An Asymptotic Approach to Second-kind Similarity Solutions of the Modified Porous-medium Equation

The problem of a spreading ground-water mound of liquid in a porous medium, situated on an impermeable horizontal solid layer is revisited. The mathematical formulation for this problem is given by the modified porous medium equation. A global condition in form of an energy integral is derived, describing the loss of liquid in the porous medium. This yields the necessary condition that enables the aymptotic derivation of the similarity exponents for the similarity solution of second kind. The method developed here, is further applied to the corresponding dipole problem, when instead of an energy integral another conservation law, the first moment integral, is considered.