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Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. (English) Zbl 1169.35313

Summary: We study the long-time behavior of non-negative solutions to the Cauchy problem
\[ \rho(x)\partial_tu=\Delta u^m\quad\text{in }Q:=\mathbb R^n\times\mathbb R_+, \qquad u(x,0)=u_0 \tag{P} \]
in dimension \(n\geq 3\). We assume that \(m>1\) (slow diffusion) and \(\rho(x)\) is positive, bounded and behaves like \(\rho(x)\sim |x|^{-\gamma}\) as \(|x|\to\infty\), with \(0\leq\gamma<2\). The data \(u_0\) are assumed to be nonnegative and such that \(\int\rho(x)u_0\,dx<\infty\).
Our asymptotic analysis leads to the associated singular equation \(|x|^{-\gamma}u_t=\Delta u^m\), which admits a one-parameter family of selfsimilar solutions \(U_E(x,t)= t^{-\alpha}F_E(xt^{-\beta})\), \(E>0\), which are source-type in the sense that \(|x|^{-\gamma}u(x,0)= E\delta(x)\). We show that these solutions provide the first term in the asymptotic expansion of generic solutions to problem (P) for large times, both in the weighted \(L^1\) sense
\[ u(t)=U_E(t)+o(1) \quad\text{in }L_\rho^1 \]
and in the uniform sense \(u(t)= U_E(t)+ o(t^{-\alpha})\) in \(L^\infty\) as \(t\to\infty\) for the explicit rate \(\alpha= \alpha(m,n,\gamma)>0\) which is precisely the time-decay rate of \(U_E\). For a given solution, the proper choice of the parameter is \(E=\int\rho(x)u_0\,dx\).

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
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