The Cauchy problem for the inhomogeneous porous medium equation
نویسندگان
چکیده
We consider the initial value problem for the filtration equation in an inhomogeneous medium ρ(x)ut = ∆u, m > 1. The equation is posed in the whole space R, n ≥ 2, for 0 < t < ∞ ; ρ(x) is a positive and bounded function with a certain behaviour at infinity. We take initial data u(x, 0) = u0(x) ≥ 0, and prove that this problem is well-posed in the class of solutions with finite “energy”, that is, in the weighted space Lρ, 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 ρ decays like a non-integrable power as |x| → ∞ : ρ(x) |x|α ∼ 1, with α ∈ (0, 2) (infinite mass medium). We show that the intermediate asymptotics is given by the unique selfsimilar solution U2(x, t; E) of the singular problem { |x|−αut = ∆u in R × R+ |x|−αu(x, 0) = Eδ(x), E = ‖u0‖L1ρ 2000 AMS Subject Classification. 35B05, 35B40, 35D05, 35K55, 35K60, 35K65, 47H20.
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عنوان ژورنال:
- NHM
دوره 1 شماره
صفحات -
تاریخ انتشار 2006