The periodic-parabolic logistic equation on \(\mathbb {R}^N\)
The authors consider the periodic-parabolic logistic equation on \(\mathbb{R}^N\) \((N\geq 1)\): \[ \begin{cases} \partial_tu-\Delta u=a(x,t)u-b(x,t)u^p& \text{in }\mathbb{R}^N\times (0,T),\\ u(x,0)=u(x,T)&\text{in }\mathbb{R}^N,\end{cases} \] where the constants \(T>0\) and \(p>1\), and \(a\), \(b\) are smooth with \(b>0\) in \(\mathbb{R}^n\times\mathbb{R}\) and \(T\)-periodic in time. Under the assumptions that \(\frac{a(x,t)}{|x|^{\gamma}}\) and \(\frac{b(x,t)}{|x|^{\tau}}\) are bounded from zero and infinity for all large \(|x|\), where \(\gamma>-2\) and \(\tau>0\), they give the existence and uniqueness of a positive \(T\)-periodic solution. They further discuss the asymptotic behavior of this unique \(T\)-periodic solution as \(|x|\to\infty\). The result indicates that the sign of \(\gamma\) affects the asymptotic behavior of the unique \(T\)-periodic solution definitely.
scientific article; zbMATH DE number 6015430