The periodic-parabolic logistic equation on $\mathbb{R}^N$

Rui Peng; Dong Wei

doi:10.3934/dcds.2012.32.619

Article Contents

2012, Volume 32, Issue 2: 619-641. Doi: 10.3934/dcds.2012.32.619

This issue Previous Article On strange attractors in a class of pinched skew products Next Article Symmetric interval identification systems of order three

The periodic-parabolic logistic equation on $\mathbb{R}^N$

Rui Peng^1, and
Dong Wei^2,

1.
Department of Mathematics, Xuzhou Normal University, Xuzhou, 221116
2.
Hebei University of Engineering, Handan City, Hebei Province, 056038

Received: August 2010

Revised: June 2011

Published: February 2012

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Abstract

In this article, we investigate the periodic-parabolic logistic equation on the entire space $\mathbb{R}^N\ (N\geq1)$: $$ \begin{equation} \left\{\begin{array}{ll} \partial_t u-\Delta u=a(x,t)u-b(x,t)u^p\ \ \ \ & {\rm in}\ \mathbb{R}^N\times(0,T),\\ u(x,0)=u(x,T) \ & {\rm in}\ \mathbb{R}^N, \end{array} \right. \end{equation} $$ where the constants $T>0$ and $p>1$, and the functions $a,\ b$ with $b>0$ are smooth in $\mathbb{R}^N\times\mathbb{R}$ and $T$-periodic in time. Under the assumptions that $a(x,t)/{|x|^\gamma}$ and $b(x,t)/{|x|^\tau}$ are bounded away from $0$ and infinity for all large $|x|$, where the constants $\gamma>-2$ and $\tau\in\mathbb{R}$, we study the existence and uniqueness of positive $T$-periodic solutions. In particular, we determine the asymptotic behavior of the unique positive $T$-periodic solution as $|x|\to\infty$, which turns out to depend on the sign of $\gamma$. Our investigation considerably generalizes the existing results.

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Mathematics Subject Classification: Primary: 35K20, 35B10; Secondary: 35K60, 35B05.

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