On the critical exponents for a fractional diffusion-wave equation with a nonlinear memory term in a bounded domain

TL;DR

This paper investigates the blow-up and global existence of solutions for a fractional diffusion-wave equation with nonlinear memory in bounded domains, providing sharp conditions and nonexistence results using eigenfunction methods.

Contribution

It offers new sharp criteria for blow-up and global existence in fractional diffusion-wave equations with nonlinear memory terms, extending previous results to Caputo derivatives.

Findings

Sharp blow-up conditions established

Global existence criteria derived

Nonexistence results for certain wave equations

Abstract

In this paper, we prove sharp blow-up and global existence results for a time fractional diffusion-wave equation with a nonlinear memory term in a bounded domain, where the fractional derivative in time is taken in the sense of Caputo type. Moreover, we also give a result for nonexistence of global solutions to a wave equation with a nonlinear memory term in a bounded domain. The proof of blow-up results is based on the eigenfunction method and the asymptotic properties of solutions for an ordinary fractional differential inequality.