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The Porous Medium EquationMathematical Theory$
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Juan Luis Vazquez

Print publication date: 2006

Print ISBN-13: 9780198569039

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198569039.001.0001

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The Porous Medium Equation

Juan Luis Vázquez

Oxford University Press

This chapter examines the class of solutions of the PME that are invariant under the scaling group in the variables (x, t, u), and take therefore the so-called self-similar form. After a detailed analysis of the application of the scaling group to the PME, it is shown that the self-similar solutions can be classified into tree different types: forward, backward, and exponential self-similarity. Section 16.4 introduces the technique of phase-plane analysis that allows for a rather complete description of these solutions for all parameters (under the restriction of radial symmetry in several dimensions) to be obtained. An alternative phase plane is introduced in Section 16.5 which clarifies the behaviour at infinity of the previous plane. The tools are completed in Section 16.6 with the study of sign-change trajectories through inversion. Oscillating signed solutions are studied in Section 16.7 and two special solutions are constructed that are important in the existence and uniqueness theory of signed solutions. The special features of self-similar solutions of Type II are examined in Section 16.8. Finally two short sections contain supplementary material.

Keywords:   PME, self-similarity, existence theory, phase-plane analysis, sign-change trajectories

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