# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  N.D. Alikakos - R. Rostamian , On the uniformization of the solution of the porous medium equation in Rn, israel J. Math. , 47 ( 1984 ), pp. 270 - 290 . MR 764297 | Zbl 0562.35050 · Zbl 0562.35050 · doi:10.1007/BF02760601  A. Friedman , Partial differential equations of parabolic type , Prentice-Hall , Englewood Cliffs, N.J. , 1964 . MR 181836 | Zbl 0144.34903 · Zbl 0144.34903  B.H. Gilding , Hölder continuity of solutions of parabolic equations, J . London Math. Soc. , 13 ( 1976 ), pp. 103 - 106 . MR 399658 | Zbl 0319.35045 · Zbl 0319.35045 · doi:10.1112/jlms/s2-13.1.103  L. Gmira - L. Veron , Large time behaviour of the solutions of a semilinear parabolic equation in Rn , J. Diff. Equ. , 53 ( 1984 ), pp. 258 - 276 . MR 748242 | Zbl 0529.35041 · Zbl 0529.35041 · doi:10.1016/0022-0396(84)90042-1  S. Kamin (Kamenomostkaya) , The asymptotic behaviour of the solution of the filtration equation , Israel J. Math. , 14 ( 1973 ), pp. 76 - 87 . MR 315292 | Zbl 0254.35054 · Zbl 0254.35054 · doi:10.1007/BF02761536  O.A. Oleinik - S.N. Kruzhkov , Quasilinear second order parabolic equations with many independent variables , Russian Math. , Surveys, 16 ( 1961 ), pp. 105 - 146 . Zbl 0112.32604 · Zbl 0112.32604 · doi:10.1070/rm1961v016n05ABEH004114  V.A. Galaktionov - S.P. Kurdjumov - A.A. Samarskii , On asymptotic eigenfunctions of the Cauchy problem for a nonlinear parabolic equation , Mat. Sbornik , 126 , 4 ( 1985 ), pp. 435 - 472 (in Russian). Zbl 0607.35049 · Zbl 0607.35049 · doi:10.1070/SM1986v054n02ABEH002979 · eudml:71656
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.