An irregular conjugation problem for the system of the parabolic equations in the holder space

Abstract

We consider the conjugation problem for the system of parabolic equations with two small parameters k> 0, ε> 0 in the boundary conditions. There are proved the existence, uniqueness and uniform coercive estimates of the solution with respect to the small parameters in the Holder space. This problem is linearized one of the nonlinear problem with the free boundary of Florin type and it is in the base of the proof of the solidified of this nonlinear problem in the Holder space. We study the problem with the free boundary of the Florin type in the Holder space Ċ_x^2+l,1+l/2_t (W_jT), j= 1,2, where l is non-integer positive number. Existence, uniqueness, estimates for solution of the problem with constants independent of small parameters in the Holder space are proved. It gives us the opportunity to establish the existence, uniqueness and estimates of the solution of the problem without loss of smoothness of given functions for k = 0, ε > 0; k> 0; ε =0 and k = 0, ε = 0.