Asymptotic behavior of solutions for some semilinear heat equations in R^N

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July 2009, 8(4): 1351-1371. doi: 10.3934/cpaa.2009.8.1351

Asymptotic behavior of solutions for some semilinear heat equations in $R^N$

Kazuhiro Ishige ^1, and Tatsuki Kawakami ^1,

1.	Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan, Japan

Received January 2008 Revised November 2008 Published March 2009

Abstract
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We consider the Cauchy problem of the semilinear heat equation,

$\partial_t u = \Delta u +f(u)$ in $R^N \times (0,\infty),$

$u (x,0) = \phi (x) \ge 0$ in $R^N,\quad\quad$

where $N \geq 1$, $f \in C^1([0,\infty))$, and $\phi \in L^1(R^N) \cap L^{\infty}(R^N)$. We study the asymptotic behavior of the solutions in the $L^q$ spaces with $q \in [1,\infty]$, by using the relative entropy methods.

Keywords: Cauchy problem, semilinear heat equation, Asymptotic behavior, relative entropy methods..

Mathematics Subject Classification: 35K55, 35B4.

Citation: Kazuhiro Ishige, Tatsuki Kawakami. Asymptotic behavior of solutions for some semilinear heat equations in $R^N$. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1351-1371. doi: 10.3934/cpaa.2009.8.1351

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