Stability, instability, and blowup for time fractional and other non-local in time semilinear subdiffusion equations

TL;DR

This paper investigates the behavior of solutions to non-local in time semilinear subdiffusion equations, analyzing conditions for stability, instability, and blowup, with implications for fractional dynamics in physics.

Contribution

It provides well-posedness results, stability analysis, and blowup conditions for a broad class of non-local in time equations, including fractional derivatives.

Findings

Well-posedness of bounded weak solutions

Stability and instability criteria for zero solution

Blowup conditions for superlinear nonlinearities

Abstract

We consider non-local in time semilinear subdiffusion equations on a bounded domain, where the kernel in the integro-differential operator belongs to a large class, which covers many relevant cases from physics applications, in particular the important case of fractional dynamics. The elliptic operator in the equation is given in divergence form with bounded measurable coefficients. We prove a well-posedness result in the setting of bounded weak solutions and study the stability and instability of the zero function in the special case where the nonlinearity vanishes at 0. We also establish a blowup result for positive convex and superlinear nonlinearities.