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.
