On the correctness of one class of fourth order singular differential equations

Abstract

The differential equation of fourth-order with variable smooth coefficients, given on the real axis, is investigated. The coefficients of the equation can be unbounded, and the sign of the potential (the lowest coefficient) is not defined. The case is considered that the intermediate term containing the second derivative of the desired function, as an operator, does not obey the operator of the equation. It is known that fourth-order differential equations with variable coefficients are used in various problems of mathematical physics. The simplest biharmonic equation with minor terms is important for its applications in the theory of elasticity, the mechanics of elastic plates and the slow flow of viscous liquids. However, it is a very special case of a general equation with variable coefficients. Some problems of stochastic analysis, oscillation theory, biology and mathematical finance lead to the equation we are studying. In addition, fourth–order singular differential
equations are often used as regularizers in the study of lower-order equations, for example, reaction-diffusion equations. In the paper, sufficient solvability conditions of the equation and the maximal regularity inequality of a strong generalized solution are obtained. The restrictions are formulated in terms of the coefficients themselves and rather weak. They do not contain conditions for any derivatives of coefficients and represent relations between the orders of growth of coefficients of different orders at infinity. The weight estimation of the norms of a solution established by us allows us to additionally apply the methods of the theory of functional spaces to the study of further properties of the solution, for example, the approximation of its elements by finite-dimensional spaces.