The Gevrey problem for a third-order difference-differential mixed-parabolic equation (Q1874824)

The paper deals with the following third-order parabolic equation with delay \[ u_{xxx}(x,y) - \text{sgn } x(\tau-x)u_y(x,y) = H(x-\tau)u(x-\tau,y) + H(\tau-x)f(x,y),\tag{1} \] where \(H(\xi)\) is the Heaviside function and (1) is considered in the domain \(D = \{ (x,y): -\tau < x < 2\tau, \;0 < y < 1\}\), together with appropriate boundary conditions. The author proves uniqueness of the solutions provided that \(0 < \tau \leq 3^{1/3}\) and existence of the solutions under certain conditions on \(f\). The main tool of the author is the reduction of the problem to a system of three second-order singular integral equations of normal type.