An identification problem related to a parabolic integrodifferential equation with non commuting spatial operators (Q2709825)

This paper intends to deal with identification problems in nonsmooth space domains as well as with convolution kernels depending on time and one space variable. NEWLINENEWLINENEWLINEThe authors consider the integrodifferential equation NEWLINE\[NEWLINE D_t u(t,x,y) = \text{div} \mathcal{E} u(t,x,y)+ \int_0^t \text{div} \{h(t-s,x)\mathcal{F} u(s,x,y) \}ds +f(t,x,y),NEWLINE\]NEWLINE NEWLINE\[NEWLINE (t,x,y) \in [0,T] \times \Omega,NEWLINE\]NEWLINE \(\mathcal{E}\) and \(\mathcal{F}\) being linear first-order operators. They show that \(\text{div}\mathcal{E} u(t,x,y) \) generates a semigroup of linear bounded operators in the space. NEWLINENEWLINENEWLINEThe paper also deals with a more general identification problem, which is described by the following integrodifferential equation NEWLINE\[NEWLINE D_t u(t,x,y) = \mathcal{A} u(t,x,y)+ \int_0^t h(t-s,x)\mathcal{B} u(s,x,y) ds NEWLINE\]NEWLINE NEWLINE\[NEWLINE+ \int_0^t D_th(t-x,x)\mathcal{C}u(s,x,y) ds+f(t,x,y),NEWLINE\]NEWLINE NEWLINE\[NEWLINE (t,x,y) \in [0,T] \times \Omega,NEWLINE\]NEWLINE where \( \mathcal{A,B,C}\) are operators. In particular, the operator \( \mathcal{A}\) turns out to be the sum of two differential operators \( \mathcal{A}_1\) and \( \mathcal{A}_2\), which do not commute. NEWLINENEWLINENEWLINEThe authors prove an existence and uniqueness theorem for these identification problems.