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

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}=\Delta _{\alpha } u + \psi (t)f(u),\,\,&{} \text{ in } \mathbb {R}^{N}\times (0,\infty ),\\ u(\cdot ,0)=u_{0}\ge 0,\,\,&{} \text{ in } \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$$

has not been known and remained as an open problem for a few decades, where \(N\ge 2\), \(\Delta _{\alpha }=-\left( -\Delta \right) ^{\alpha /2}\) denotes the fractional Laplace operator with \(0<\alpha \le 2\), \(\psi \) 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:

$$\begin{aligned} \begin{aligned}&\text{ There } \text{ is } \text{ a } \text{ global } \text{ solution } \text{ to } \text{ the } \text{ equation } \text{ if } \text{ and } \text{ only } \text{ if }\\&\hspace{20mm}\int _{1}^{\infty }\psi (t)t^{\frac{N}{\alpha }}f\left( \epsilon \, t^{-\frac{N}{\alpha }}\right) dt<\infty ,\\&\text{ for } \text{ some } \epsilon >0. \end{aligned} \end{aligned}$$