Conditions of coercive solvability of third-order differential equation with unbounded intermediate coefficients

In this paper we study the following equation −y′′′ + r (x) y′′ + q (x) y′ + s (x) y = f (x), where the intermediate coefficients r and q do not depend on s. We give the conditions of the coercive solvability for f ∈ L2 (−∞, +∞) of this equation. For the solution y, we obtained the following maximal regularity estimate: ‖y‖2 + ‖ry ‖2 + ‖qy ‖2 + ‖sy‖2 ≤ C ‖f‖2, where ‖ · ‖2 is the norm of L2 (−∞, +∞). This estimate is important for study of the qwasilinear third-order differential equation in (−∞, +∞). We investigate some binomial degenerate differential equations and we prove that they are coercive solvable. Here we apply the method of the separability theory for differential operators in a Hilbert space, wich was developed by M. Otelbaev. Using these auxillary statements and some well-known Hardy type weighted integral inequalities, we obtain the desired result. In contrast to the preliminary results, we do not assume that the coefficient s is strict positive, the results are also valid in the case that s = 0.