On the Classical Solvability of Mixed Problems for a Second-Order One-Dimensional Parabolic Equation

We prove the existence and uniqueness of classical solutions to mixed problems for the equation partial derivative u/partial derivative t (x, t) - partial derivative(2)u/partial derivative t(2) (x, t) + q(x)u(x, t) = f(x, t) on a rectangle (Omega) over bar = [a; b] x [0; T], with arbitrary self- adjoint homogenous boundary conditions. We assume that q and f are continuous functions, that f(x; .) satisfies a H <spacing diaeresis> older condition uniformly with respect to x, and the initial function belongs to the class W-p((1)) (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 <(Omega(infinity))over bar> = [a, b] x [0, + 1). The existence and uniqueness of classical solutions are established, and some properties of the solutions are considered.