Nonlocal problem for a parabolic-hyperbolic equation in a rectangular domain

For an equation of mixed type, namely, (1 - sgnt)u(tt) + (1 - sgnt)u(t) - 2u(xx) - 0 in the domain {(x, t) | 0 < x < 1, -alpha < t < beta}, where alpha, beta are given positive real numbers, we study the problem with boundary conditions u(0, t) = u(1, t) = 0, -alpha <= t <= beta, u(x, -alpha) - u(x, beta) = phi(x), 0 <= x <= 1. We establish a uniqueness criterion for the solution constructed as the sum of Fourier series. We establish the stability of the solution with respect to its nonlocal condition phi(x).