-C_{7}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\gamma)\gamma_{1}}{1+\gamma_{1}}}, &m < 0,\ &t\in S_{\phi}, \\ &\frac{k_{1}\chi }{4}\psi(t)-C_{5}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\frac{2}{n}-\gamma)\gamma_{1}-\frac{2}{n}}{1+\gamma_{1}}}-C_{7}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\gamma)\gamma_{1}}{1+\gamma_{1}}},&m = 0,\ &t\in S_{\phi},\\ &\frac{k_{1}\chi }{4}\psi(t)-2^{m}\frac{n^{2}d}{m}s_{0}^{3-\frac{2}{n}-\gamma}-C_{6}\psi^{\frac{m}{1+\gamma_{1}}}(t)s_{0}^{\frac{(1+\gamma_{1}-m)(3-\gamma)-\frac{2}{n}(1+\gamma_{1})}{1+\gamma_{1}}}-C_{7}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\gamma)\gamma_{1}}{1+\gamma_{1}}},&m > 0,\ &t\in S_{\phi}. \end{split} \right. \nonumber \end{equation}

    If m = 0 , by Young's inequality we can find positive constants C_{8}, C_{9} such that

    \begin{equation} \nonumber C_{5}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\frac{2}{n}-\gamma)\gamma_{1}-\frac{2}{n}}{1+\gamma_{1}}}\leq \frac{k_{1}\chi }{16}\psi(t)+C_{8}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}, \quad \forall t\in S_{\phi}, \end{equation}

    while as m > 0 we have

    \begin{equation} \nonumber C_{6}\psi^{\frac{m}{1+\gamma_{1}}}(t)s_{0}^{\frac{(1+\gamma_{1}-m)(3-\gamma)-\frac{2}{n}(1+\gamma_{1})}{1+\gamma_{1}}}\leq\frac{k_{1}\chi }{16}\psi(t)+C_{9}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}, \quad \forall t\in S_{\phi}. \end{equation}

    On the other hand, we use Young's inequality again

    \begin{equation} \nonumber C_{7}\psi^{\frac{1}{1+\gamma_{1}}}(t)s_{0}^{\frac{(3-\gamma)\gamma_{1}}{1+\gamma_{1}}}\leq\frac{k_{1}\chi }{16}\psi(t)+C_{10}s_{0}^{3-\gamma}, \quad \forall t\in S_{\phi}. \end{equation}

    In the case m < 0 , because of s_{0}\in(0, \frac{R^{n}}{6}] , we have

    \begin{equation} \nonumber s_{0}^{3-\frac{2}{n}-\gamma} = s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}\cdot s_{0}^{\frac{2}{n\gamma_{1}}}\leq\big(\frac{R^{n}}{6}\big)^{\frac{2}{n\gamma_{1}}}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}, \end{equation}

    when m > 0 we have

    \begin{equation} \nonumber s_{0}^{3-\frac{2}{n}-\gamma} = s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}\cdot s_{0}^{\frac{2m}{n(1+\gamma_{1}-m)}}\leq\big(\frac{R^{n}}{6}\big)^{\frac{2m}{n(1+\gamma_{1}-m)}}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}. \end{equation}

    All in all, we have

    \begin{equation} \phi'(t)\geq\left\{ \begin{split} &\frac{k_{1}\chi }{8}\psi(t)-C_{11}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}-C_{10}s_{0}^{3-\gamma}, &m\leq0, \\ &\frac{k_{1}\chi }{8}\psi(t)-C_{12}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}-C_{10}s_{0}^{3-\gamma},& m > 0,\\ \end{split} \right. \end{equation} (3.17)

    for all t\in S_{\phi} with C_{11} = C_{8}-\frac{n^{2}d}{m}\big(\frac{R^{n}}{6}\big)^{\frac{2}{n\gamma_{1}}} and C_{12} = C_{9}+\frac{2^{m}n^{2}d}{m}\big(\frac{R^{n}}{6}\big)^{\frac{2m}{n(1+\gamma_{1}-m)}} . When 0 < m\leq1 , we have \frac{1+\gamma_{1}}{1+\gamma_{1}-m}\leq\frac{1+\gamma_{1}}{\gamma_{1}} such that s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}} = s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}s_{0}^{\frac{2}{n}(\frac{1+\gamma_{1}}{\gamma_{1}}-\frac{1+\gamma_{1}}{1+\gamma_{1}-m})} \leq\big(\frac{R^{n}}{6}\big)^{\frac{2(1-m)(1+\gamma_{1})}{n\gamma_{1}(1+\gamma_{1}-m)}}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}} . In the case m\leq1

    \begin{equation} s_{0}^{3-\gamma} = s_{0}^{\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}\cdot s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}\leq\big(\frac{R^{n}}{6}\big)^{\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}\cdot s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}, \quad \nonumber \end{equation}

    and if m > 1 we have

    \begin{equation} s_{0}^{3-\gamma} = s_{0}^{\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}\cdot s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}\leq\big(\frac{R^{n}}{6}\big)^{\frac{2}{n}\cdot\frac{1+\gamma_{1}}{(1+\gamma_{1}-m)}}\cdot s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}. \nonumber \end{equation}

    Thus (3.17) turns into (3.14).

    Next, we need to build a connection between \phi(t) and \psi(t) . Let us define

    \begin{equation} S_{\psi}: = \bigg\{t\in(0,T_{max})|\psi(t)\geq s_{0}^{3-\gamma}\bigg\}. \end{equation} (3.18)

    Lemma 3.7. Let \gamma\in(-\infty, 1) satisfying \gamma > 1-\gamma_{1} and (3.13) . Then for any choice of s_{0}\in(0, \frac{R^{n}}{6}] , the following inequality

    \begin{equation} \phi'(t)\geq\left\{ \begin{split} &Cs_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t)-Cs_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}, &m\leq1, \\ &Cs_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t)-Cs_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}},&m > 1, \ \end{split} \right. \end{equation} (3.19)

    holds for all t\in S_{\phi}\cap S_{\psi} with C > 0 .

    Proof. We first split

    \begin{align} U(s,t) = \int^{s}_{0}U_{s}(\sigma,t)d\sigma& = \int^{s}_{0}\boldsymbol{\chi}_{\{U_{s}(\cdot,t) < 1\}}(\sigma)\cdot U_{s}(\sigma,t)d\sigma+\int^{s}_{0}\boldsymbol{\chi}_{\{U_{s}(\cdot,t)\geq1\}}(\sigma)\cdot U_{s}(\sigma,t)d\sigma\\ &\leq s+\int^{s}_{0} \boldsymbol{\chi}_{\{U_{s}(\cdot,t)\geq1\}}(\sigma)\cdot\big\{\sigma^{1-\gamma}(s_{0}-\sigma)U_{s}^{1+\gamma_{1}}\big\}^{\frac{1}{1+\gamma_{1}}}\cdot\sigma^{-\frac{1-\gamma}{1+\gamma_{1}}}(s_{0}-\sigma)^{-\frac{1}{1+\gamma_{1}}}d\sigma\\ &\leq s+(s_{0}-s)^{-\frac{1}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t)\cdot\Big\{\int^{s}_{0}\sigma^{-\frac{1-\gamma}{1+\gamma_{1}}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}d\sigma\Big\}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\\ & = s+\Big(\frac{\gamma_{1}}{\gamma+\gamma_{1}-1}\Big)^{\frac{\gamma_{1}}{1+\gamma_{1}}}(s_{0}-s)^{-\frac{1}{1+\gamma_{1}}}s^{\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t), \end{align} (3.20)

    for all s\in(0, s_{0}) and t\in(0, T_{max}) where \frac{\gamma_{1}}{\gamma+\gamma_{1}-1} > 0 . According to the definition of S_{\psi} , we can find

    \begin{align} \frac{s}{(s_{0}-s)^{-\frac{1}{1+\gamma_{1}}}s^{\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t)}& = s^{\frac{2-\gamma}{1+\gamma_{1}}}(s_{0}-s)^{\frac{1}{1+\gamma_{1}}}\psi^{-\frac{1}{1+\gamma_{1}}}(t)\\ &\leq s_{0}^{\frac{2-\gamma}{1+\gamma_{1}}}\cdot s_{0}^{\frac{1}{1+\gamma_{1}}}\cdot(s_{0}^{3-\gamma})^{-\frac{1}{1+\gamma_{1}}} = 1, \end{align} (3.21)

    for all s\in(0, s_{0}) and t\in S_{\psi} . Combining (3.20) and (3.21) we have

    \begin{equation} \nonumber U(s,t)\leq C_{1}s^{\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}(s_{0}-s)^{-\frac{1}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t), \end{equation}

    where C_{1} = 1+\Big(\frac{\gamma_{1}}{\gamma+\gamma_{1}-1}\Big)^{\frac{\gamma_{1}}{1+\gamma_{1}}} for all s\in(0, s_{0}) and t\in S_{\psi} . Invoking Hölder's inequality, we get

    \begin{align} \phi(t)& = \int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U(s,t)ds \\ &\leq C_{1}\int^{s_{0}}_{0}s^{-\gamma+\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}(s_{0}-s)^{1-\frac{1}{1+\gamma_{1}}}ds\cdot\psi^{\frac{1}{1+\gamma_{1}}}(t)\\ &\leq C_{1}s_{0}^{\frac{\gamma_{1}}{1+\gamma_{1}}}\int^{s_{0}}_{0}s^{-\gamma+\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}ds\cdot\psi^{\frac{1}{1+\gamma_{1}}}(t)\\ & = C_{2}s_{0}^{\frac{\gamma_{1}(3-\gamma)}{1+\gamma_{1}}}\cdot\psi^{\frac{1}{1+\gamma_{1}}}(t), \end{align} (3.22)

    where C_{2} = C_{1}\frac{1+\gamma_{1}}{\gamma_{1}(2-\gamma)} for all s\in(0, s_{0}) and t\in S_{\psi} . Employing these conclusion we deduce (3.19).

    These preparations above will enable us to establish a superlinear ODI for \phi as mentioned earlier, and we prove our main result on blow-up based on a contradictory argument.

    Proof of Theorem 1.1. Step 1. Assume on the contrary that T_{max} = +\infty , and we define the function

    \begin{equation} S: = \bigg\{T\in(0,+\infty)\Big|\phi(t) > \frac{nM-s_{0}}{\omega_{n}(1-\gamma)(2-\gamma)}\cdot s_{0}^{2-\gamma} {\rm\ for \ all \ } t\in[0,T]\bigg\}. \end{equation} (3.23)

    Let us choose s_{0} > 0 such that

    \begin{equation} s_{0}\leq \min\bigg\{\frac{R^{n}}{6},\frac{nM}{2},\frac{nM\gamma_{1}}{2(1-\gamma)\omega_{n}[(C_{3}+1)(1+\gamma_{1})-1]}\bigg\}, \end{equation} (3.24)

    where M and \omega_{n} were defined in (2.7) and C_{3} = \big(\frac{\gamma_{1}}{\gamma+\gamma_{1}-1}\big)^{\frac{\gamma_{1}}{1+\gamma_{1}}} has been mentioned in (3.20). Then we pick 0 < \varepsilon(\gamma_{1}, M, R) = \varepsilon < \frac{s_{0}}{n} and s^{\star}(\gamma_{1}, M, R)\in(0, s_{0}) wtih r^{\star}(\gamma_{1}, M, R) = (s^{\star})^{\frac{1}{n}}\in(0, R) such that

    \begin{equation} \nonumber U(s,0)\geq U(s^{\star},0) = \frac{n}{\omega_{n}}\int_{B_{r^{\star}(0)}}u_{0}dx\geq\frac{n}{\omega_{n}}(M-\varepsilon), \quad \forall s\in(s^{\star},R^{n}). \end{equation}

    Therefore it is possible to estimate

    \begin{align} \phi(0)& = \int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U(s,0)ds\\ &\geq\int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)U(s^{\star},0)ds\\ & > \frac{n}{\omega_{n}}(M-\frac{s_{0}}{n})\int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)ds\\ & = \frac{nM-s_{0}}{\omega_{n}(1-\gamma)(2-\gamma)}\cdot s_{0}^{2-\gamma}. \end{align} (3.25)

    Then S is non-empty and denote T = \sup S\in(0, \infty] . Next, we need to prove (0, T)\subset S_{\phi}\cap S_{\psi}\neq\emptyset . Note that

    \begin{equation} \phi(t) > \frac{nM-s_{0}}{\omega_{n}(1-\gamma)(2-\gamma)}\cdot s_{0}^{2-\gamma}, \quad \forall t\in(0,T), \end{equation} (3.26)

    we obtain (0, T)\subset S_{\phi} . From (3.20) we have

    \begin{align} \phi(t)&\leq \int^{s_{0}}_{0}s^{-\gamma}(s_{0}-s)\big[s+C_{3}s^{\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}(s_{0}-s)^{-\frac{1}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t)\big]ds\\ &\leq s_{0}\int^{s_{0}}_{0}s^{1-\gamma}ds+C_{3}\int^{s_{0}}_{0}s^{-\gamma+\frac{\gamma+\gamma_{1}-1}{1+\gamma_{1}}}(s_{0}-s)^{\frac{\gamma_{1}}{1+\gamma_{1}}}\psi^{\frac{1}{1+\gamma_{1}}}(t)ds\\ & = \frac{s^{3-\gamma}_{0}}{2-\gamma}+\frac{C_{3}(1+\gamma_{1})}{\gamma_{1}(2-\gamma)}s_{0}^{\frac{\gamma_{1}(3-\gamma)}{1+\gamma_{1}}}\cdot \psi^{\frac{1}{1+\gamma_{1}}}(t). \end{align}

    It follows from (3.24) and (3.26) that

    \begin{equation} \phi(t)\geq \frac{nM}{2(1-\gamma)(2-\gamma)\omega_{n}}\cdot s^{2-\gamma}_{0} \quad {\rm for\ all \ } t\in(0,T). \end{equation} (3.27)

    Then

    \begin{equation} \nonumber\frac{C_{3}(1+\gamma_{1})}{\gamma_{1}(2-\gamma)}s^{\frac{\gamma_{1}(3-\gamma)}{1+\gamma_{1}}}_{0}\cdot \psi^{\frac{1}{1+\gamma_{1}}}(t)\geq\frac{nM}{2(1-\gamma)(2-\gamma)\omega_{n}}\cdot s^{2-\gamma}_{0}-\frac{s^{3-\gamma}_{0}}{2-\gamma}. \end{equation}

    Note that (3.24) implies

    \begin{equation} \nonumber \frac{nM\gamma_{1}}{2C_{3}(1-\gamma)\omega_{n}(1+\gamma_{1})s_{0}}-\frac{\gamma_{1}}{C_{3}(1+\gamma_{1})}\geq1, \end{equation}

    then we have

    \begin{align} \psi(t)&\geq\bigg[\Big(\frac{nMs^{2-\gamma}_{0}}{2(1-\gamma)(2-\gamma)\omega_{n}} -\frac{s^{3-\gamma}_{0}}{2-\gamma}\Big)\cdot \frac{\gamma_{1}(2-\gamma)}{C_{3}(1+\gamma_{1})}s_{0}^{-\frac{\gamma_{1}(3-\gamma)}{1+\gamma_{1}}}\bigg]^{1+\gamma_{1}}\\ &\geq\bigg[\frac{nM\gamma_{1}}{2C_{3}(1-\gamma)\omega_{n}(1+\gamma_{1})s_{0}}-\frac{\gamma_{1}}{C_{3}(1+\gamma_{1})}\bigg]^{1+\gamma_{1}}\cdot s_{0}^{3-\gamma}\\ &\geq s_{0}^{3-\gamma}. \end{align}

    Therefore, (0, T)\subset S_{\phi}\cap S_{\psi}\neq\emptyset.

    Step 2. Applying Lemma 3.7 we can find \gamma\in(-\infty, 1) and C_{1}, C_{2} > 0 such that for all s_{0}\in(0, \frac{R^{n}}{6}]

    \begin{equation} \phi'(t)\geq\left\{ \begin{split} &C_{1}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t)-C_{2}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}, &m\leq1, \\ &C_{1}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t)-C_{2}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}},&m > 1, \ \end{split} \right. \nonumber \end{equation}

    for all t\in S_{\phi}\cap S_{\psi} and with (3.22) we have

    \begin{equation} \nonumber\psi(t)\geq C_{3}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t), \quad \forall t\in S_{\psi}. \end{equation}

    To specify our choice of s_{0} , for given M > 0 we choose s_{0}\in(0, \frac{R^{n}}{6}] small enough such that

    \begin{equation} s_{0}\leq\frac{nM}{2}, \end{equation} (3.28)

    and also

    \begin{equation} s_{0}^{\gamma_{1}} < \frac{TC_{1}\gamma_{1}}{4}\bigg(\frac{nM}{2\omega_{n}(1-\gamma)(2-\gamma)}\bigg)^{\gamma_{1}}, \end{equation} (3.29)

    as well as

    \begin{equation} s_{0}^{1+\gamma_{1}}\leq C_{3}\bigg(\frac{nM}{2\omega_{n}(1-\gamma)(2-\gamma)}\bigg)^{1+\gamma_{1}}. \end{equation} (3.30)

    From (3.23), (3.28) and (3.30) we have

    \begin{equation} \nonumber\psi(t)\geq C_{3}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t) > C_{3}\bigg(\frac{nM-s_{0}}{\omega_{n}(1-\gamma)(2-\gamma)}\cdot \frac{1}{s_{0}}\bigg)^{1+\gamma_{1}}\cdot s_{0}^{3-\gamma}\geq s_{0}^{3-\gamma}, \quad \forall t\in S_{\psi}, \end{equation}

    which shows that S\subset S_{\phi}\cap S_{\psi} . Since 1+\gamma_{1}-m > \frac{2}{n} , we have (1+\gamma_{1})(1-\frac{2}{n(1+\gamma_{1}-m)}) > 0 if m > 1 so that we can choose s_{0} sufficiently small satisfying (3.28) (3.30) such that

    \begin{equation} \nonumber s_{0}^{(1+\gamma_{1})(1-\frac{2}{n(1+\gamma_{1}-m)})}\leq\frac{C_{1}}{2C_{2}}\bigg(\frac{nM}{2\omega_{n}(1-\gamma)(2-\gamma)}\bigg)^{1+\gamma_{1}}, \end{equation}

    while in the case m\leq1 , the condition \gamma_{1} > m-1+\frac{2}{n}\geq\frac{2}{n} which infers that (1+\gamma_{1})(1-\frac{2}{n\gamma_{1}}) > 0 and we select s_{0} small enough fulfilling (3.28) (3.30) such that

    \begin{equation} \nonumber s_{0}^{(1+\gamma_{1})(1-\frac{2}{n\gamma_{1}})}\leq\frac{C_{1}}{2C_{2}}\bigg(\frac{nM}{2\omega_{n}(1-\gamma)(2-\gamma)}\bigg)^{1+\gamma_{1}}. \end{equation}

    It is possible to obtain

    \begin{equation} \nonumber \frac{\frac{C_{1}}{2}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(0)}{C_{2}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{1+\gamma_{1}-m}}}\geq\frac{C_{1}}{2C_{2}}\bigg(\frac{nM}{2\omega_{n}(1-\gamma)(2-\gamma)}\bigg)^{1+\gamma_{1}}\cdot s_{0}^{-(1+\gamma_{1})+\frac{2}{n}\cdot\frac{1+\gamma_{1}}{(1+\gamma_{1}-m)}}\geq1, \quad \forall m > 1, \end{equation}

    and we have

    \begin{equation} \nonumber \frac{\frac{C_{1}}{2}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(0)}{C_{2}s_{0}^{3-\gamma-\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}}\geq\frac{C_{1}}{2C_{2}}\bigg(\frac{nM}{2\omega_{n}(1-\gamma)(2-\gamma)}\bigg)^{1+\gamma_{1}}\cdot s_{0}^{-(1+\gamma_{1})+\frac{2}{n}\cdot\frac{1+\gamma_{1}}{\gamma_{1}}}\geq1, \quad \forall m\leq1. \end{equation}

    All in all, for any m\in\mathbb{R} , we apply an ODI comparison argument to obtain that

    \begin{equation} \nonumber \phi'(t)\geq\frac{C_{1}}{2}s_{0}^{-\gamma_{1}(3-\gamma)}\phi^{1+\gamma_{1}}(t), \quad \forall t\in(0,T). \end{equation}

    By a direct calculation we obtain

    \begin{equation} \nonumber -\frac{1}{\gamma_{1}}\bigg(\frac{1}{\phi^{\gamma_{1}}(t)}-\frac{1}{\phi^{\gamma_{1}}(0)}\bigg)\geq\frac{C_{1}}{2}s_{0}^{-\gamma_{1}(3-\gamma)}t, \quad \forall t\in(0,T). \end{equation}

    Hence, according to (3.25) and (3.29) we conclude

    \begin{equation} \nonumber t < \frac{2}{C_{1}\gamma_{1}}s_{0}^{\gamma_{1}}\bigg(\frac{2\omega_{n}(1-\gamma)(2-\gamma)}{nM}\bigg)^{\gamma_{1}}\leq\frac{T}{2}, \end{equation}

    for all t\in(0, T) . As a consequence, we infer that T_{max} must be finite.

    In this section, we are preparing to prove Theorem 1.2 by providing the L^{p} estimate of u and the Moser-type iteration.

    Lemma 4.1. Let (u, v, w) be a classical solution of the system (1.5) under the condition of Theorem 1.2 . Suppose that

    \begin{equation} \gamma_{2} < 1+\gamma_{1} < \frac{2}{n}+m. \end{equation} (4.1)

    Then for any p > \max\big\{1, 2-m, \gamma_{2}\big\} , there exists C = C(p) > 0 such that

    \begin{equation} \int_{\Omega}(1+u)^{p}(x,t)dx\leq C \quad on \ (0,T_{max}). \end{equation} (4.2)

    Proof. Notice f_{1}(u)\leq k_{1}(1+u)^{\gamma_{1}}, \ f_{2}(u) = k_{2}(1+u)^{\gamma_{2}} for all u\geq0 . Multiplying the first equation of (1.5) by p(1+u)^{p-1} and integrating by parts with the boundary conditions for u, v and w , we have

    \begin{align} \frac{d}{dt}\int_{\Omega}&(1+u)^{p}dx+p(p-1)\int_{\Omega}(1+u)^{p-2}D(u)|\nabla u|^{2}dx\\ & = \chi p(p-1)\int_{\Omega}u(1+u)^{p-2}\nabla u\cdot\nabla vdx-\xi p(p-1)\int_{\Omega}u(1+u)^{p-2}\nabla u\cdot\nabla wdx\\ & = -\chi(p-1)\int_{\Omega}(1+u)^{p}\Delta vdx+\chi p\int_{\Omega}(1+u)^{p-1}\Delta vdx\\ & \quad +\xi(p-1)\int_{\Omega}(1+u)^{p}\Delta wdx-\xi p\int_{\Omega}(1+u)^{p-1}\Delta wdx\\ \quad &\leq\chi(p-1)\int_{\Omega}(1+u)^{p}f_{1}(u)dx+\chi p\int_{\Omega}(1+u)^{p-1}\mu_{1}(t)dx+\xi(p-1)\int_{\Omega}(1+u)^{p}\mu_{2}(t)dx\\ & \quad -\xi(p-1)\int_{\Omega}(1+u)^{p}f_{2}(u)dx+\xi p\int_{\Omega}(1+u)^{p-1}f_{2}(u)dx\\ &\leq k_{1}\chi(p-1)\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+\chi p\int_{\Omega}(1+u)^{p-1}\mu_{1}(t)dx+\xi(p-1)\int_{\Omega}(1+u)^{p}\mu_{2}(t)dx\\ & \quad -k_{2}\xi(p-1)\int_{\Omega}(1+u)^{p+\gamma_{2}}dx+k_{2}\xi p\int_{\Omega}(1+u)^{p+\gamma_{2}-1}dx, \quad \forall t\in(0,T_{max}). \end{align} (4.3)

    Firstly,

    \begin{align} p(p-1)\int_{\Omega}(1+u)^{p-2}D(u)|\nabla u|^{2}dx&\geq dp(p-1)\int_{\Omega}(1+u)^{p+m-3}|\nabla u|^{2}dx\\ & = \frac{4dp(p-1)}{(p+m-1)^{2}}\int_{\Omega}|\nabla(1+u)^{\frac{p+m-1}{2}}|^{2}dx. \end{align}

    By Young's inequality and Hölder's inequality, we obtain

    \begin{align} \chi p\int_{\Omega}(1+u)^{p-1}\mu_{1}(t)dx&\leq C_{1}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{2}\mu_{1}^{\frac{p+\gamma_{1}}{1+\gamma_{1}}}(t)\\ & = C_{1}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{2}\bigg(\frac{1}{|\Omega|}\int_{\Omega}f_{1}(u)dx\bigg)^{\frac{p+\gamma_{1}}{1+\gamma_{1}}}\\ &\leq C_{1}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{3}\bigg(\int_{\Omega}(1+u)^{1+\gamma_{1}}dx\bigg)^{\frac{p+\gamma_{1}}{1+\gamma_{1}}}\\ &\leq C_{1}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{3}\bigg\{\Big(\int_{\Omega}(1+u)^{p+\gamma_{1}}dx\Big)^{\frac{1+\gamma_{1}}{p+\gamma_{1}}}\cdot|\Omega|^{\frac{p-1}{p+\gamma_{1}}}\bigg\}^{\frac{p+\gamma_{1}}{1+\gamma_{1}}}\\ & = C_{1}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{3}|\Omega|^{\frac{p-1}{1+\gamma_{1}}}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx. \end{align}

    for all t\in(0, T_{max}) . Then by Hölder's inequality we obtain

    \begin{align} \xi(p-1)\int_{\Omega}(1+u)^{p}\mu_{2}(t)dx& = \frac{k_{2}\xi(p-1)}{|\Omega|}\int_{\Omega}(1+u)^{\gamma_{2}}dx\int_{\Omega}(1+u)^{p}dx\\ &\leq\frac{k_{2}\xi(p-1)}{|\Omega|}\bigg\{\int_{\Omega}(1+u)^{p+\gamma_{2}}dx\bigg\}^{\frac{\gamma_{2}}{p+\gamma_{2}}}|\Omega|^{\frac{p}{p+\gamma_{2}}}\times\bigg\{\int_{\Omega}(1+u)^{p+\gamma_{2}}dx\bigg\}^{\frac{p}{p+\gamma_{2}}}|\Omega|^{\frac{\gamma_{2}}{p+\gamma_{2}}}\\ & = k_{2}\xi(p-1)\int_{\Omega}(1+u)^{p+\gamma_{2}}dx, \quad \forall t\in(0,T_{max}). \end{align}

    Furthermore, by using Young's inequality and (4.1) we have

    \begin{equation} \nonumber k_{2}\xi p\int_{\Omega}(1+u)^{p+\gamma_{2}-1}dx\leq C_{4}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{5}, \end{equation}

    for all t\in(0, T_{max}) . Therefore, combining these we conclude

    \begin{equation} \nonumber\frac{d}{dt}\int_{\Omega}(1+u)^{p}dx+\frac{4dp(p-1)}{(p+m-1)^{2}}\int_{\Omega}|\nabla(1+u)^{\frac{p+m-1}{2}}|^{2}dx\leq C_{6}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx+C_{5}, \quad \forall t\in(0,T_{max}), \end{equation}

    where C_{6} = C_{1}+C_{3}|\Omega|^{\frac{p-1}{1+\gamma_{1}}}+C_{4}+k_{1}\chi(p-1) . By means of Gagliardo-Nirenberg inequality we can find C_{7} such that

    \begin{align} C_{6}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx& = C_{6}\|(1+u)^{\frac{p+m-1}{2}}\|^{\frac{2(p+\gamma_{1})}{p+m-1}}_{L^{\frac{2(p+\gamma_{1})}{p+m-1}}(\Omega)}\\ &\leq C_{7}\|\nabla(1+u)^{\frac{p+m-1}{2}}\|^{\frac{2(p+\gamma_{1})}{p+m-1}\cdot a}_{L^{2}(\Omega)}\cdot \|(1+u)^{\frac{p+m-1}{2}}\|^{\frac{2(p+\gamma_{1})}{p+m-1}\cdot(1-a)}_{L^{\frac{2}{p+m-1}}(\Omega)}\\ & \quad +C_{7}\|(1+u)^{\frac{p+m-1}{2}}\|^{\frac{2(p+\gamma_{1})}{p+m-1}}_{L^{\frac{2}{p+m-1}}(\Omega)} \end{align}

    for all t\in(0, T_{max}) , where

    \begin{equation} \nonumber a = \frac{\frac{p+m-1}{2}-\frac{p+m-1}{2(p+\gamma_{1})}}{\frac{p+m-1}{2}-(\frac{1}{2}-\frac{1}{n})}\in(0,1). \end{equation}

    Since 1-m+\gamma_{1} < \frac{2}{n} , we have \frac{2(p+\gamma_{1})}{p+m-1}\cdot a < 2 , and we use Young's inequality to see that for all t\in(0, T_{max})

    \begin{equation} \nonumber C_{6}\int_{\Omega}(1+u)^{p+\gamma_{1}}dx\leq\frac{2dp(p-1)}{(p+m-1)^{2}}\|\nabla(1+u)^{\frac{p+m-1}{2}}\|^{2}_{L^{2}(\Omega)}+C_{8}. \end{equation}

    In quite a similar manner, we obtain C_{9} = C_{9}(p) > 0 fulfilling

    \begin{equation} \nonumber\int_{\Omega}(1+u)^{p}dx\leq\frac{2dp(p-1)}{(p+m-1)^{2}}\|\nabla(1+u)^{\frac{p+m-1}{2}}\|^{2}_{L^{2}(\Omega)}+C_{9} \quad {\rm for\ all\ } t\in(0,T_{max}). \end{equation}

    Finally, combining these to (4.3) we obtain

    \begin{equation} \nonumber\frac{d}{dt}\int_{\Omega}(1+u)^{p}dx+\int_{\Omega}(1+u)^{p}dx\leq C_{5}+C_{8}+C_{9} \quad {\rm for\ all\ } t\in(0,T_{max}). \end{equation}

    Thus,

    \begin{equation} \nonumber\int_{\Omega}(1+u)^{p}dx\leq\max\Big\{\int_{\Omega}(1+u_{0})^{p}dx,C_{5}+C_{8}+C_{9}\Big\} \quad {\rm for\ all\ } t\in(0,T_{max}). \end{equation}

    We have done the proof.

    Under the condition of Lemma 4.1 we can use the above information to prove Theorem 1.2.

    Proof of Theorem 1.2. From Lemma 4.1, we let p > \max\big\{\gamma_{1}n, \gamma_{2}n, 1\big\} . By the elliptic L^{p} -estimate to the two elliptic equations in (1.5), we get that for all t\in(0, T_{max}) there exists some C_{10}(p) > 0 such that

    \begin{equation} \|v(\cdot,t)\| _{w^{2,\frac{p}{\gamma_{1}}}(\Omega)}\leq C_{10}(p), \quad \|w(\cdot,t)\| _{w^{2,\frac{p}{\gamma_{2}}}(\Omega)}\leq C_{10}(p), \end{equation} (4.4)

    and hence, by the Sobolev embedding theorem, we get

    \begin{equation} \|v(\cdot,t)\|_{C^{1}(\overline{\Omega})}\leq C_{10}(p), \quad \|w(\cdot,t)\|_{C^{1}(\overline{\Omega})}\leq C_{10}(p). \end{equation} (4.5)

    Now the Moser iteration technique ([3,51]) ensures that \|u(\cdot, t)\|_{L^{\infty}(\Omega)}\leq C for any t\in(0, T_{max}) .

    This concludes by Lemma 2.1 that T_{max} = \infty .

    The paper is supported by the Research and Innovation Team of China West Normal University (CXTD2020–5).

    The authors declare that there is no conflict of interest.



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