Existence for a time‐dependent heat equation with non‐local radiation terms

Abstract

The paper deals with the time-dependent linear heat equation with a non-linear and non-local boundary condition that arises when considering the radiation balance. Solutions are considered to be functions with values in V := {v ∈ H1(Ω)∣γv ∈ L5(∂Ω)}. As a consequence one has to work with non-standard Sobolev spaces. The existence of solutions was proved by using a Galerkin-based approximation scheme. Because of the non-Hilbert character of the space V and the non-local character of the boundary conditions, convergence of the Galerkin approximations is difficult to prove. The advantage of this approach is that we don't have to make assumptions about sub- and supersolutions. Finally, continuity of the solutions with respect to time is analysed. Copyright © 1999 John Wiley & Sons, Ltd.