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Existence for a time-dependent heat equation with non-local radiation terms. (English) Zbl 0933.35104

The paper deals with the time-dependent linear heat equation with a nonlinear and nonlocal boundary condition that arises when considering the radiation balance. Solutions are considered to be functions with values in \(V:=\{v \in H^1(\Omega) \mid\gamma v\in L_5 (\partial \Omega)\}\). As a consequence one has to work with nonstandard 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 nonlocal 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.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
80A20 Heat and mass transfer, heat flow (MSC2010)
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