Abstract
We study the porous medium equation ut=(um). 0＜x＜∞, t＞0 with a singular boundary condition (um) (0,t)=u-β(0,t). We prove finite time quenching for the solution at the boundary χ=0. We also establish the quenching rate and asymptotic behavior on the quenching point.
Highlights
The nonlinear diffusion equation ut = um with exponent m > 1, is usually called the porous medium equation, written here PME for short
The PME equation is one of the simplest examples of a nonlinear evolution equation of parabolic type. It appears in the description of different natural phenomena, and its theory and properties depart strongly from the heat equation ut = u, its most famous relative
There are a number of physical applications where this simple model appears in a natural way, mainly to describe processes involving fluid flow, heat transfer or diffusion
Summary
The study of quenching (in general the solution is defined up to t = T but some term in the problem ceases to make sense) began with the work of Kawarada [4] appeared in 1975. In that paper he studied the semilinear heat equation as a singular reaction at level u = 1. He proved that the reaction term, and the time derivative blows up wherever u reaches this value, see [5].
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