The maximal regularity of the third‐order differential equation and its applications

Abstract

In this work, we study the solvability and maximal regularity questions for the singular third‐order differential equation with unbounded coefficients and some applications. Unlike previously studied cases, the leading and intermediate coefficients of this equation can grow independently. We obtain sufficient coefficient conditions for correctness of this equation and compactness for inverse of the corresponding differential operator. We also prove the maximal regularity estimate for a generalized solution. Using these results, we obtain upper and lower estimates for the number of Kolmogorov ‐diameters of the set associated with the linear Korteweg–de Vries equation's solutions. We give an example and show that from the above inequalities follow two‐sided estimates of the Kolmogorov ‐diameters.