A necessary and sufficient conditions for the global existence of solutions to fractional reaction-diffusion equations on \(\mathbb {R}^{N}\)

A necessary and sufficient condition for the existence or nonexistence of global solutions to the following fractional reaction-diffusion equations \(\begin{aligned} {\left\{ \begin{array}{ll} u_{t}=\Delta _{\alpha } u + \psi (t)f(u),\,\,&{} \text{ in } \mathbb {R}^{N}\times (0,\infty ),\\ u(\cdot ,0)=u_{0}\ge 0,\,\,&{} \text{ in } \mathbb {R}^{N}, \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t={\mathrm{\Delta }}_{\alpha }u+\psi (t)f(u),\hfill & \text{in}{\mathbb{R}}^N\times (0,\infty ),\hfill \\ u(·,0)=u_0\ge 0,\hfill & \text{in}{\mathbb{R}}^N,\hfill \end{array}\right)\end{array}$$ has not been known and remained as an open problem for a few decades, where \(N\ge 2\) $N\ge 2$ , \(\Delta _{\alpha }=-\left( -\Delta \right) ^{\alpha /2}\) ${\mathrm{\Delta }}_{\alpha }=-\left(-,\mathrm{\Delta }\right)_{}^{\alpha /2}$ denotes the fractional Laplace operator with \(0<\alpha \le 2\) $0<\alpha \le 2$ , \(\psi \) $\psi$ is a nonnegative and continuous function, and f is a convex function. The purpose of this paper is to resolve this problem completely as follows: \(\begin{aligned} \begin{aligned}&\text{ There } \text{ is } \text{ a } \text{ global } \text{ solution } \text{ to } \text{ the } \text{ equation } \text{ if } \text{ and } \text{ only } \text{ if }\\&\hspace{20mm}\int _{1}^{\infty }\psi (t)t^{\frac{N}{\alpha }}f\left( \epsilon \, t^{-\frac{N}{\alpha }}\right) dt<\infty ,\\&\text{ for } \text{ some } \epsilon >0. \end{aligned} \end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{cc}& \text{There}\text{is}\text{a}\text{global}\text{solution}\text{to}\text{the}\text{equation}\text{if}\text{and}\text{only}\text{if}\hfill \\ \hfill & {\int }_1^{\infty }\psi (t)t^{\frac{N}{\alpha }}f\left(ϵ,,t_{}^{-\frac{N}{\alpha }}\right)dt<\infty ,\hfill \\ \hfill & \text{for}\text{some}ϵ>0.\hfill \end{array}\end{array}$$