Nonlocal Problem for a Parabolic-Hyperbolic Equation in a Rectangular Domain

For an equation of mixed type, namely,
$$ (1-\operatorname{sgn}t)u_{tt}+(1-\operatorname{sgn}t)u_{t}-2u_{xx}=0 $$
in the domain $\{(x,t)\mid0<x<1,\,-\alpha<t<\beta\}$, where $\alpha$, $\beta$ are given positive real numbers, we study the problem with boundary conditions
$$ u(0,t)=u(1,t)=0,\quad -\alpha\le t\le\beta,\qquad u(x,-\alpha)-u(x,\beta)=\varphi(x),\quad 0\le x\le1. $$
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 $\varphi(x)$.