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The Cauchy problem for the inhomogeneous porous medium equation
| 1. | Depto. de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avda. de la Universidad 30, Leganés 28911, Madrid |
| 2. | Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid |
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(x,0) = E\delta(x), E = ||u_0||_{L^1_p}$
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