On a semilinear parabolic problem with non-local (Bitsadze–Samarskii type) boundary conditions in more dimensions

Abstract

This paper studies a semilinear parabolic equation in a bounded domain Ω⊂Rd along with nonlocal boundary conditions. The boundary values are linked to the values of a solution on an interior (d−1)-dimensional manifold lying inside Ω. Firstly, the solvability of a steady-state problem is addressed. Secondly, involving the semi-discretization in time, a constructive algorithm for approximation of a solution to a transient problem is developed. The well-posedness of the problem in a weighted Hilbert space is shown. Convergence of approximations is addressed and the error estimated are derived. Numerical experiments support the theoretical algorithms.