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 ∈ L ₂(-∞ , +∞) 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 L ₂(-∞ , +∞). 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.