On a class of similarity solutions of the porous media equation II (original) (raw)

The authors consider the nonlinear diffusion equation u_t = (u^m)_xx where u is the density of a polytropic gas in one-dimensional motion through a homogeneous porous medium, t is the time, and m a constant greater than 1. Self-similar solutions of three types are looked for. Substitution of the self-similar forms into the partial differential equation leads to a class of ordinary differential equations containing two parameters. Weak solutions with compact support of this class of equation were studied in an earlier paper. In the present paper the authors investigate weak solutions that do not have compact support. First it is shown that any such solution is a bounded, positive, classical solution. Then some nonexistence results are obtained, and existence and uniqueness theorems for different values of the parameters are proven.