## Asymptotic behaviour of solutions of some semilinear parabolic problems.(English)Zbl 0918.35025

Summary: We consider the Cauchy problem: $u_t-\Delta u+ u^p= 0\quad\text{for }x\in \mathbb{R}^N,\quad t>0,\tag{1}$
$u(x,0)= u_0(x)\quad\text{for }x\in \mathbb{R}^N.\tag{2}$ Here $$p>1$$, $$N\geq 1$$ and $$u_0(x)$$ is a continuous, nonnegative and bounded function such that: $u_0(x)\sim A| x|^{-\alpha},\quad\text{as }| x|\to \infty,\tag{3}$ for some $$A>0$$ and $$\alpha>0$$. In this paper, we discuss the asymptotic behaviour of solutions to (1)–(3) in terms of the various values of the parameters $$p$$, $$N$$, $$\alpha$$ and $$A$$. A common pattern that emerges from our analysis is the existence of an external zone where $$u(x,t)\sim u_0(x)$$ and one (or several) internal regions, where the influence of diffusion and absorption is most strongly felt. We present a complete classification of the size of these regions, as well as that of the stabilization profiles that unfold therein, in terms of the aforementioned parameters.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35K55 Nonlinear parabolic equations 35K15 Initial value problems for second-order parabolic equations 35K57 Reaction-diffusion equations
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### References:

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