Existence of stable periodic solutions of a semilinear parabolic problem under Hammerstein-type conditions

We prove the solvability of the parabolic problem $$\partial_t u-\sum_{i,j=1}^N \partial_{x_i}(a_{i,j}(x,t)\partial_{x_j}u)+\sum_{i=1}^N b_i(x,t)\partial_{x_i}u=f(x,t,u)\hbox{ in }\Omega\times R$$ $$u(x,t)=0\hbox{ on }\partial\Omega\times R$$ $$u(x,t)=u(x,t+T)\hbox{ in }\Omega\times R$$ assuming certain conditions on the ratio $2\int_0^s f(x,t,\sigma) d\sigma/s^2$ with respect to the principal eigenvalue of the associated linear problem. The method of proof, whcih is based on the construction of upper and lower solutions, also yields information on the localization and the stability of the solution.