A sufficient condition for the existence of periodic solution for a reaction diffusion equation with infinite delay. (Q1417016)

The following integro-partial differential equation is considered: \[ \begin{aligned}\frac{\partial u}{\partial t} -\Delta u&=u(t,x) \left(a(t,x)-b(t,x)u(t,x)- c(t,x) \int_0^\infty u(t-\tau, x)\,d\mu(\tau)\right),\\ \frac{\partial u(t,x)}{\partial n}&=0,\quad (t,x)\in \mathbb R^+\times\partial\Omega,\\ u(t,x)&=\phi(t,x),\quad (t,x)\in\mathbb R^-\times\bar\Omega,\end{aligned} \] where \(a\), \(b\), \(c\), are functions which are strictly positive, smooth and periodic with the same period in the \(t\) variable. \(\Omega\) is a bounded domain in \(\mathbb R^n\). Sufficient conditions for the existence of a unique periodic solution are derived; furthermore, it is shown that a solution with appropriate initial function \(\phi(t,x)\) approaches the unique periodic solution as \(t\to \infty\).