On the mixed problem E for second-order parabolic equations degenerating on the boundary of a domain
The paper investigates the question of the unique solvability of the first mixed problem for a degenerate second-order parabolic equation in the case when the boundary and initial functions belong to spaces of type $L_2$. This topic originates from the classical works of F. Riesz [1] and Littlewood and Paley [2] devoted to the boundary values of analytic functions. Under the weakest restrictions on the smoothness of the boundary (and on the coefficients of the equation), the criterion for the existence of a boundary value was established in [7-9]. The boundary smoothness condition ($\partial Q\in C^2$) can be weakened (see [10]). Moreover, as shown in [9], all directions of the adoption of boundary values for uniformly elliptic equations turn out to be equal; the solution has the property similar to the property of continuity in the set of variables. In the case of degeneration of the equation at the boundary of the region, when the directions are not equal, the situation is more complicated. Moreover, the formulation of the first boundary value problem is determined by the type of degeneracy. In the case when the values of the corresponding quadratic form of the degenerate elliptic equation on the normal vector are nonzero (Tricomi type degeneracy), the Dirichlet problem is correct, and the properties of such degenerate equation are very close to the properties of a uniformly elliptic equation. In particular, in this situation, analogs of the Riesz theorems [1] and Littlewood-Paley [2, 3] are valid. In the case of degeneracy of the Keldysh type, the situation is more complicated. The statement of the first boundary-value problem and the behavior of the solution near the boundary are determined by the order of degeneracy of the equation, and in the case of "strong" degeneracy, by the coefficients of the lower terms. A large number of papers have been devoted to the solvability of the first boundary-value problem for degenerate elliptic and parabolic equations. Note the works of F. Tricomi [11], M. V. Keldysh [12], A. V. Bitsadze [13], S. A. Tersenov [14], I. M. Petrushko [15], O. A. Oleinik and E. V. Radkevich [16], G. Fichera [17], etc. From recent works it is worth noting [18]. In this paper, we consider the case of strong degeneracy of a second-order parabolic equation when the corresponding quadratic form decreases as $r(x)$ and the formulation of the first mixed problem is determined by the coefficient of the first normal derivative.