ON A DOUBLY NONLINEAR DIFFUSION EQUATION IN HYDROLOGY.

The authors consider the nonlinear partial differential equation u//t equals (D(u) phi (u//x))//x, where the functions D and phi satisfy the hypotheses H// phi : phi an element of C**1 ( left bracket minus 1, 1 right bracket ) intersection C**2(( minus 1, 1)), phi (0) equals 0, phi prime ( minus 1) equals phi prime (1) equals 0, phi prime greater than 0 on ( minus 1, 1). and H//D: D an element of C**1 ( left bracket 0, 1 right bracket ) intersection C**2 ((0, 1)), D greater than 0 on (0, 1), D(0) equals D(1) equals 0, D double prime less than equivalent to 0 on (0, 1). An equation of this type (1. 1) arises in the theory of hydrology, with D(s) equals s(1 minus s) and phi (s) equals s/(1 plus s**2). The authors investigate three problems related to the equation: the Neumann problem on ( minus 1, 1) with the natural boundary conditions D(u) phi (u//x) ( plus or minus 1, t) equals 0 for t greater than 0, the Cauchy problem and a related Cauchy-Dirichlet problem on (0, infinity ) with the boundary condition u(0, t) equals A for t greater than 0, A equals 0 or 1.