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

We consider a Cauchy semilinear problem for a time-fractional diffusion systempartial derivative(alpha)u partial derivative t(alpha) + Au = F(u, v), partial derivative(alpha)v partial derivative t(alpha) + Bv = G(u, v),which involves symmetric uniformly elliptic operators A, B on a bounded domain Omega in R-d with sufficiently smooth boundary. The problem is equipped with final value conditions (FVCs), i. e., (u; v)vertical bar(t=T) are given. We derive a spectral representation of solutions with FVCs where the solution operators are not bounded on L-2(Omega). Our work focuses on establishing existence and uniqueness of a solution in a suitable space. (C) 2019 Elsevier B.V. All rights reserved.