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Large time behaviour of solutions of the heat equation with absorption. (English) Zbl 0598.35050
The authors consider the asymptotic behaviour of solutions of the Cauchy problem for semilinear heat equations \[ u_ t=\Delta u-u^ p\quad in\quad {\mathbb{R}}^ n\times (0,\infty);\quad u(x,0)=\phi (x)\quad in\quad {\mathbb{R}}^ n. \] Here \(p>1+2/n\), \(\phi \in L^{\infty}({\mathbb{R}}^ n)\) with \(\phi\) nonnegative and prescribed decay rate \(\phi\) (x)\(\sim A | x|^{-\alpha}\) as \(| x| \to \infty\). The main result of the paper studies the case \(\alpha <n\). It is shown that if \(2/(p-1)<\alpha\) then the effect of the nonlinear term vanishes as \(t\to +\infty\), more precisely \(t^{\alpha /2} | u(x,t)-w(x,t)| \to 0\) as \(t\to \infty\) where w is the solution of the corresponding linear problem. If, however, \(2/(p-1)=\alpha\) the behaviour is different, namely one has \[ | t^{\alpha /2} u(x,t)-f(| x| /\sqrt{t})| \to 0\quad as\quad t\to \infty, \] where f is a similarity solution of the full nonlinear equation, i.e. \[ f''+((n-1)/\eta +\eta /2)f'+(1/(p-1))f-f^ p=0,\quad f'(0)=0,\quad \lim_{\eta \to \infty} \eta^{\alpha}f(\eta)=A\quad (\eta =| x| /\sqrt{t}). \] Corresponding results in the case \(\alpha <n\), \(\alpha <2/(p-1)\) were previously given by A. Gmira and L. Veron [J. Differ. Equations 53, 258-276 (1984; Zbl 0506.35058)].
Reviewer: H.Pecher

MSC:
35K05 Heat equation
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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