On the Problem of Local-Nonequilibrium Heat Transfer with High Order Derivatives

We study a nonlocal problem for new types of hyperbolic heat equations, which is obtained by introducing relaxation corrections into the Fourier formula for both heat flux and temperature gradient. The resulting hyperbolic equations contain the third and fourth order (depending on the number of relaxation coefficients) derivatives with respect to the spatial variable and time (mixed derivatives). A new nonlinear mathematical problem of locally nonequilibrium transport processes is investigated, taking into account relaxation phenomena, based on hyperbolic and parabolic equations. A method for establishing a priori estimates of the Schauder type is proposed. The unique solvability of the problem with internal boundary nonlocal boundary conditions is proved.