The periodic-parabolic logistic equation on $\mathbb{R}^N$
Rui Peng, Wei Fu Dong · Discrete and Continuous Dynamical Systems · 2011
In this article, we investigate the periodic-parabolic logisticequation 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. Underthe assumptions that $a(x,t)/{|x|^\gamma}$ and $b(x,t)/{|x|^\tau}$are bounded away from $0$ and infinity for all large $|x|$, wherethe constants $\gamma>-2$ and $\tau\in\mathbb{R}$, we study the existenceand uniqueness of positive $T$-periodic solutions. In particular, wedetermine the asymptotic behavior of the unique positive$T$-periodic solution as $|x|\to\infty$, which turns out to dependon the sign of $\gamma$. Our investigation considerably generalizesthe existing results.