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摘要
We study the long-time behavior of non-negative, finite-energy solutions to the initial value problem for the Porous Medium Equation with variable density, i.e. solutions of the problem %26lt;br%26gt;{rho(x)partial derivative(t)u = Delta u(m), in Q := R-n x R+, %26lt;br%26gt;u(x, 0) = u(0)(x), in R-n, %26lt;br%26gt;where m %26gt; 1, u(0) is an element of L-1 (R-n, rho(x)dx) and n %26gt;= 3. We assume that rho(x) similar to C|x|(-2) as |x| -%26gt; infinity in R-n. Such a decay rate turns out to be critical. We show that the limit behavior can be described in terms of a family of source-type solutions of the associated singular equation |x|(-2)u(t) = Delta u(m). The latter have a self-similar structure and exhibit a logarithmic singularity at the origin.
- 出版日期2013-3
