Asymptotic behaviour for the porous medium equation posed in the whole space
Juan Luis Vázquez · Birkhäuser Basel eBooks · 2003
This paper is devoted to present a detailed account of the asymptotic behaviour as t → ∞ of the solutions u(x, t) of the equation (0.1) $$ {u_{{t = }}}\Delta ({u^{m}}) $$ with exponent m > 1, a range in which it is known as the porous medium equation written here PME for short. The study extends the well-known theory of the classical heat equation (HE, the case m = 1) into a nonlinear situation, which needs a whole set of new tools. The space dimension can be any integer n ≥ 1. We will also present the extension of the results to exponents m 0} which lives in L 1 (ℝ n ) ∩ L ∞(ℝ n ) and describes the evolution of the process. The solution is not classical for m > 1, but it is proved that there exists a unique weak solution for all m > 0.