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Large time behaviour of solutions of the porous media equation with absorption. (English) Zbl 0625.35048

The paper is concerned with the qualitative properties of the solution u of the problem \[ u\in L^{\infty}({\mathbb{R}}^ n\times]0,T[)\quad \forall T>0; \]
\[ \frac{\partial u}{\partial t}=\Delta (u^ m)-u^ p\quad in\quad {\mathbb{R}}^ n\times]0,+\infty [\quad u(x,t)\geq 0\quad in\quad {\mathbb{R}}^ n\times]0,+\infty [;\quad u(x,0)=\phi (x)\quad in\quad {\mathbb{R}}^ n \] where \(m\geq 1\), \(p>1\), \(n\geq 1\), \(\phi \in L^{\infty}({\mathbb{R}}^ n)\) and \(\phi\geq 0\). Under the assumption that \(\lim_{| x| \to \infty} | x|^{\alpha} \phi (x)=A\) in a suitable sense for some \(\alpha\), \(A>0\), the authors describe the asymptotic behaviour of u(x,t) as \(t\to +\infty\) in dependence of the parameters m, p and \(\alpha\).
More precisely, the following cases are treated:
i) \(p>m\), \(0<\alpha <2/(p-m);\)
ii) \(p>m+2/n\), \(m>1\), \(2/(p-m)<\alpha <n;\)
iii) \(p>m+2/n\), \(m>1\), \(\alpha >n.\)
Finally, in the last section the authors show some generalizations to the case in which \(u^ p\) is substituted by a function g(u) with \(g(0)=0\) and \(g(s)>0\) for \(s>0\).
Reviewer: M.Degiovanni

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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