Large Time Behaviour of Solutions of the Porous Medium Equation with Convection

摘要

The paper presents some results about the large time behavior of solutions of the equation (u sub t) + ((u sup lambda) sub x) = ((u sup m) sub xx), u = or > 0, (1.1) where m > 1 and lambda > 0 are constants and where the subscripts t and x denote differentiation with respect to the variable t (time) and x (space coordinate), respectively. Equation (1.1) arises in a model describing the unsaturated flow of a fluid through a homogeneous porous column (x denotes the vertical distance) under influence of capillary pressure and gravity. The unknown u denotes the moisture content in the porous material. When u > 0, both liquid and gas (air) are present; when u = 0, no liquid is present and the material is dry. The case lambda > 1 describes downward infiltration and the case 0 < lambda < 1 describes evaporation of liquid from the column e.g. see Diaz & Kersner (2), further references are given there. (Copyright (c) 1987 by Faculty of Mathematics and Informatics, Delft, The Netherlands.)