Degenerate Diffusion in Heterogeneous Media

When
Start: 10/02/2013 - 4:15pm
End  : 10/02/2013 - 5:15pm

Category
Colloquium

Speaker
Guillermo Reyes, University of California, Irvine

Abstract

In this talk, I will present some recent results on the long-time behavior of non-negative solutions to the Cauchy problem for the Porous Medium Equation in the presence of variable density vanishing at infinity. More precisely, we consider the initial value problem

(P)   $ \rho(x)\, \partial_t u= \Delta u^m $ in $ Q:=\mathbb{R}^n\times\mathbb{R}_+ $$ u(x,\, 0)=u_0(x) $ in $ \mathbb{R}^n $,

where we assume $ m $>$ 1 $, $ n $$ 3 $  and $ \, \rho(x)\,  $ is positive, smooth and has a power-like decay at infinity, $ \rho(x)\sim|x|^{-\gamma} $  as $  |x|\to\infty $ for some $ \gamma $>$ 0 $. The data $ u_0 $ are assumed to be non-negative and such that $ \int_{\mathbb{R}^n} \rho(x)u_0(x)\, dx $< $ \infty $ (finite-energy solutions).

For $ m $>$ 1 $, the behavior of solutions depends on whether $ 0 $<$ \gamma $<$ 2 $ or $ \gamma $>$ 2 $. In the first case, the asymptotic behavior is described in terms of a one-parameter family of source-type, self-similar solutions of a related singular problem (so called Barenblatt-type solutions).  For $ \gamma $>$ 2 $, however,  solutions to (P)  have a universal long-time behavior in separate variables, typical of initial-boundary problems on bounded domains.
If $ \rho(x) $ has an intermediate decay, $ \rho(x)\sim |x|^{-\gamma} $ with $ 2 $<$ \gamma $<$ \gamma_2:=N-(N-2)/m $, solutions still enjoy the finite propagation property (as in the case of lower $ \gamma $). In this range a more precise description may be given at the diffusive scale in terms of the Barenblatt-type solutions $ U_E(x,t) $ of the related singular equation $ |x|^{-\gamma}u_t= \Delta u^m $.

Where
Davidson Lecture Hall, Claremont McKenna College

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