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

Abstract

We consider nonlocal 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.