Periodic solutions of semilinear partial differential equations of parabolic type (Q1105120)

The authors investigate the periodic Dirichlet problem for the equation \[ u_ t-Au-\lambda_ 1u+g(u)=h(x,t), \] where h is T-periodic in t, A is a uniformly elliptic operator, \(\lambda_ 1\) is the first eigenvalue of the operator \(L=D_ t-A\) and \(g: R\to R\) is a nondecreasing function. They prove that \(\omega\in Int (Range g)\) is a sufficient condition for the problem to have a solution, where \[ \omega =(h,\phi^*)| \int^{T}_{0}\int_{\Omega}\phi^*(x,t)dx dt,\quad L^*\phi^*=\lambda_ 1\phi^*,\quad \phi^*>0. \] Moreover, if \(\omega\in Bdry (Range g)\), then there exists a solution if and only if \(g(0)=\omega\). Further they show that for g bounded with \(\omega\in Int (Range g)\) the solution set is compact and connected and if \(\omega\in Bdry (Range g)\), \(g(0)=\omega\) then the solution set is closed, unbounded and connected. Asymptotic behavior for the initial boundary value problem is also discussed.