Correctness conditions for high-order differential equations with unbounded coefficients

Abstract

We give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

Well posedness for one class of elliptic equations with drift

“…In [18], the regularity estimate was applied to the study of nonlinear Schrödinger equation. In the case n = 1, the correctness of equation ( 1) with a rapidly growing drift was shown in [19] (see also [20][21][22]).…”

“…We denote by W 1 2, √ λa (R) the completion of C (1) (R) with respect to the norm θ W = √ λaθ 2,R + θ 2,R . ( 21) and (22) imply that the sequence {v s } ∞ s=1 is a Cauchy sequence in W 1 2, √ λa (R). Hence, there exists v ∈ W 1 2, √ λa (R) such that v sv W → 0 (s → ∞).…”

“…Hence, there exists v ∈ W 1 2, √ λa (R) such that v sv W → 0 (s → ∞). Then, passing to the limit in (22) and using (21), we obtain (18). Remark 2.3 Lemma 2.3 remains true if a ≥ δ > 0, where δ < 1.…”