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

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.