Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 619 - 641, Volume 32, Issue 2, February 2012      doi:10.3934/dcds.2012.32.619

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Rui Peng - Department of Mathematics, Xuzhou Normal University, Xuzhou, 221116, China (email)
Dong Wei - Hebei University of Engineering, Handan City, Hebei Province, 056038, China (email)

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.

Keywords:  Periodic-parabolic logistic equation, entire space, positive periodic solution, uniqueness, asymptotic behavior.
Mathematics Subject Classification:  Primary: 35K20, 35B10; Secondary: 35K60, 35B05.

Received: August 2010;      Revised: June 2011;      Available Online: September 2011.