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

Abstract

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 E L2 (–8; +8) of this equation. For the solution y, we obtained the following maximal regularity estimate: IIy’’’II2 + IIry’’II2 + IIqy’II2 + IIsyII2 ≤ C IIfII2; where II . II2 is the norm of L2 (-8; +8). This estimate is important for study of the qwasilinear third-order differential equation in (-8; +8). 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.