The existence of a periodic solution of a parabolic equation with the Bessel operator

summary:In this paper, the existence of an $\omega$-periodic weak solution of a parabolic equation (1.1) with the boundary conditions (1.2) and (1.3) is proved. The real functions $f(t,r),h(t),a(t)$ are assumed to be $\omega$-periodic in $t,f\in L_2(S,H),a,h$ such that $a'\in L_\infty (R), h'\in L_\infty (R)$ and they fulfil (3). The solution $u$ belongs to the space $L_2(S,V)\cap L_\infty (S,H)$, has the derivative $u'\in L_2(S,H)$ and satisfies the equations (4.1) and (4.2). In the proof the Faedo-Galerkin method is employed