An Introduction to the Mathematical Theory of the Porous Medium Equation

These notes contain an introduction to the mathematical treatment of the porous medium equation ut = △ (um) , one of the simplest examples of nonlinear evolution equation of parabolic type , which appears in the description of different natural phenomen a rela ted to diffusion, filtration or heat propagation. The notes begin with a discussion of the relevance of the equation and some of its applications. The main body of the notes is devoted to the study of the existence , uniqueness and regularity of a (generalized) solution for the two main problems, i.e. the initial-value problem and the Dirichlet boundary-value problem. Special attention is paid to the appearance of a free boundary, a consequence of the finite propagation property.

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