is a bounded domain in with a where is an -periodic (in ) function and is Concerning we assume that defined and regular (say, class) boundary . nondecreasing on $R$ . We may assume also $¥beta(0)=0$ without loss of generality. If we impose the Dirichlet boundary condition instead of the Neumann the existence of classical periodic solutions to nonlinear parabolic equations is well known under appropriate assumptions on the nonlinear terms (cf. Smulev [9], , Nakao Kusano [7], Kolesov etc.). However, the Neumann case seems not to be investigated enough (cf. Amann [2]). For further references on periodic solutions of nonlinear parabolic equations see Vejovoda [9; Chapt. 3]. The difficulty in the Neumann case lies in the fact that the problem (1) does not admit a periodic solution for all $f(x, t)$ even if $f$ is smooth. Indeed, if be an -periodic solution of (1) it must hold that $f$
On the basis of Theorem 0 we shall prove here-the existence of a classical periodic solution of (1) under slightly stronger assumption than (3). Our result reads as follows: Theorem 1. pose that
