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

✍ Scribed by Michael Metzger

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 192 KB
• Volume: 22
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/(sici)1099-1476(19990910)22:13<1101::aid-mma74>3.0.co;2-n

✦ Synopsis

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 < :

"+v3H( )" v3¸(R ),. 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 < and the non-local character of the boundary conditions, convergence of the Galerkin approximations is di$cult 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.