Unique solvability in the class \(L_ 2\) of the Tricomi problem with a nonlocal condition on the line of change of type for a model parabolic-hyperbolic equation (Q1363945)

In studies on the propagation of heat in a system composed of two bodies, heat transfer is described by parabolic and hyperbolic heat transfer equations depending on the value of the relaxation time \[ {1\over c^2_q} u_{tt}+ {1\over a_2} u_t- u_{xx}= f(x,t),\quad x>0,\quad {1\over a_1} u_t- u_{xx}= f(x,t),\quad x<0.\tag{1} \] A class of continuous solutions to this equation of mixed type appears due to the equality of the temperatures at the common boundary of the bodies, and a class for partial derivatives is obtained from the requirement to equate the heat fluxes at this boundary: \[ \lambda_1{\partial u^-(0,t)\over\partial x}= {\lambda_2\over\tau_q} \int^t_0 \Biggl[{\partial u^+(0,s)\over\partial x}\Biggr] e^{-(t-s)/\tau_q}ds,\quad t\in(0,T),\tag{2} \] i.e., relation (2) is the equality of the flux for usual heat transfer equation to the flux with ``memory'' for a hyperbolic heat transfer equation. The coefficients in equation (1) and equality (2) are given positive numbers. In this paper, we consider the question of unique solvability in the class \(L_2\) of the Tricomi problem with nonlocal condition (2) for a simplest parabolic-hyperbolic equation.