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- W2325213690 abstract "We study the long-time behavior of non-negative, finite-energysolutions to the initial value problem for the Porous MediumEquation with variable density, i.e. solutions of the problembegin{eqnarray*}rho (x) partial_{t} u = Delta u^{m}, quad in quad Q:= R^n times R_+, u(x,0)=u_{0}(x), quad inquad R^n,end{eqnarray*}where $m>1$, $u_0in L^1(R^n, rho(x)dx)$ and $nge 3$.We assume that $rho (x)sim C|x|^{-2}$ as $|x|toinfty$ in$R^n$. Such a decay rate turns out to be critical. Weshow that the limit behavior can be described in terms of a familyof source-type solutions of the associated singular equation$|x|^{-2}u_t = Delta u^{m}$. The latter have a self-similarstructure and exhibit a logarithmic singularity at the origin." @default.
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- W2325213690 date "2012-09-01" @default.
- W2325213690 modified "2023-10-16" @default.
- W2325213690 title "Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density" @default.
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- W2325213690 doi "https://doi.org/10.3934/cpaa.2013.12.1123" @default.
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