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A central limit theorem for solutions of the porous medium equation. (English) Zbl 1082.35091

The large-time behaviour of the second moment (energy) \[ E(t)= {1\over 2}\int_{\mathbb{R}^n} |x|^2 v(x,t)\,dx \] of solutions to the problem: \[ {\partial v\over\partial t}=\Delta v^m,\quad x\in\mathbb{R}^N,\;t> 0\tag{1} \] \(v(x,0)= v_0(x)\geq 0\), \(x\in\mathbb{R}^N\) is studied. This Cauchy problem describes the flow of a gas in a \(N\)-dimensional porous medium, the function \(v\) represents the density of the gas and \(m> 1\) is a physical constant. It is well known that equation (1) admits a family of self-similar solutions (in a weak sense) called Barenblatt-Pattle solutions.
Assuming that \(\int_{\mathbb{R}^N} (v^m_0(x)+ |x|^{2+\delta} v_0(x))\,dx<\infty\) for some \(\delta> 0\), the energy \(E(t)\) is proved to behave asymptotically as \(t\to\infty\) like the energy \(E_B(t)\) of the Barenblatt-Pattle solution. More precisely \(E(t)|_{E_B(t)}\) is shown to converge to 1 at the (optimal) rate \(t^{-2/[N(m,-1)+2]}\). As a direct consequence of this result is a central limit theorem for the scaled solution \[ E(t)^{N/2} v(E(t)^{1/2} x,t). \]

MSC:

35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
60F05 Central limit and other weak theorems
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