Localization of Solutions of a Nonlinear Fokker-Planck Equation with Dirichlet Boundary Conditions

摘要

The Cauchy-Dirichlet problem for a nonlinear degenerate second-order parabolic equation in one space dimension is considered. The equation possesses the property that if the initial data function for the Cauchy-Dirichlet problem is nonnegative and has compact support, then at all later times the (generalized) solution of the problem also is nonnegative and has compact support as a function of the spatial variable. Such equations arise in the theory of soil moisture infiltration. A solution of the Cauchy-Dirichlet problem is said to be localized if its support with respect to the spatial variable is uniformly bounded in time. The influence of the coefficients in the equation and of the lateral boundary data on the localization of solutions is investigated. With regard to the physical background to the problem, the results relate to the role of soil characteristics, gravity, and boundary conditions in determining the finite penetration of a wetting front during soil-moisture infiltration.