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

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.