The porous medium equation $\dv{t}u=\dv{x}(k(u)\dv{x}u)$ is a non-linear degenerate parabolic partial differential equation. Consequently, existence and uniqueness of its solutions is not immediately evident.
This bachelor thesis presents a detailed discussion of Atkinson's a
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The porous medium equation $\dv{t}u=\dv{x}(k(u)\dv{x}u)$ is a non-linear degenerate parabolic partial differential equation. Consequently, existence and uniqueness of its solutions is not immediately evident.
This bachelor thesis presents a detailed discussion of Atkinson's and Peletier's 1971 article ``Similarity profiles of flows through porous media" on existence and uniqueness of self-similar solutions to the porous medium equation.
First, in chapter 2 the general version of the porous medium equation along with some applications will be discussed. Then, in chapter 3 the proofs and statements of the Picard-Lindelöf theorem, Peano's existence theorem and Gronwall's inequality will be presented. These standard theorems concern differential equations and will be used in the next chapter. Finally, in chapter 4 Atkinson's and Peletier's article will be worked out in detail.
