We prove the existence and uniqueness of classical solutions to mixed problems for the equation ∂u ∂t (x,t) − ∂ 2 u ∂x 2 (x, t) + q(x) u(x, t) = f (x, t) on a rectangle Ω = [a, b] × [0, T] , with arbitrary self–adjoint homogenous boundary conditions. We assume that q and f are continuous functions, that f (x, ·) satisfies a H ¨ older condition uniformly with respect to x , and the initial function belongs to the class •W (1) p (a, b) (1 < p ≤ 2). Also, an upper–bound estimate for the solution and, as a consequence, a kind of stability of the solution with respect to the initial function are established. Moreover, some convergence rate estimates for the series defining solutions (and their first derivatives) are given. A modification of the Fourier method is used. Based on the obtained results, we also study the mixed problems on an unbounded rectangle Ω ∞ = [a, b] × [0, +∞). The existence and uniqueness of classical solutions are established, and some properties of the solutions are considered.