Bounded invertibility and separability of a parabolic type singular operator in the space L2(R2 )

Abstract

In this paper, we consider the operator of parabolic type Lu = ∂u ∂t − ∂ 2u ∂x2 + q(x)u, in the space L2(R 2 ) with a greatly growing coefficient at infinity. The operator is originally defined on C ∞0 (R 2 ), where C ∞0 (R 2 ) is the set of infinitely differentiable and compactly supported functions. Assume that the coefficient q(x) is a continuous function in R = (−∞, ∞), and it can be a strongly increasing function at infinity. The operator L admits closure in space L2(R 2 ), and the closure is also denoted by L. In the paper, we proved the bounded invertibility of the operator L in the space L2(R 2 ) and the existence of the estimate ∂u ∂t L2(R2) + ∂ 2u ∂x2 L2(R2) + ∥q(x)u∥L2(R2) ≤ C(∥Lu∥L2(R2) + ∥u∥L2(R2) ), under certain restrictions on q(x) in addition to the conditions indicated above. Example. q(x) = e 100|x| , −∞ < x < ∞.