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Zbl 0933.35104
Metzger, Michael
Existence for a time-dependent heat equation with non-local radiation terms. (English)
[J] Math. Methods Appl. Sci. 22, No.13, 1101-1119 (1999). ISSN 0170-4214; ISSN 1099-1476/e

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.
[N.A.Watson (Christchurch)]

MSC 2000:: *35K60 (Nonlinear) BVP for (non)linear parabolic equations
80A20 Heat and mass transfer

Keywords: nonlinear and nonlocal boundary condition; nonstandard Sobolev spaces; Galerkin based approximation scheme

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