On Solvability in the Sense of Sequences for some Non-Fredholm Operators with Drift and Anomalous Diffusion
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Journal of Mathematical Sciences, Vol. 250, No. 2, October, 2020
ON SOLVABILITY IN THE SENSE OF SEQUENCES FOR SOME NON-FREDHOLM OPERATORS WITH DRIFT AND ANOMALOUS DIFFUSION V. Vougalter University of Toronto 27 King’s College Circle Toronto, Ontario M5S 1A1, Canada [email protected]
UDC 517.9
We study the solvability of certain linear nonhomogeneous elliptic equations and establish that, under some technical assumptions, the L2 -convergence of the right-hand sides yields the existence and convergence of solutions in an appropriate Sobolev space. The problems involve differential operators with or without Fredholm property, in particular, the one-dimensional negative Laplacian in a fractional power, on the whole real line or on a finite interval with periodic boundary conditions. We prove that the presence of the transport term in these equations provides regularization of the solutions. Bibliography: 23 titles.
1
Introduction
We consider the equation −Δu + V (x)u − au = f,
(1.1)
where u ∈ E = H 2 (Rd ), f ∈ F = L2 (Rd ), d ∈ N, a is a constant, and V (x) converges to 0 at infinity. For a 0 the essential spectrum of the operator A : E → F corresponding to the left-hand side of (1.1) contains the origin. Consequently, the operator does not possess the Fredholm property. For such operators the image is not closed and for d > 1 the kernel dimension and the image codimension are not finite. In the present paper, we study some properties of such operators. We note that elliptic equations with non-Fredholm operators were treated extensively in recent years (cf. [1]–[6]) along with their potential applications to the theory of reaction-diffusion problems (cf. [7, 8]). In the particular case a = 0, the operator A satisfies the Fredholm property in some properly chosen weighted spaces (cf. [6] and [9]–[12]). However, the case a = 0 is significantly different and the approach developed in the cited works cannot be applied. One of the important issues about equations with non-Fredholm operators concerns their solvability. We address it in the following setting. Let fn be a sequence of functions in the image of the operator A such that fn → f in L2 (Rd ) as n → ∞. Denote by un a sequence of Translated from Problemy Matematicheskogo Analiza 105, 2020, pp. 89-100. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2502-0285
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functions from H 2 (Rd ) such that Aun = fn , n ∈ N. Since the operator A fails to satisfy the Fredholm property, the sequence un may not be convergent. Let us call a sequence un such that Aun → f a solution in the sense of sequences of the problem Au = f (cf. [13]). If this sequence converges to a function u0 in the E-norm, then u0 is a solution to this problem. A solution in the sense of sequences is equivalent in this sense to the usual solution. However, in the case of non-Fredholm operators, the convergence may not hold or it can occur in some weaker sense. Then the solution in the sense of sequences may not imply the existence of the usual solution. In the this paper, we obtain su
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