American Institute of Mathematical Sciences

June  2006, 1(2): 337-351. doi: 10.3934/nhm.2006.1.337

The Cauchy problem for the inhomogeneous porous medium equation

Received  February 2006 Revised  April 2006 Published  April 2006

We consider the initial value problem for the filtration equation in an inhomogeneous medium
$p(x)u_t = \Delta u^m, m>1$.

The equation is posed in the whole space $\mathbb R^n$ , $n \geq 2$, for $0 < t < \infty$; $p(x)$ is a positive and bounded function with a certain behaviour at infinity. We take initial data $u(x,0) = u_0(x) \geq 0$, and prove that this problem is well-posed in the class of solutions with finite "energy", that is, in the weighted space $L^1_p$, thus completing previous work of several authors on the issue. Indeed, it generates a contraction semigroup.
We also study the asymptotic behaviour of solutions in two space dimensions when $p$ decays like a non-integrable power as $|x| \rightarrow \infty$ : $p(x)$ $|x|^\alpha$ ~ $1$ with $\alpha \epsilon (0,2)$ (infinite mass medium). We show that the intermediate asymptotics is given by the unique selfsimilar solution $U_2(x, t; E)$ of the singular problem
$|x|^{- \alpha} u_t = \Delta u_m$ in $\mathbb R^2 \times \mathbb R_+$
$|x|^{- \alpha} u(x,0) = E\delta(x), E = ||u_0||_{L^1_p}$
Citation: Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks & Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337
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