A nonlinear parabolic integro-differential problem with an unknown Dirichlet boundary condition

A nonlinear parabolic integro-differential equation ∂tg(u)−Δu=F+∫0tf(s,u(s))ds with a known Neumann boundary condition on a part of the boundary and an unknown Dirichlet boundary condition α(t)α(t) on the other part of the boundary is studied. The inverse problem of identifying the unknown time-dependent function α(t)α(t) from an additional integral measurement E(t)=∫Ωg(u(t,x))dx is investigated. The well-posedness of the problem in suitable function spaces is shown and a numerical time-discrete scheme for approximations is designed. Convergence of the proposed scheme is supported by a numerical experiment.