Bounded and almost periodic solutions of semilinear parabolic equations (Q1120733)

Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ n\), \(n\geq 1,\) (1) \(u_ t-\Delta u=F(t,x,u(t,x),u(t-r,x))\) in \({\mathbb{R}}\times \Omega\), \(r\geq 0,\) (2) \(u(t,x)=0\) on \({\mathbb{R}}\times \partial \Omega,\) (3) \(\partial u/\partial \nu (t,x)=0\) on \({\mathbb{R}}\times \partial \Omega.\) Under some smoothness assumptions on \(\partial \Omega\) and on F the existence of classical almost periodic solutions of (1), (2) or (1), (3) is obtained by the method of upper and lower solutions.