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 ∂ α u ∂ t α + A u = F ( u , v ) , ∂ α v ∂ t α + B v = G ( u , v ) , which involves symmetric uniformly elliptic operators A , B on a bounded domain Ω in R d 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.

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