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.


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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