Conditions of coercive solvability of third-order differential equation with unbounded intermediate coefficients - Ospanov, K.N.; Yeskabylova, Zh.B. (Bulletin of the Karaganda University. Mathematics Series, 2019)
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.