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

M. R. Grossinho, Technical University of Lisboa, Portugal
P. Omari, University of Trieste, Italy

E. J. Qualitative Theory of Diff. Equ., No. 9. (1999), pp. 1-24.

Abstract: 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.

You can download the full text of this paper in DVI, PostScript or PDF format, or have a look at the Zentralblatt or the Mathematical Reviews entry of this paper.