November  2008, 7(6): 1275-1294. doi: 10.3934/cpaa.2008.7.1275

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

1. 

Departamento de Matemática Aplicada, E.T.S.I. de Caminos, Canales y Puertos, Universidad Politécnica de Madrid. 28040 Madrid, Spain

2. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid

Received  January 2008 Revised  June 2008 Published  August 2008

We study the questions of existence and uniqueness of non-negative solutions to the Cauchy problem

$\rho(x)\partial_t u= \Delta u^m\qquad$ in $Q$:$=\mathbb R^n\times\mathbb R_+$

$u(x, 0)=u_0$

in dimensions $n\ge 3$. We deal with a class of solutions having finite energy

$E(t)=\int_{\mathbb R^n} \rho(x)u(x,t) dx$

for all $t\ge 0$. 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 $u_0\ge 0$ with finite energy. We show that uniqueness takes place if $\rho$ has a moderate decay as $|x|\to\infty$ that essentially amounts to the condition $\rho\notin L^1(\mathbb R^n)$. 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.

Citation: Guillermo Reyes, Juan-Luis Vázquez. The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1275-1294. doi: 10.3934/cpaa.2008.7.1275
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