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 is an element of L-2 (-infinity, +infinity) of this equation. For the solution y, we obtained the following maximal regularity estimate: parallel to y '''parallel to(2) + parallel to ry ''parallel to(2) + parallel to qy 'parallel to(2) + parallel to sy parallel to(2) <= C parallel to f parallel to(2),where parallel to.parallel to(2) is the norm of L-2 (-infinity, +infinity). This estimate is important for study of the qwasilinear third -order differential equation in (-infinity, +infinity). 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
