A Class of Quasilinear Parabolic P.D.E.s

Abstract

This chapter deals with (possibly degenerate) quasilinear⁽¹⁾ parabolic inclusions of the form

$$(\frac{{\partial u}}{{\partial t}})\alpha (u) - \nabla u \mathrel\backepsilon f(\Delta : = \sum\nolimits_{i = 1}^N {\frac{{\partial ^2 }}{{\partial x_i^2 }}} ),$$

where α is a maximal monotone graph. Existence of a solution is proved via approximation by time-discretization, derivation of a priori estimates, and passage to the limit by compactness and monotonicity procedures. Uniqueness and regularity results are derived via L²- and L¹ -techniques.