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 u t ( t , x ) = Δ u ( t , x ) + ∫ 0 t k ( t − s ) Δ u ( s , x ) d s + ∫ Ω u t ( t , x ) d x + F ( u ( t , x ) , ∇ u ( t , x ) ) given suitable initial-boundary conditions. Here F is a given function and in the case F ( u ( t , x ) , ∇ u ( t , x ) ) = e u ( t , x ) the evolution equation has applications in the theory of combustion. we identify the convolution memory kernel k and the temperature u we associate an additional measurement on the temperature of type ∫ Ω φ ( x ) u ( t , x ) d x = g ( t ) , where φ and g are given functions. velty with respect to the existing literature is the presence of the term ∫ Ω u t ( t , x ) d x 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.