Existence of extremal periodic solutions for discontinuous quasilinear parabolic equations

In this paper we consider a quasilinear parabolic equation in a bounded domain under periodic Dirichlet boundary conditions with a discontinuous nonlinear right-hand side. Our main goal is to prove the existence of extremal solutions within a sector formed by appropriately defined upper and lower solutions. The main tool used in the proof is a recently obtained extremality result for 'continuous' quasilinear parabolic equations which is due to the first author combined with an abstract fixed point result in partially ordered sets which is due to the second author. These tools are based on truncation and comparison techniques using suitably constructed special testfunctions as well as on abstract existence results for nonlinear evolution equations. AMS (MOS) Subject Classification: 35B05, 35B10, 35K60, 47N20.