Hyperbolic Stefan problem with nonlocal boundary conditions

In this paper, we consider problems with unknown boundaries for hyperbolic equations and systems with free boundaries with two independent variables. The boundary conditions for such equations in the linear or quasilinear cases are given in nonlocal (non-separable and integral) form. The hyperbolic Stefan and Darboux-Stefan problems (the line of initial conditions degenerates to a point) are considered. There are proved the existence and uniqueness theorems of generalized solution, which are continuous solutions of equivalent systems of the second kind Volterra integral equations. The method of characteristics based on a combination of the Banach fixed point theorem allows us to obtain global generalized solutions in terms of the time variable in the case of linear hyperbolic equations with free boundaries and local solutions for quasilinear equations. Nonlocal (non-separable and integral) conditions require additional solvability conditions that are not present in the case of generally accepted boundary conditions for hyperbolic equations and systems. The paper provides examples indicating the significance of the conditions for the solvability of the corresponding problems. The corresponding solutions may have discontinuities along the characteristics of the hyperbolic equations. This additionally requires setting the conditions for matching the initial data of the problems at the corner points of the considered domains. This paper extends the results on the problems with nonlocal conditions for hyperbolic equations and systems to the case of hyperbolic equations with free boundaries.