An inverse problem for a parabolic integrodifferential model in the theory of combustion

In this paper we investigate an abstract inverse problem that can be applied to the evolution equation ut(t,x)=Δu(t,x)+∫0tk(t−s)Δu(s,x)ds+∫Ωut(t,x)dx+F(u(t,x),∇u(t,x)) given suitable initial-boundary conditions. Here FF is a given function and in the case F(u(t,x),∇u(t,x))=eu(t,x) the evolution equation has applications in the theory of combustion.Since we identify the convolution memory kernel kk and the temperature uu we associate an additional measurement on the temperature of type ∫Ωφ(x)u(t,x)dx=g(t), where φφ and gg are given functions.The novelty with respect to the existing literature is the presence of the term ∫Ωut(t,x)dx in the evolution equation that is motivated by a model in the theory of combustion. We prove a local in time existence theorem and a global in time uniqueness result.