The first boundary value problem with deviation from the characteristics for a second-order parabolic-hyperbolic equation

where k(y) ≥ k1 > 0, f = f(x, y) are given functions, and u = u(x, y) is an unknown one. For y < 0, equation (1) coincides with the Chaplygin equation [1; 21], and for y > 0, it is a parabolic equation backward in time (with x standing for a time variable). Thus, equation (1) is a second-order parabolic-hyperbolic equation with non-characteristic line of degeneracy [2]. A great number of scientific researches are devoted to the study of boundary-value problems for secondorder parabolic-hyperbolic equations with non-characteristic line of degeneracy. For example, in [3], by the spectral method, a priori estimates in the Lpand C-classes for solution of the Tricomi problem for an equation of the form (1), are obtained. In [4–8], boundary value problems with deviation from the characteristics for parabolic-hyperbolic equations are studied. In [9], a method enabling one to formulate well-posed boundary value problems for a class of linear parabolichyperbolic equations of the form