Existence and uniqueness of mild solutions for a final value problem for nonlinear fractional diffusion systems

Abstract

We consider a Cauchy semilinear problem for a time-fractional diffusion system∂αu∂tα+Au=F(u,v),∂αv∂tα+Bv=G(u,v),which involves symmetric uniformly elliptic operators A,B on a bounded domain Ω in Rd with sufficiently smooth boundary. The problem is equipped with final value conditions (FVCs), i.e., (u;v)|t=T are given. We derive a spectral representation of solutions with FVCs where the solution operators are not bounded on L2(Ω). Our work focuses on establishing existence and uniqueness of a solution in a suitable space.