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The Cauchy problem for the inhomogeneous porous medium equation

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  • 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}$
    Mathematics Subject Classification: Primary: 35B40, 35D05, 35K55, 35K60, 35K65, 47H20.


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