## 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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### References:

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