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

Abstract

For an equation of mixed type, namely,

$$ \left( {1 - \operatorname{sgn} t} \right)u_{tt} + \left( {1 - \operatorname{sgn} t} \right)u_t - 2u_{xx} = 0 $$

in the domain {(x, t) | 0 < x < 1, −α < t < β}, where α, β are given positive real numbers, we study the problem with boundary conditions

$$ u\left( {0,t} \right) = u\left( {1,t} \right) = 0, - \alpha \leqslant t \leqslant \beta , u\left( {x, - \alpha } \right) - u\left( {x,\beta } \right) = \phi \left( x \right), 0 \leqslant x \leqslant 1. $$

. 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 φ(x).