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

• Main Authors: Maria do Rosário Grossinho, Pierpaolo Omari
• Format: Article
• Language: English
• Published: University of Szeged 1999-01-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=18

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.