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Both authors funded by Project MTM2005-08760-C01 and -C02 (Spain) and European Science Foundation Programme "Global And Geometric Aspects of Nonlinear Partial Differential Equations".

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The inhomogeneous PME in several space dimensions.: Existence and uniqueness of finite energy solutions

Publicado en:COMMUNICATIONS ON PURE AND APPLIED ANALYSIS. 7 (6): 1275-1294 - 2008-11-01 7(6), DOI: 10.3934/cpaa.2008.7.1275

Autores: Reyes, Guillermo; Vazquez, Juan Luis

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We study the questions of existence and uniqueness of non-negative solutions to the Cauchy problem {rho(x)partial derivative(u)(t) = Delta u(m) in Q := R-n x R+ u(x, 0) = u(0) in dimensions n >= 3. We deal with a class of solutions having finite energy E(t) = integral(Rn) rho(x)u(x, t) dx for all t >= 0. We assume that m > 1 (slow diffusion) and the density rho(x) is positive, bounded and smooth. We prove existence of weak solutions starting from data u(0) >= 0 with finite energy. We show that uniqueness takes place if rho has a moderate decay as vertical bar x vertical bar -> infinity that essentially amounts to the condition rho is not an element of L-1(R-n). We also identify conditions on the density that guarantee finite speed of propagation and energy conservation, E(t) = const. Our results are based on a new a priori estimate of the solutions.

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