The Second Critical Exponent for a Time-Fractional Reaction-Diffusion Equation

Abstract

In this paper, we consider the Cauchy problem of a time-fractional nonlinear diffusion equation. According to Kaplan’s first eigenvalue method, we first prove the blow-up of the solutions in finite time under some sufficient conditions. We next provide sufficient conditions for the existence of global solutions by using the results of Zhang and Sun. In conclusion, we find the second critical exponent for the existence of global and non-global solutions via the decay rates of the initial data at spatial infinity.

1. Introduction

We study the Cauchy problem for a time-fractional reaction-diffusion equation, as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_t^{\alpha }u=\Delta u+{\left|u\right|}^{p-1}u,\hfill & x\in {\mathbf{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right)\ge 0,\hfill & x\in {\mathbf{R}}^n,\hfill \end{array}\right.\end{array}$$

(1)

where $n\ge 1$, $0<\alpha <1$, $p>1$, $u_0\in C_0\left({\mathbf{R}}^n\right):=\{f\in C\left({\mathbf{R}}^n\right);{lim}_{\left|x\right|\to \infty }f\left(x\right)=0\}$, and ${\partial }_t^{\alpha }$ denotes the Caputo time-fractional derivative of order $\alpha$ defined by the following:

$$\begin{array}{c}\hfill {\partial }_t^{\alpha }u(x,t)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{(t-s)}^{-\alpha }{\displaystyle \frac{\partial u}{\partial s}}(x,s)ds,0<\alpha <1.\end{array}$$

(2)

Here, $\Gamma (·)$ is the Gamma function. Moreover, the Caputo time-fractional derivative (2) is related to the Riemann–Liouville derivative by the following:

$$\begin{array}{c}\hfill {\partial }_t^{\alpha }u(x,t)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dt}}(t^{-\alpha }\ast [u-u_0])\left(t\right)={\displaystyle \frac{\partial }{\partial t}}{[}_0{I}_t^{1-\alpha }(u(x,t)-u_0\left(x\right))],\end{array}$$

(3)

where ${}_{0}I_t^{1-\alpha }$ denotes left Riemann–Liouville fractional integrals of order $1-\alpha$ and is defined by the following:

$$\begin{array}{c}\hfill {}_{0}I_t^{1-\alpha }u={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{(t-s)}^{-\alpha }u\left(s\right)ds.\end{array}$$

(4)

The motivation for studying the time-fractional diffusion Equation (1) comes from its application in modeling the anomalous diffusion of contaminants in soil. Fractional calculus has seen considerable development and is widely used across various fields of science and engineering. Its applications include modeling diffusion processes, signal processing, porous media, and various phenomena in physics and chemistry. It also provides mathematical tools for describing the hereditary properties and diffusion processes of various materials (see, e.g., [1,2] for more details). Time-fractional diffusion equations have been widely used in physics and engineering for memory effects, porous media, anomalous diffusion, quantum mechanics, etc. (see, e.g., [2]). Hence, in recent years, time-fractional differential equations have received extensive attention.

For the given initial data $u_0$, let $T^{\ast }=T^{\ast }\left(u_0\right)$ be the maximal existence time of the solution of (1). If $T^{\ast }=\infty$, the solution is global in time. However, if $T^{\ast }<\infty$, then the solution is not global in time in the sense that it blows up at $t=T^{\ast }$, such that we have the following:

$$\begin{array}{c}\hfill \underset{t\to T^{\ast }}{lim\; sup}{\parallel u(·,t)\parallel }_{L^{\infty }\left({\mathbf{R}}^n\right)}=\infty .\end{array}$$

Many significant results on the critical exponents of nonlinear parabolic equations have been obtained in the past decades. Fujita [3] considered the following Cauchy problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\Delta u+u^p,\hfill & x\in {\mathbf{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right)\ge 0,\hfill & x\in {\mathbf{R}}^n.\hfill \end{array}\right.\end{array}$$

(5)

In [3], it is shown that (5) possesses the critical Fujita exponent $p^{\ast }=1+2/n$, such that we have the following:

• If $1<p<p^{\ast }$, then the solution blows up in finite time for any nontrivial initial data.
• If $p>p^{\ast }$, then there are both global solutions and non-global solutions corresponding to small and large initial data, respectively.

According to Hayakawa [4], Kobayashi et al. [5], and Weissler [6], it is known that $p=p^{\ast }=1+2/n$ belongs to the blow-up case. In some situations, the sizes of the initial data required by the global and non-global solutions can be determined through the so-called second critical exponent, with respect to the decay rates of the initial data, as $\left|x\right|\to \infty$. When $p>p^{\ast }=1+2/n$, Lee and Ni [7] established the second critical exponent, $a^{\ast }=2/(p-1)$, for (5) with the initial data, $u_0\left(x\right)=\lambda \psi \left(x\right)$, where $\lambda >0$, and $\psi \left(x\right)$ is a bounded continuous function in ${\mathbf{R}}^n$, such that the following conditions hold:

• If ${\displaystyle \underset{\left|x\right|\to \infty }{lim\; inf}{\left|x\right|}^a\psi \left(x\right)>0}$ for some $a\in (0,a^{\ast })$ and any $\lambda >0$, then the solution blows up in finite time.
• If ${\displaystyle \underset{\left|x\right|\to \infty }{lim\; sup}{\left|x\right|}^a\psi \left(x\right)<\infty }$ for some $a\in (a^{\ast },n)$, then there exists ${\lambda }_0>0$, such that the solution is global in time whenever $\lambda \in (0,{\lambda }_0)$.

Lee and Ni [7] proved that $a=a^{\ast }=2/(p-1)$ belongs to the global case.

The weighted source case, i.e.,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\Delta u+K\left(x\right)u^p,\hfill & x\in {\mathbf{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right)\ge 0,\hfill & x\in {\mathbf{R}}^n.\hfill \end{array}\right.\end{array}$$

with $K\left(x\right)>0$ of the order ${\left|x\right|}^{\sigma }$ for $\sigma >-1$ if $n=1$ or for $\sigma >-2$ if $n\ge 2$ was considered with the critical Fujita exponent $p^{\ast }=1+(2+\sigma )/n$ by Pinsky [8].

The degenerate case, i.e.,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\Delta u^m+u^p,\hfill & x\in {\mathbf{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right)\ge 0,\hfill & x\in {\mathbf{R}}^n,\hfill \end{array}\right.\end{array}$$

(6)

with $m>1$ and $max(0,1-2/n)<m<1$ was thoroughly studied with the critical Fujita exponent $p^{\ast }=m+2/n$ by Galaktionov et al. [9], Qi [10], and Mochizuki and Mukai [11]. Furthermore, Galaktionov [12], Mochizuki and Mukai [11], Kawanago [13], and Mochizuki and Suzuki [14] have shown that $p=p^{\ast }=m+2/n$ belongs to the blow-up case. When $p>p^{\ast }=m+2/n$, Mukai et al. [15] and Guo and Guo [16] obtained the second critical exponent, $a^{\ast }=2/(p-m)$, for (6).

The extended case, i.e.,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\Delta u^m+K(x,t)u^p,\hfill & x\in {\mathbf{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right)\ge 0,\hfill & x\in {\mathbf{R}}^n.\hfill \end{array}\right.\end{array}$$

(7)

with $K(x,t)=t^s{\left|x\right|}^{\sigma }$ for $s\ge 0$, $m>max(0,1-2/n)$, $p>max(1,m)$, $\sigma >-1$ if $n=1$, or $\sigma >-2$ if $n\ge 2$ was obtained with the critical Fujita exponent $p^{\ast }=m+s(m-1)+(2+2s+\sigma )/n$ by Qi [17]. In the case where $K(x,t)=K\left(x\right)$ of the order ${\left|x\right|}^{\sigma }$ as $\left|x\right|\to \infty$ with $\sigma \in \mathbf{R}$ in some cone D, and $K(x,t)=0$ otherwise, Suzuki [18] considered (7) for $1\le m<p$ and obtained the critical Fujita exponent $p^{\ast }=m+\{2+max(\sigma ,-n)\}/n$ and the second critical exponent $a^{\ast }=\{2+max(\sigma ,-n)\}/(p-m)$.

Winkler [19] considered the following nonlinear diffusion equation not in divergence form:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=u^p\Delta u+u^q,\hfill & x\in {\mathbf{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right)>0,\hfill & x\in {\mathbf{R}}^n,\hfill \end{array}\right.\end{array}$$

(8)

and obtained the critical Fujita exponent $q^{\ast }=p+1$ for $p\ge 1$; we have the following:

• Suppose that $1\le q<q^{\ast }$ ($1\le q<3/2$ if $p=1$). If $u_0$ decreases sufficiently fast in space, all positive solutions of (8) are global and unbounded.
• Suppose that $q=q^{\ast }$. Then, all positive solutions of (8) blow up in finite time.
• Suppose that $q>q^{\ast }$. If $u_0$ is sufficiently large, then any positive solution of (8) blows up in finite time. If $u_0\left(x\right)\le f\left(\right|x\left|\right)$ in ${\mathbf{R}}^n$, then the solutions of (8) are global, where f satisfies for $u(x,t)={(1+t)}^{-\alpha }{f((1+t)}^{-\beta }\left|x\right|)$

  $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{f}^{″}+\left({\displaystyle \frac{n-1}{r}}+\beta r{f}^{-p}\right)f^{\prime }+\alpha f^{1-p}+f^{q-p}=0,\hfill & r\in (0,\infty ),\hfill \\ f\left(0\right)=f_0,f^{\prime }\left(0\right)=0,\hfill \end{array}\right.\end{array}$$

  with $\alpha =1/(q-1)$, $\beta =(q-p-1)/(2q-2)$, $f_0>0$, $r={(1+t)}^{-\beta }\left|x\right|$.

Furthermore, Li and Mu [20] also considered (8) and obtained the second critical exponent $a^{\ast }=2/(q-p-1)$ for $p>1$ and $q>p+1+2/n$ with initial data $u_0\left(x\right)=\lambda \psi \left(x\right)$, where $\lambda >0$, and $\psi \left(x\right)$ is a bounded continuous function in ${\mathbf{R}}^n$, such that we have the following:

• Let $n\ge 2$. Assume that ${\displaystyle \underset{\left|x\right|\to \infty }{lim\; inf}{\left|x\right|}^a\psi \left(x\right)>0}$. If $0<a<a^{\ast }$, or $a=a^{\ast }$ and $\lambda$ are large enough, then the solution $u(x,t)$ of (8) blows up in finite time.
• Assume that ${\displaystyle \underset{\left|x\right|\to \infty }{lim\; sup}{\left|x\right|}^a\psi \left(x\right)<\infty }$. If $a>a^{\ast }$, then there exists ${\lambda }_0>0$, such that the solution $u(x,t)$ of (8) is global in time whenever $\lambda \in (0,{\lambda }_0)$.

Yang et al. [21] and the author [22] studied the following extended case:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=u^p\Delta u+K\left(x\right)u^q,\hfill & x\in {\mathbf{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right)>0,\hfill & x\in {\mathbf{R}}^n,\hfill \end{array}\right.\end{array}$$

(9)

with the positive weight function $K\in C^0\left({\mathbf{R}}^n\right)$ satisfying the following:

$$\begin{array}{c}\hfill c_1{\left|x\right|}^{\sigma }\le K\left(x\right)\le c_2{\left|x\right|}^{\sigma },\left|x\right|>R_0,\sigma >-2\end{array}$$

for some $R_0,c_1,c_2>0$. Then, Yang et al. [21] obtained the critical Fujita exponent $q^{\ast }=p+1$ for $p\ge 1$; thus, we have the following:

• Suppose that $1<q<q^{\ast }$ ($1<q<3/2$ if $p=1$). If $u_0$ decreases sufficiently fast in space, all positive solutions of (9) are global and unbounded.
• Suppose that $q=q^{\ast }$. Then, all positive solutions of (9) blow up in finite time.
• Suppose that $q>q^{\ast }$. If $u_0$ is sufficiently large, then any positive solution of (9) blows up in finite time. If $u_0\left(x\right)\le f\left(\right|x\left|\right)$ in ${\mathbf{R}}^n$, then the solutions of (9) are global, where f satisfies $u(x,t)\le {(1+t)}^{-\alpha }{f((1+t)}^{-\beta }\left|x\right|)$, and we have the following:

  $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{f}^{″}+\left({\displaystyle \frac{n-1}{r}}+\beta r{f}^{-p}\right)f^{\prime }+\alpha f^{1-p}+C_1{r}^{\sigma }{f}^{q-p}=0,\hfill & r\in (0,\infty ),\hfill \\ f\left(0\right)=f_0,f^{\prime }\left(0\right)=0,\hfill \end{array}\right.\end{array}$$

  with $-2<\sigma <0$, $C_1={sup}_{{\mathbf{R}}^n}{\left(K\left(x\right)\right|x|}^{-\sigma })$, $\alpha =(\sigma +2)/(2q-2+p\sigma )$, $\beta =(q-p-1)/(2q-2+p\sigma )$, $f_0>0$, $r={(1+t)}^{-\beta }\left|x\right|$, and

  $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{f}^{″}+\left({\displaystyle \frac{n-1}{r}}+\beta r{f}^{-p}\right)f^{\prime }+\alpha f^{1-p}+C_{2}max(r^{\sigma },1)f^{q-p}=0,\hfill & r\in (0,\infty ),\hfill \\ f\left(0\right)=f_0,f^{\prime }\left(0\right)=0,\hfill \end{array}\right.\end{array}$$

  with $\sigma \ge 0$, $C_2={sup}_{{\mathbf{R}}^n}{(K\left(x\right)min(\left|x\right|}^{-\sigma },1\left)\right)$.

Moreover, Yang et al. [21] and the author [22] obtained the second critical exponent $a^{\ast }=(2+\sigma )/(q-p-1)$ for $p\ge 1$ and $q>p+1$ with initial data $u_0\left(x\right)=\lambda \psi \left(x\right)$, where $\lambda >0$, and $\psi \left(x\right)$ is a bounded continuous function in ${\mathbf{R}}^n$, such that we have the following:

• Assume that ${\displaystyle \underset{\left|x\right|\to \infty }{lim\; inf}{\left|x\right|}^a\psi \left(x\right)>0}$. If $0<a<a^{\ast }$ with $\sigma >-2$, or $a=a^{\ast }$ with $\sigma >-2$ and $\lambda$ is large enough, then the solution $u(x,t)$ of (9) blows up in finite time.
• Assume that ${\displaystyle \underset{\left|x\right|\to \infty }{lim\; sup}{\left|x\right|}^a\psi \left(x\right)<\infty }$. If $a>a^{\ast }$ with $\sigma \ge 0$, or $a\ge a^{\ast }$ with $-2<\sigma <0$, then there exists ${\lambda }_0>0$, such that the solution $u(x,t)$ of (9) is global in time whenever $\lambda \in (0,{\lambda }_0)$.

Zhang and Sun [23] studied the Cauchy problem (1), and determined the critical Fujita exponent $p^{\ast }=1+2/n$, such that we have the following:

• For $u_0\in C_0\left({\mathbf{R}}^n\right)$, $u_0\left(x\right)\ge 0$ and $u_0\left(x\right)\not\equiv 0$, if $1<p<1+2/n$, then the solution of (1) blows up in finite time.
• For $u_0\in C_0\left({\mathbf{R}}^n\right)\cap L^{q_c}\left({\mathbf{R}}^n\right)$, where $q_c=n(p-1)/2$, if $p\ge 1+2/n$ and $\parallel u_0{\parallel }_{L^{q_c}\left({\mathbf{R}}^n\right)}$ is sufficiently small, then (1) has a global solution.

The difference between the time-fractional Equation (1) and the heat Equation (5) is that $p=p^{\ast }=1+2/n$ belongs to the global case.

Zhang and Li [24] considered the following time-fractional subdiffusion equation with nonlinear memory:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_t^{\alpha }u=\Delta u{+}_0{I}_t^{1-\gamma }{\left(\right|u|}^{p-1}u),\hfill & x\in {\mathbf{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right),\hfill & x\in {\mathbf{R}}^n,\hfill \end{array}\right.\end{array}$$

(10)

where $0<\alpha <1$, $0\le \gamma <1$, $\gamma \le \alpha$, $p>1$ and $u_0\in C_0\left({\mathbf{R}}^n\right)$. Here, ${\partial }_t^{\alpha }$ and ${}_{0}I_t^{1-\alpha }$ are defined by (2) and (4), respectively. Then they determined the critical Fujita exponent $p^{\ast }=max\{1+[2(\alpha +1-\gamma )]/{[2+\alpha n-2(\alpha +1-\gamma )]}_{+},1/\gamma \}$, such that we have the following:

• If $1<p\le p^{\ast }$, then any nontrivial positive solution of (10) blows up in finite time.
• If $p>p^{\ast }$ and $\parallel u_0{\parallel }_{L^{q_c}\left({\mathbf{R}}^n\right)}$ is sufficiently small, where $q_c=n\alpha (p-1)/[2(\alpha +1-\gamma )]$, then the solution of (10) exists globally.

Furthermore, Zhang and Li [25] also considered the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_t^{\alpha }u=\Delta u{+}_0{I}_t^{1-\gamma }{\left(\right|u|}^{p-1}u),\hfill & x\in \Omega ,t>0,\hfill \\ u(x,t)=0,\hfill & x\in \partial \Omega ,t>0,\hfill \\ u(x,0)=u_0\left(x\right),\hfill & x\in \Omega ,\hfill \end{array}\right.\end{array}$$

(11)

where $\Omega$ is a bounded domain in ${\mathbf{R}}^n$ with a smooth boundary $\partial \Omega$, $u_0\in C_0(\Omega )$, $0<\alpha <1$, $0\le \gamma <1$, and $p>1$. Then they proved the following results:

• Let $\gamma \le \alpha$.

  (i)

  If $p\gamma \le 1$ and $u_0\left(x\right)\ge 0$, $u_0\left(x\right)\not\equiv 0$, then all solutions of (11) blow up in finite time.

  (ii)

  If $p\gamma >1$ and $\parallel u_0{\parallel }_{L^{\infty }(\Omega )}$ is sufficiently small, then the solution of (11) exists globally.

• Let $\gamma >\alpha$.

  (i)

  If $p<1+(1-\gamma )/\alpha$ and $u_0\left(x\right)\ge 0$, $u_0\left(x\right)\not\equiv 0$, then all solutions of (11) blow up in finite time.

  (ii)

  If $p\ge 1+(1-\gamma )/\alpha$ and $\parallel u_0{\parallel }_{L^{\infty }(\Omega )}$ is sufficiently small, then the solution of (11) exists globally.

Asogwa et al. [26] studied the following space–time-fractional reaction-diffusion type equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_t^{\alpha }u=-{(-\Delta )}^{\beta /2}u{+}_0{I}_t^{1-\alpha }\left(u^p\right),\hfill & x\in {\mathbf{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right),\hfill & x\in {\mathbf{R}}^n,\hfill \end{array}\right.\end{array}$$

(12)

where $0<\alpha <1$, $0<\beta <2$ and $p>1$. Here, ${\partial }_t^{\alpha }$ and ${}_{0}I_t^{1-\alpha }$ are defined by (2) and (4), respectively. Then they obtained the critical Fujita exponent $p^{\ast }=1+\beta /\left(\alpha n\right)$, such that we have the following:

• If $1<p\le p^{\ast }$, $\parallel u_0{\parallel }_{L^1\left({\mathbf{R}}^n\right)}<\infty$ and $u_0\left(x\right)$ is strictly positive on a set of positive measure, then for any fixed $x\in {\mathbf{R}}^n$, the solution to (12) blows up in finite time.
• If $p>p^{\ast }$, $u_0\left(x\right)\not\equiv 0$, and $\parallel u_0{\parallel }_{L^{q_c}\left({\mathbf{R}}^n\right)}$ is small, where $q_c=\alpha n(p-1)/\beta$, then the solutions to (12) exist globally in the sense that ${\parallel u(·,t)\parallel }_{L^{\infty }\left({\mathbf{R}}^n\right)}<\infty$ for all $t>0$.

Zhao and Tang [27] studied the following time-fractional semilinear diffusion equation with a forcing term, as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_t^{\alpha }u=\Delta u+{\left|u\right|}^p+t^{\sigma }w\left(x\right),\hfill & x\in {\mathbf{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right),\hfill & x\in {\mathbf{R}}^n,\hfill \end{array}\right.\end{array}$$

(13)

where $n\ge 2$, $0<\alpha <1$, $p>1$, $-1<\sigma <0$, $\alpha +\sigma >0$, $w\not\equiv 0$, and $u_0,w\in C_0\left({\mathbf{R}}^n\right)$. Here, ${\partial }_t^{\alpha }$ is defined by (2). Then, they give the critical Fujita exponent $p^{\ast }=1+2\alpha /(\alpha n-2\alpha -2\sigma )$, such that we have the following:

• If $1<p<p^{\ast }$, $u_0\left(x\right)\ge 0$, and ${\int }_{{\mathbf{R}}^n}w\left(x\right)dx>0$, then the mild solution of (13) blows up in finite time.
• If $p\ge p^{\ast }$, $u_0\in L^{q_c}\left({\mathbf{R}}^n\right)$, and $w\in L^k\left({\mathbf{R}}^n\right)$ with $\parallel u_0{\parallel }_{L^{q_c}\left({\mathbf{R}}^n\right)}+{\parallel w\parallel }_{L^k\left({\mathbf{R}}^n\right)}$ being sufficiently small, where $q_c=n(p-1)/2$ and $k=\alpha q_c/[p(\alpha +\sigma )-\sigma ]$, then the solution of (13) exists globally.

Motivated by the above results, in this paper, we shall study the behavior of solutions $u(x,t)$ to (1) when the initial data $u_0\left(x\right)$ exhibit slow decay at spatial infinity. In particular, we have the following:

$$\begin{array}{c}\hfill u_0\left(x\right)\sim \lambda {\left|x\right|}^{-a}\mathrm{near}x=\infty \end{array}$$

with $\lambda >0$ and $a>0$, we are interested in global existence and blow-up of solutions for (1) in terms of $\lambda$ and a. By reviewing the literature on time-fractional nonlinear diffusion equations, we found that there are no studies on the second critical exponent for the Cauchy problem (1), so we provide the second critical exponent for the Cauchy problem (1) based on the aforementioned literature.

The rest of this paper is organized as follows. In Section 2, we provide some preliminaries for the Cauchy problem (1). In Section 3, two sufficient conditions for the blow-up of solutions in finite time are presented in Theorem 1. In Section 4, we state the existence of global solutions under certain conditions in Theorem 2. In Section 5, conclusions are presented.

2. Preliminaries

In this section, we present some preliminaries.

We need the following Wright-type function:

$$\begin{array}{c}\hfill {\varphi }_{\alpha }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{(-z)}^k}{k!\Gamma (-\alpha k+1-\alpha )}},0<\alpha <1,z\in \mathbf{C}.\end{array}$$

(14)

The function ${\varphi }_{\alpha }$ is an entire function and satisfies the following properties:

(a)

${\varphi }_{\alpha }\left(\theta \right)\ge 0$ for $\theta \ge 0$ and ${\displaystyle {\int }_0^{\infty }{\varphi }_{\alpha }\left(\theta \right)d\theta =1}$.

(b)

${\displaystyle {\int }_0^{\infty }{\varphi }_{\alpha }\left(\theta \right){\theta }^{r}d\theta =\frac{\Gamma (1+r)}{\Gamma (1+\alpha r)}}$ for $r>-1$.

The operator $A=\Delta$ generates a semigroup ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ on $C_0\left({\mathbf{R}}^n\right)$ with the domain, as follows:

$$\begin{array}{c}\hfill D\left(A\right)=\{u\in C_0\left({\mathbf{R}}^n\right):\Delta u\in C_0\left({\mathbf{R}}^n\right)\}.\end{array}$$

Then $T\left(t\right)$ is an analytic and contractive semigroup on $C_0\left({\mathbf{R}}^n\right)$, and we have the following:

$$\begin{array}{cc}\hfill [T\left(t\right)u_0]\left(x\right)& ={\int }_{{\mathbf{R}}^n}G(x-y,t)u_0\left(y\right)dy,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill G(x,t)& ={\displaystyle \frac{1}{{\left(4\pi t\right)}^{n/2}}}{e}^{-{\left|x\right|}^2/\left(4t\right)},x\in {\mathbf{R}}^n,t>0.\hfill \end{array}$$

For $t\ge 0$, we define the operators $P_{\alpha }\left(t\right)$ and $S_{\alpha }\left(t\right)$ as follows:

$$\begin{array}{cc}\hfill [P_{\alpha }\left(t\right)u_0]\left(x\right)& ={\int }_0^{\infty }{\varphi }_{\alpha }\left(\theta \right)[T\left(t^{\alpha }\theta \right)u_0]\left(x\right)d\theta \hfill \\ \hfill & ={\int }_0^{\infty }{\varphi }_{\alpha }\left(\theta \right){\int }_{{\mathbf{R}}^n}G(x-y,t^{\alpha }\theta )u_0\left(y\right)dyd\theta ,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill [P_{\alpha }\left(t\right)u_0]\left(x\right)& =\alpha {\int }_0^{\infty }\theta {\varphi }_{\alpha }\left(\theta \right)[T\left(t^{\alpha }\theta \right)u_0]\left(x\right)d\theta \hfill \\ \hfill & =\alpha {\int }_0^{\infty }\theta {\varphi }_{\alpha }\left(\theta \right){\int }_{{\mathbf{R}}^n}G(x-y,t^{\alpha }\theta )u_0\left(y\right)dyd\theta ,\hfill \end{array}$$

(16)

where ${\varphi }_{\alpha }\left(\theta \right)$ is the function defined by (14). Note that for a given $x\in {\mathbf{R}}^n\setminus \left\{0\right\}$ and $t>0$, the function $G(x,t^{\alpha }\theta )\to 0$ as $\theta \to 0$. Hence, ${\int }_0^{\infty }{\varphi }_{\alpha }\left(\theta \right)G(x,t^{\alpha }\theta )d\theta$ is well-defined. Since ${\int }_0^{\infty }{\varphi }_{\alpha }\left(\theta \right)d\theta =1$ and ${\int }_{{\mathbf{R}}^n}G(x,t)dx=1$, we know the following:

$$\begin{array}{c}\hfill {\int }_0^{\infty }{\varphi }_{\alpha }\left(\theta \right){\int }_{{\mathbf{R}}^n}G(x,t^{\alpha }\theta )dxd\theta =1\mathrm{for}t>0.\end{array}$$

Consider the following linear equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_t^{\alpha }u=\Delta u+f(x,t),\hfill & x\in {\mathbf{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right)\ge 0,\hfill & x\in {\mathbf{R}}^n,\hfill \end{array}\right.\end{array}$$

(17)

where $u_0\in C_0\left({\mathbf{R}}^n\right)$ and $f\in L^1((0,T),C_0\left({\mathbf{R}}^n\right))$. If u is a solution of (17), then by [23], it satisfies the following:

$$\begin{array}{c}\hfill u(x,t)=[P_{\alpha }\left(t\right)u_0]\left(x\right)+{\int }_0^t{(t-s)}^{\alpha -1}[S_{\alpha }(t-s)f(·,s)]\left(x\right)ds,\end{array}$$

where $P_{\alpha }\left(t\right)$ and $S_{\alpha }\left(t\right)$ are given by (15) and (16), respectively.

Zhang and Sun [23] obtained the following lemmas related to the operators $P_{\alpha }\left(t\right)$ and $S_{\alpha }\left(t\right)$.

Lemma 1 

([23]). If $u_0\left(x\right)\ge 0$, $u_0\left(x\right)\not\equiv 0$, then $[P_{\alpha }\left(t\right)u_0]\left(x\right)>0$, $[S_{\alpha }\left(t\right)u_0]\left(x\right)>0$ and

$$\begin{array}{c}\hfill \parallel P_{\alpha }\left(t\right)u_0{\parallel }_{L^1\left({\mathbf{R}}^n\right)}=\parallel u_0{\parallel }_{L^1\left({\mathbf{R}}^n\right)},\parallel S_{\alpha }\left(t\right)u_0{\parallel }_{L^1\left({\mathbf{R}}^n\right)}={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\parallel u_0\parallel }_{L^1\left({\mathbf{R}}^n\right)}.\end{array}$$

Proof. 

See Lemmas 2.1 (a) and 2.2 (a) in [23]. □

Lemma 2 

([23]). Let $1\le p\le q\le +\infty$ and $1/r=1/p-1/q$.

(a)

If $1/r<2/n$, then we have the following:

$$\begin{array}{c}\hfill \parallel P_{\alpha }\left(t\right)u_0{\parallel }_{L^q\left({\mathbf{R}}^n\right)}\le {\left(4\pi t^{\alpha }\right)}^{-{\displaystyle \frac{n}{2r}}}{\displaystyle \frac{\Gamma (1-\frac{n}{2r})}{\Gamma (1-\frac{\alpha n}{2r})}}{\parallel u_0\parallel }_{L^p\left({\mathbf{R}}^n\right)}.\end{array}$$

(b)

If $1/r<4/n$, then we have the following:

$$\begin{array}{c}\hfill \parallel S_{\alpha }\left(t\right)u_0{\parallel }_{L^q\left({\mathbf{R}}^n\right)}\le \alpha {\left(4\pi t^{\alpha }\right)}^{-{\displaystyle \frac{n}{2r}}}{\displaystyle \frac{\Gamma (2-\frac{n}{2r})}{\Gamma (1+\alpha -\frac{\alpha n}{2r})}}{\parallel u_0\parallel }_{L^p\left({\mathbf{R}}^n\right)}.\end{array}$$

Proof. 

See Lemmas 2.1 (b) and 2.2 (b) in [23]. □

Next, we define a mild solution of the Cauchy problem (1), as follows:

Definition 1.

Let $u_0\in C_0\left({\mathbf{R}}^n\right)$ and $T>0$. We call $u\in C([0,T],C_0\left({\mathbf{R}}^n\right))$ a mild solution of the problem (1) if u satisfies the following integral equation:

$$\begin{array}{c}\hfill u(x,t)=[P_{\alpha }\left(t\right)u_0]\left(x\right)+{\int }_0^t{(t-s)}^{\alpha -1}[S_{\alpha }{(t-s)\left|u(·,s)\right|}^{p-1}u(·,s)]\left(x\right)ds,t\in [0,T].\end{array}$$

For the Cauchy problem (1), Zhang and Sun [23] established the following local existence result:

Proposition 1 

(Theorem 3.2 in [23]). Let $0<\alpha <1$. For a given $u_0\in C_0\left({\mathbf{R}}^n\right)$, there exists a maximal existence time $T^{\ast }>0$, such that the problem (1) has a unique mild solution $u\in C([0,T^{\ast }],C_0\left({\mathbf{R}}^n\right))$ and either $T^{\ast }=+\infty$ or $T^{\ast }<+\infty$ and ${\parallel u\parallel }_{L^{\infty }((0,t),C_0\left({\mathbf{R}}^n\right))}\to \infty$. In addition, if $u_0\left(x\right)\ge 0$ and $u_0\left(x\right)\not\equiv 0$, then $u(x,t)\ge [P_{\alpha }\left(t\right)u_0]\left(x\right)>0$ for $t\in (0,T^{\ast })$. Moreover, if $u_0\in L^r\left({\mathbf{R}}^n\right)$ for some $r\in [1,\infty )$, then $u\in C([0,T^{\ast }),L^r\left({\mathbf{R}}^n\right))$.

Furthermore, Zhang and Sun [23] also obtained the following blow-up and global existence results:

Proposition 2 

(Theorem 4.3 in [23]). Let $0<\alpha <1$, $u_0\in C_0\left({\mathbf{R}}^n\right)$ and $u_0\left(x\right)\ge 0$. If

$$\begin{array}{c}\hfill {\int }_{{\mathbf{R}}^n}{u}_0\left(x\right)\chi \left(x\right)dx>1,where\chi \left(x\right)={\left({\int }_{{\mathbf{R}}^n}{e}^{-\sqrt{n^2+{\left|x\right|}^2}}dx\right)}^{-1}{e}^{-\sqrt{n^2+{\left|x\right|}^2}},\end{array}$$

then the mild solutions of (1) blow up in finite time.

Proposition 3 

(Theorem 4.4 in [23]). Let $0<\alpha <1$, $u_0\in C_0\left({\mathbf{R}}^n\right)$, $u_0\left(x\right)\ge 0$ and $u_0\left(x\right)\not\equiv 0$.

(a)

If $1<p<1+2/n$, then the mild solution of (1) blows up in finite time.

(b)

If $p\ge 1+2/n$ and $\parallel u_0{\parallel }_{L^{q_c}\left({\mathbf{R}}^n\right)}$ is sufficiently small, where $q_c=n(p-1)/2$, then the mild solution of (1) exists globally.

3. Blow-Up of the Solution

In this section, we state the blow-up result. The main methods used to analyze the blow-up phenomenon are Kaplan’s first eigenvalue method [28] and the concavity method [29]. Kaplan’s first eigenvalue method [28] is simpler than the concavity method [29] and has been successfully used to study fractional reaction-diffusion equations, so we use Kaplan’s first eigenvalue method [28].

Next, we shall prove the following result:

Theorem 1.

Let $n\ge 1$ and $0<\alpha <1$. Assume that the initial data $u_0\left(x\right)=\lambda \psi \left(x\right)\ge 0$, where $\lambda >0$ and $\psi \in C_0\left({\mathbf{R}}^n\right)$. Suppose that one of the following two conditions holds:

(a)

$\lambda >0$ is large enough;

(b)

$0<a<2/(p-1)$ and

$$\begin{array}{c}\hfill {\displaystyle \underset{\left|x\right|\to \infty }{lim\; inf}{\left|x\right|}^a\psi \left(x\right)>0.}\end{array}$$

(18)

Then, the solution of (1) blows up in finite time.

Proof. 

We take a similar strategy, such as Theorem 1 in [22] and Theorem 3.7 in [30], using Kaplan’s first eigenvalue method [28].

Let

$$\begin{array}{c}\hfill B_m=\left\{x\in {\mathbf{R}}^n;|x-x_m|<{\displaystyle \frac{1}{2}}m\right\}\end{array}$$

(19)

for a sequence ${\left\{x_m\right\}}_{m=1}^{\infty }$ satisfying $|x_m|=m$ for any $m\in \mathbf{N}$. □

Remark 1.

The method using the sequence of balls $B_m$ in (19) was used in [22,31,32,33,34,35].

Let ${\lambda }_m>0$ denote the first eigenvalue of $-\Delta$ with the Dirichlet problem in $B_m$, and let ${\varphi }_m\left(x\right)>0$ denote the corresponding eigenfunction, normalized by the following:

$$\begin{array}{c}\hfill {\int }_{B_m}{\varphi }_m\left(x\right)dx=1.\end{array}$$

(20)

Let $T\in (0,T^{\ast })$ be arbitrarily fixed. We define the following:

$$\begin{array}{c}\hfill F_m\left(t\right)={\int }_{B_m}u(x,t){\varphi }_m\left(x\right)dx.\end{array}$$

(21)

From (1), we have the following:

$$\begin{array}{c}\hfill {\partial }_t^{\alpha }{\int }_{B_m}u(x,t){\varphi }_m\left(x\right)dx={\int }_{B_m}\Delta u(x,t){\varphi }_m\left(x\right)dx+{\int }_{B_m}u{(x,t)}^p{\varphi }_m\left(x\right)dx\\ \hfill \mathrm{for}t\in (0,T),\end{array}$$

supplemented with the following initial condition:

$$\begin{array}{c}\hfill F_m\left(0\right)={\int }_{B_m}{u}_0\left(x\right){\varphi }_m\left(x\right)dx.\end{array}$$

(22)

By integrating by parts, and the fact that ${\varphi }_m\left(x\right)=0$ and $\partial {\varphi }_m/\partial \nu \le 0$ on $\partial B_m$, where $\nu$ denotes the outward unit normal vector to $B_m$ at $x\in \partial B_m$, and applying Green’s formula, we have the following:

$$\begin{array}{c}\hfill {\partial }_t^{\alpha }{\int }_{B_m}u(x,t){\varphi }_m\left(x\right)dx\ge {\int }_{B_m}u(x,t)\Delta {\varphi }_m\left(x\right)dx+{\int }_{B_m}u{(x,t)}^p{\varphi }_m\left(x\right)dx.\end{array}$$

Since the first eigenvalue ${\lambda }_m>0$ and the eigenfunction ${\varphi }_m\left(x\right)$ satisfy the following:

$$\begin{array}{c}\hfill \Delta {\varphi }_m\left(x\right)=-{\lambda }_m{\varphi }_m\left(x\right),\end{array}$$

we obtain the following:

$$\begin{array}{c}\hfill {\partial }_t^{\alpha }{\int }_{B_m}u(x,t){\varphi }_m\left(x\right)dx\ge -{\lambda }_m{\int }_{B_m}u(x,t){\varphi }_m\left(x\right)dx+{\int }_{B_m}u{(x,t)}^p{\varphi }_m\left(x\right)dx,\end{array}$$

(23)

By (20), (21), and Hölder’s inequality, we have the following:

$$\begin{array}{cc}\hfill & F_m\left(t\right)={\int }_{B_m}u(x,t){\varphi }_m{\left(x\right)}^{{\displaystyle \frac{1}{p}}}{\varphi }_m{\left(x\right)}^{1-{\displaystyle \frac{1}{p}}}dx\le {\left({\int }_{B_m}u{(x,t)}^p{\varphi }_m\left(x\right)dx\right)}^{1/p}.\hfill \end{array}$$

So, we obtain the following:

$$\begin{array}{c}\hfill {\int }_{B_m}u{(x,t)}^p{\varphi }_m\left(x\right)dx\ge F_m{\left(t\right)}^p.\end{array}$$

(24)

Using (21) and (24) in (23) yields the following:

$$\begin{array}{cc}\hfill {\partial }_t^{\alpha }{F}_m\left(t\right)& \ge -{\lambda }_m{F}_m\left(t\right)+F_m{\left(t\right)}^p\mathrm{for}t\in (0,T).\hfill \end{array}$$

Since $B_m$ is an n-dimensional ball of radius ${\displaystyle \frac{1}{2}}m$, it follows that ${\lambda }_m$ satisfies the following:

$$\begin{array}{c}\hfill {\lambda }_m\le {\displaystyle \frac{c_1}{m^2}},\end{array}$$

(25)

where $c_1>0$ depends only on the dimension n. Thus, we have the following:

$$\begin{array}{cc}\hfill {\partial }_t^{\alpha }{F}_m\left(t\right)& \ge -c_1{m}^{-2}{F}_m\left(t\right)+F_m{\left(t\right)}^p.\hfill \end{array}$$

(26)

Setting $H\left(\zeta \right):=-c_1{m}^{-2}\zeta +{\zeta }^p$, then the function $H\left(\zeta \right)$ is convex in $\zeta >0$ since $H\in C^2(0,\infty )$ and $H^{″}\left(\zeta \right)\ge 0$. By (3), writing ${\displaystyle \frac{d}{dt}}(k\ast [F_m-F_m\left(0\right)])\left(t\right)$ instead of ${\partial }_t^{\alpha }{F}_m\left(t\right)$ with $k\left(t\right)={\displaystyle \frac{t^{-\alpha }}{\Gamma (1-\alpha )}}$ in (26), we obtain the following:

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}(k\ast [F_m-F_m\left(0\right)])\left(t\right)\ge H\left(F_m\left(t\right)\right)\mathrm{for}t\in (0,T).\end{array}$$

(27)

It is clear that $H\left(\zeta \right)>0$ and $H^{\prime }\left(\zeta \right)>0$ for all $\zeta >{\left(c_1{m}^{-2}\right)}^{{\displaystyle \frac{1}{p-1}}}$.

Suppose we have the following:

$$\begin{array}{c}\hfill F_m\left(0\right)>{\left(c_1{m}^{-2}\right)}^{{\displaystyle \frac{1}{p-1}}}.\end{array}$$

(28)

We claim that (27) implies that $F_m\left(t\right)>{\left(c_1{m}^{-2}\right)}^{{\displaystyle \frac{1}{p-1}}}$ for all $t\in (0,T)$ (the fact is stated in the proof of Theorem 3.7 in [30]). Knowing that $F_m\left(t\right)\ge F_m\left(0\right)>{\left(c_1{m}^{-2}\right)}^{{\displaystyle \frac{1}{p-1}}}$ for all $t\in (0,T)$; from (27), we have the following:

$$\begin{array}{c}\hfill {\partial }_t^{\alpha }{F}_m\left(t\right)={\displaystyle \frac{d}{dt}}(k\ast [F_m-F_m\left(0\right)])\left(t\right)\ge H\left(F_m\left(t\right)\right)>0\mathrm{for}\mathrm{all}t\in (0,T).\end{array}$$

(29)

Therefore, the function $F_m\left(t\right)$ satisfying (29) is an upper solution of the following problem:

$$\begin{array}{c}\hfill {\partial }_t^{\alpha }\zeta =H\left(\zeta \right)=-c_1{m}^{-2}\zeta +{\zeta }^p,\zeta \left(0\right)=F_m\left(0\right),\end{array}$$

(30)

According to the comparison principle, we have $F_m\left(t\right)\ge \zeta \left(t\right)$ (see Theorem 4.10 in [36]).

On the other hand, since $H\left(0\right)=0$, $H\left(\zeta \right)>0$, and $H^{\prime }\left(\zeta \right)>0$ for all $\zeta \ge F_m\left(0\right)>{\left(c_1{m}^{-2}\right)}^{{\displaystyle \frac{1}{p-1}}}$, it follows from Lemma 3.8 in [30] that $v\left(t\right)=w\left({\displaystyle \frac{t^{\alpha }}{\Gamma (\alpha +1)}}\right)$ is a lower solution for (30), where $v\left(t\right)$ satisfies the following:

$$\begin{array}{c}\hfill {\partial }_t^{\alpha }v\le H\left(v\right)=-c_1{m}^{-2}v+v^p,v\left(0\right)\le F_m\left(0\right),\end{array}$$

and $w\left(t\right)$ solves the ordinary differential equation as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{dw}{dt}}=H\left(w\right)=-c_1{m}^{-2}w+w^p,w\left(0\right)=F_m\left(0\right).\end{array}$$

(31)

By the comparison principle (see Theorem 4.10 in [36]), we obtain $\zeta \left(t\right)\ge v\left(t\right)$. Solving the initial value problem (31), we have the following solution:

$$\begin{array}{c}\hfill w\left(t\right)={\left[F_m{\left(0\right)}^{1-p}-{\displaystyle \frac{1-exp\left\{(1-p)c_1{m}^{-2}t\right\}}{c_1{m}^{-2}}}\right]}^{{\displaystyle \frac{1}{1-p}}}exp\left(-c_1{m}^{-2}t\right).\end{array}$$

By the comparison principle (see Theorem 4.10 in [36]), we conclude the following:

$$\begin{array}{cc}\hfill F_m\left(t\right)\ge v\left(t\right)& =w\left({\displaystyle \frac{t^{\alpha }}{\Gamma (\alpha +1)}}\right)\hfill \\ \hfill & ={\left[F_m{\left(0\right)}^{1-p}-{\displaystyle \frac{1-exp\left\{(1-p)c_2{t}^{\alpha }\right\}}{c_2\Gamma (\alpha +1)}}\right]}^{{\displaystyle \frac{1}{1-p}}}exp\left(-c_2{t}^{\alpha }\right).\hfill \end{array}$$

(32)

with $c_2={\displaystyle \frac{c_1{m}^{-2}}{\Gamma (\alpha +1)}}$. Therefore, from (32), we obtain $v\left(t\right)\to \infty$ as follows:

$$\begin{array}{c}\hfill t\to {\left[{\displaystyle \frac{log\left(1-c_2\Gamma (\alpha +1)F_m{\left(0\right)}^{1-p}\right)}{(1-p)c_2}}\right]}^{{\displaystyle \frac{1}{\alpha }}},\end{array}$$

and that $F_m\left(t\right)\to \infty$. This implies that the solution $u(x,t)$ blows up in finite time when (28) holds.

As a result of these arguments, we have the following lemma:

Lemma 3.

Let $F_m\left(t\right)$ be defined by (21). If $F_m\left(0\right)$ as in (22) satisfies (28) for some $m\in \mathbf{N}$, i.e.,

$$\begin{array}{c}\hfill F_m\left(0\right)>A{m}^{-{\displaystyle \frac{2}{p-1}}}\mathit{for}\mathit{some}m\in \mathbf{N},\end{array}$$

where $A=c_1^{{\displaystyle \frac{1}{p-1}}}$ with $c_1$ as in (25), then $u(x,t)$ blows up in finite time.

Proof for Theorem 1. 

Here, we shall state the rest of the proof for Theorem 1.

Assuming that $u(x,t)$ represents a nontrivial global solution, we proceed to prove by reductio ad absurdum.

By Lemma 3, it follows that for any $m\in \mathbf{N}$, we have the following:

$$\begin{array}{c}\hfill F_m\left(0\right)\le A{m}^{-{\displaystyle \frac{2}{p-1}}}.\end{array}$$

Then, by (22) and $u_0\left(x\right)=\lambda \psi \left(x\right)\ge 0$, we obtain

$$\begin{array}{c}\hfill \lambda {\int }_{B_m}\psi \left(x\right){\varphi }_m\left(x\right)dx\le A{m}^{-{\displaystyle \frac{2}{p-1}}}.\end{array}$$

(33)

Here, if we choose $\lambda$ to be large enough for any $m\in \mathbf{N}$, then the left-hand side of (33) is larger than the right-hand side of (33). Thus, we arrive at a contradiction. This completes the proof for the condition (a).

Next, if $\psi \in C_0\left({\mathbf{R}}^n\right)$ satisfies (18), then there is a positive constant L, such that $\psi \left(x\right)\ge {L\left|x\right|}^{-a}$ for sufficiently large $\left|x\right|$. Then, we have the following for the sufficiently large m:

$$\begin{array}{c}\hfill \lambda L{\int }_{B_m}{\left|x\right|}^{-a}{\varphi }_m\left(x\right)dx\le A{m}^{-{\displaystyle \frac{2}{p-1}}}.\end{array}$$

By noting that $\left|x\right|\le {\displaystyle \frac{3}{2}}m$ in $B_m$ by (19), we obtain the following:

$$\begin{array}{c}\hfill \lambda L{\left({\displaystyle \frac{3}{2}}m\right)}^{-a}{\int }_{B_m}{\varphi }_m\left(x\right)dx\le A{m}^{-{\displaystyle \frac{2}{p-1}}},\end{array}$$

and then by (20), we have the following:

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{3}{2}}\right)}^{-a}\lambda L{m}^{-a}\le A{m}^{-{\displaystyle \frac{2}{p-1}}}.\end{array}$$

(34)

By multiplying both sides of (34) by $m^a$, we obtain the following:

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{3}{2}}\right)}^{-a}\lambda L\le A{m}^{a-{\displaystyle \frac{2}{p-1}}}.\end{array}$$

(35)

Then, if ${\displaystyle 0<a<\frac{2}{p-1}}$ and m is sufficiently large, the left-hand side of (35) is larger than the right-hand side of (35). Thus, we arrive at a contradiction. This completes the proof for condition (b). □

Remark 2.

The key features of the first eigenvalue method by Kaplan are as follows:

• Let $\mu >0$ denote the first eigenvalue of $-\Delta$ with the Dirichlet problem in Ω, and let $\rho \left(x\right)>0$ denote the corresponding eigenfunction, normalized by the following:

  $$\begin{array}{c}\hfill {\int }_{\Omega }\rho \left(x\right)dx=1,\end{array}$$

  where Ω is a bounded smooth domain in ${\mathbf{R}}^n$. We define the following:

  $$\begin{array}{c}\hfill F\left(t\right)={\int }_{\Omega }u(x,t)\rho \left(x\right)dx,\end{array}$$

  where $u(x,t)$ is a nonnegative solution of the equation ${\partial }_t^{\alpha }u=\Delta u+{\left|u\right|}^{p-1}u$ in $\Omega \times [0,\infty )$. By Green’s formula and Hölder’s inequality, we have the following:

  $$\begin{array}{cc}\hfill {\partial }_t^{\alpha }F\left(t\right)& \ge -\mu F\left(t\right)+F{\left(t\right)}^{p}fort>0.\hfill \end{array}$$

  If $F\left(0\right)>{\mu }^{1/(p-1)}$, then $F\left(t\right)$ blows up in finite time. In other words, the global existence of $F\left(t\right)$ requires the following:

  $$\begin{array}{c}\hfill F\left(t\right)\le {\mu }^{1/(p-1)}forallt\ge 0\end{array}$$

  (36)

  holds. Hence, supposing that $u(x,t)$ is a nontrivial global solution, we can prove by reductio ad absurdum that $u(x,t)$ blows up in finite time by leading (36) to a contradiction. A weakness is that the first eigenvalue method proposed by Kaplan cannot be applied to problems where the comparison principle cannot be used.

Remark 3.

Zhang and Sun (Theorem 4.4 in [23]) proved that if $1<p<1+2/n$, then the mild solution of the Cauchy problem (1) blows up in finite time. On the other hand, the main novelty of this paper is from Theorem 1, which states that when the initial data $u_0\left(x\right)$ are large enough or decay more slowly than ${\left|x\right|}^{-2/(p-1)}$ at spatial infinity, the mild solution of (1) blows up in finite time for any $n\ge 1$ and $p>1$, even if $p\ge 1+2/n$.

4. Global Existence

In this section, we state the following global existence result.

Theorem 2.

Let $n\ge 1$ and $0<\alpha <1$. Assume that the initial data $u_0\left(x\right)=\lambda \psi \left(x\right)\ge 0$, where $\lambda >0$ and $\psi \in C_0\left({\mathbf{R}}^n\right)$. Suppose that $p\ge 1+2/n$, and that $a>2/(p-1)$, and we have the following:

$$\begin{array}{c}\hfill {\displaystyle \underset{\left|x\right|\to \infty }{lim\; sup}{\left|x\right|}^a\psi \left(x\right)<\infty .}\end{array}$$

(37)

Then, the mild solution of (1) exists globally whenever $\lambda >0$ is small enough.

Proof. 

In what follows, by the letter C, we denote generic positive constants, and they may have different values within the same line.

Since $\psi \in C_0\left({\mathbf{R}}^n\right)$ satisfies (37), there is a constant $C>0$, such that we have the following:

$$\begin{array}{c}\hfill \psi \left(x\right)\le C{(1+|x\left|\right)}^{-a}\mathrm{for}\mathrm{all}x\in {\mathbf{R}}^n.\end{array}$$

(38)

Let $q_c=n(p-1)/2$. First, if $p>1+2/n$ and $a>2/(p-1)$, then we know $a{q}_c>n$. Hence, from (38), we have the following:

$$\begin{array}{cc}\hfill & {\parallel \psi \parallel }_{L^{q_c}\left({\mathbf{R}}^n\right)}^{q_c}={\int }_{{\mathbf{R}}^n}{\left|\psi \left(x\right)\right|}^{q_c}dx\le C{\int }_{{\mathbf{R}}^n}{(1+|x\left|\right)}^{-a{q}_c}dx\hfill \\ \hfill & \le C{\int }_0^{\infty }{(1+r)}^{-a{q}_c}{r}^{n-1}dr\le C{\int }_0^{\infty }{(1+r)}^{n-a{q}_c-1}dr\le C.\hfill \end{array}$$

(39)

Next, if $p=1+2/n$ and $a>2/(p-1)$, then we have $q_c=1$ and $a>n=2/(p-1)$. Hence, from (38), we have the following:

$$\begin{array}{cc}\hfill & {\parallel \psi \parallel }_{L^1\left({\mathbf{R}}^n\right)}={\int }_{{\mathbf{R}}^n}\left|\psi \left(x\right)\right|dx\le C{\int }_{{\mathbf{R}}^n}{(1+|x\left|\right)}^{-a}dx\hfill \\ \hfill & \le C{\int }_0^{\infty }{(1+r)}^{-a}{r}^{n-1}dr\le C{\int }_0^{\infty }{(1+r)}^{n-a-1}dr\le C.\hfill \end{array}$$

(40)

By (39) and (40), if $p\ge 1+2/n$ and $a>2/(p-1)$, then ${\parallel \psi \parallel }_{L^{q_c}\left({\mathbf{R}}^n\right)}\le C$. Since $u_0\left(x\right)=\lambda \psi \left(x\right)$, $\parallel u_0{\parallel }_{L^{q_c}\left({\mathbf{R}}^n\right)}$ is sufficiently small whenever $\lambda >0$ is small enough. Therefore, the mild solution of (1) exists globally by Proposition 3 (b).

This completes the proof. □

Remark 4.

Zhang and Sun proved (Theorem 4.4 in [23]); if $p\ge 1+2/n$ and $\parallel u_0{\parallel }_{L^{q_c}\left({\mathbf{R}}^n\right)}$ are sufficiently small, where $q_c=n(p-1)/2$, then the mild solution of the Cauchy problem (1) exists globally. In particular, we see from the above proof that the global condition of Theorem 2 satisfies that of Theorem 4.4 in [23].

5. Conclusions

In this paper, we analyze a reaction-diffusion equation with a Caputo fractional derivative in time and with initial conditions. By comparing the conclusions of Zhang and Sun [23] (Proposition 3) and the author (Theorems 1 and 2), we see that the Cauchy problem (1) possesses the critical Fujita exponent, as follows:

$$\begin{array}{c}\hfill p^{\ast }=1+{\displaystyle \frac{2}{n}},\end{array}$$

and the second critical exponent, as follows:

$$\begin{array}{c}\hfill a^{\ast }={\displaystyle \frac{2}{p-1}}.\end{array}$$

We summarize this in Table 1.

Table 1. Critical Fujita exponent $p^{\ast }$ and second critical exponent $a^{\ast }$.

	$a>a^{\ast }$	$0<a<a^{\ast }$
$1<p<p^{\ast }$	BU 1 [Proposition 3 (a)]	BU 1 [Theorem 1 (b)]
	BU 1 for large $\lambda >0$
$p\ge p^{\ast }$	GE 2 for small $\lambda >0$	BU1 [Theorem 1 (b)]
	[Theorems 1 (a) and 2]

¹ BU: Blow-up; ² GE: Global existence.

The significance of the results is that if the initial data $u_0\left(x\right)$ decay more slowly than ${\left|x\right|}^{-a^{\ast }}$ at spatial infinity, then the mild solution of (1) blows up in finite time for any $n\ge 1$ and $p>1$ by Theorem 1 (b). On the other hand, if the initial data $u_0\left(x\right)$ are small enough and decay faster than ${\left|x\right|}^{-a^{\ast }}$ at spatial infinity, then the mild solution of (1) exists globally for $p\ge p^{\ast }$ by Theorem 2.

Comparing with the classical results of the nonlinear heat Equation (5) (i.e., (1) with $\alpha =1$), the major difference between the time-fractional diffusion Equation (1) and the nonlinear heat Equation (5) is in the critical case; that is, $p=p^{\ast }$, the solution of (1) can exist globally.

In the case $a=a^{\ast }$, if $1<p<p^{\ast }$ then the mild solution of (1) blows up in finite time by Zhang and Sun [23] (Proposition 3 (a)), but if $p\ge p^{\ast }$, then there are few studies. Therefore, we will consider studying $a=a^{\ast }$ and $p\ge p^{\ast }$ in the future.

Example 1.

We present some numerical examples of Theorems 1 and 2, as follows:

(i)

When $n=1$, we have $p^{\ast }=3$.

If $p\ge p^{\ast }$, then $a^{\ast }\le 1$, and if $1<p<p^{\ast }$, then $a^{\ast }>1$.

(ii)

When $n=3$, we have $p^{\ast }=5/3$.

If $p\ge p^{\ast }$, then $a^{\ast }\le 3$, and if $1<p<p^{\ast }$, then $a^{\ast }>3$.

(iii)

When $n=10$, we have $p^{\ast }=6/5$.

If $p\ge p^{\ast }$, then $a^{\ast }\le 10$, and if $1<p<p^{\ast }$, then $a^{\ast }>10$.

In general, when $p^{\ast }=1+2/n$, if $p\ge p^{\ast }$, then $a^{\ast }\le n$, and if $1<p<p^{\ast }$, then $a^{\ast }>n$.

Suzuki [37] also studied the Cauchy problem for a time-fractional reaction-diffusion equation, as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_t^{\alpha }u=\Delta u+f\left(u\right),\hfill & x\in {\mathbf{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right)\ge 0,\hfill & x\in {\mathbf{R}}^n,\hfill \end{array}\right.\end{array}$$

(41)

where $n\ge 1$, $0<\alpha <1$. The difference between the results of Suzuki [37] and ours is that Suzuki [37] obtained integrability conditions on $u_0\left(x\right)$ with a singularity that determined the existence and nonexistence of a nonnegative “local” solution in time for f having an exponential or a super-exponential growth, e.g., $f\left(u\right)=exp\left(u^r\right)$ ($r>0$) or $f\left(u\right)=exp\left(\right|logu{|}^{r-1}logu)$ ($r>1$), whereas we obtained the conditions on $u_0\left(x\right)$ with a polynomial decay at spatial infinity that determined the existence and nonexistence of a nonnegative “global” solution in time for $f\left(u\right)={\left|u\right|}^{p-1}u$ ($p>1$).

Zhang and Sun [24], Asogwa et al. [26], and Zhao and Tang [27] studied the Cauchy problems (10), (12), and (13), respectively. The difference between their results [24,26,27] and ours is that they [24,26,27] obtained the critical “Fujita” exponent for (10), (12), and (13), whereas we obtained the “second” critical exponent for (1). Therefore, we would also like to study the second critical exponent for (10), (12), and (13) in the future.