Periodic solutions of the first boundary value problem for a linear and weakly nonlinear heat equation

Abstract

One investigates the existence of an $\omega$-periodic solution of the problem $u_t=u_{xx}+cu+g(t,x)+\epsilon f(t,x,u,u_x,\epsilon),\ u(t,0)=h_0(t)+\epsilon \chi_0(t,u(t,0),u(t,\pi)), u(t,\pi)=h_1(t)+\epsilon \chi_1(t,u(t,0), u(t,\pi))$, provided the functions $g,f,h_0,h_1,\chi_0,\chi_1$ are sufficiently smooth and $\omega$-periodic in $t$. If $c\neq k^2$, $k$ natural, such a solution always exists for sufficiently small $\epsilon >0$. On the other hand, if $c=l^2$, $l$ natural, some additional conditions have to be satisfied.