Title: The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions
Abstract: We study the questions of existence and uniqueness ofnon-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 solutionshaving 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) andthe density $\rho(x)$ is positive, bounded and smooth. Weprove 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 tothe condition $\rho\notin L^1(\mathbb R^n)$. We also identifyconditions on the density that guarantee finite speed ofpropagation and energy conservation, $E(t)=$const. Ourresults are based on a new a priori estimate of thesolutions.