Abstract
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.
Highlights
This paper studies the questions of existence and uniqueness of non-negative solutions to the Cauchy problem for the following Porous Medium Equation in an inhomogeneous medium
U(x, 0) = u0 where we assume that n ≥ 3, m > 1, and ρ(x) is positive, bounded and smooth
As to the initial data, we assume that u0 ≥ 0 and ρ(x)u0(x) dx < ∞
Summary
We show that the solution is unique in a class of functions with finite energy and bounded for positive times, for all densities which are power-like at infinity with a moderate decay ρ(x) ≥ A0(1 + |x|)−n, (3) In establishing a priori estimates, and prior to the proof of uniqueness, we need to deal with the class of minimal solutions as defined in [EK] by approximation with weak solutions of Dirichlet problems in bounded domains, see Appendix III.
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