An identification problem with evolution on the boundary of hyperbolic type (Q981950)

This paper addresses an identification (ill-posed) problem appearing in the linear second-order elliptic equation \( A(u+\int_0^t k(t-s)u(s,x)ds)=f \) in \((0,T) \times \Omega,\) \(\Omega\) is bounded domain, \(A\) is a linear second-order elliptic operator, and the kernel \(k\) is a scalar unknown function of time. It is shown that under a supplementary condition the kernel can be identified and a global existence and uniqueness theorem is formulated. This work was inspired by the direct problem of the solution of a nonlinear evolution equation with boundary operators containing time derivatives studied by \textit{T. Hintermann} [Proc. R. Soc. Edinb., Sect. A 113, No.~1--2, 43--60 (1989; Zbl 0699.35045)].