Two free boundaries problem for a parabolic equation

This paper considers a two-free-boundary Stefan-type problem for a quasi-linear parabolic equation in one dimension. Nonlinear problems with free boundaries are studied using a method based on constructing a priori estimates. Therefore, some initial a priori estimates for the solution to the problem under consideration are first established. The main difficulty in constructing a theory for second-order quasi-linear parabolic equations is obtaining an a priori estimate for the solution's derivative module, and additional arguments are required in problems with a free boundary. To address this, the problem is reduced to a fixed-boundary problem through a change of variables. The resulting problem has time- and spacedependent coefficients with nonlinear terms. Next, Schauder-type a priori estimates are constructed for the equation with nonlinear terms and a fixed boundary. Based on these estimates, the uniqueness of the solution to the problem is proven. Then, the global existence of the solution to the problem is demonstrated using the Leray-Schauder fixed-point theorem.