Geometrical properties of solutions of the Porous Medium Equation for large times

Ki-ahm Lee ^[1] ; J.L. Vazquez ^[1]
1. [1] Department of Mathematics, Univ. of Texas at Austin
Localización: XVII Congreso de Ecuaciones Diferenciales y Aplicaciones ; VII Congreso de Matemática Aplicada: Salamanca, 14-28 septiembre 2001 / coord. por Luis Ferragut Canals , Anastasio Pedro Santos Yanguas , 2001, ISBN 8469961446, págs. 221-244
Idioma: inglés
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Resumen
- We establish the behavior of the solutions of the degenerate parabolic equation ut = ∆(u m), m > 1, posed in the whole space when the initial data are nonnegative, continuous and compactly supported. We prove that after a finite time the pressure v = u m−1 becomes a concave function in the space variable which converges to all orders of differentiability to a truncated parabolic shape, so-called Barenblatt profile. In particular, the support of the solution is a convex subset of R N which converges to a ball. Estimates are optimal. The results are extended to the heat equation (log-concavity) and fast diffusion (pressure-convexity).