Abstract
Conditions of coercive solvability of third-order differential equation with unbounded intermediate coefficientsIn 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.