Existence and convergence theorems for doubly nonlinear partial differential equations of elliptic-parabolic type

Abstract

Here T> 0, V,U = (&/ax,, au/ax,, . . . . au/ax,), and B, is a map from RN x (0, T) x R x RN into R for each i = 1, . . . . N, satisfying certain conditions to be specified at a later time. A precise formulation of (1.1) is given in Section 3. For given u0 and f we study the existence of weak solutions of (1.1). Although doubly nonlinear equations of the type (1.1) are of mathematical interest in their own right, they arise as a model for a variety of diffusion problems. In particular, they comprise in a unifying scheme, free-boundary problems of different natures. An example situation is nonstationary saturated-unsaturated flow of an incompressible fluid through a porous medium in the case of time-dependent water levels. A weak formulation of this problem leads to equations of the type (1.1); see [2]. However, a large held of application for our results is the diffusion problem involving a solid-liquid phase change of the Stefan type. An architypical example is heat transfer during solidification in a nonhomogeneous 205