Existence of global solutions and blow-up of solutions for coupled systems of fractional diffusion equations

TL;DR

This paper investigates the conditions under which solutions to coupled fractional diffusion equations either exist globally or blow up in finite time, depending on the dimension and nonlinearities, and discusses their long-term behavior.

Contribution

It establishes a critical dimension for the existence of global solutions and analyzes the blow-up phenomena for coupled fractional diffusion systems.

Findings

Existence of global solutions for small data below a critical dimension.

Finite-time blow-up occurs above the critical dimension.

Long-term behavior of global solutions is characterized.

Abstract

We study the Cauchy problem for a system of semi-linear coupled fractional-diffusion equations with polynomial nonlinearities posed in $% \mathbb{R}_{+}\times \mathbb{R}^{N}$. Under appropriate conditions on the exponents and the orders of the fractional time derivatives, we present a critical value of the dimension N, for which global solutions with small data exist, otherwise solutions blow-up in finite time. Furthermore, the large time behavior of global solutions is discussed.