On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one

Marat V Markin

doi:10.1515/math-2019-0001

Abstract

Abstract It is shown that, if all weak solutions of the evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array} $$ with a scalar type spectral operator A in a complex Banach space are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded.

Highlights

IntroductionThis remarkable fact contrasts the situation when, in (1.1), a closed densely de ned linear operator A generates a C -semigroup, in which case the strong di erentiability of all weak solutions of (1.1) at alone immediately implies boundedness for A (cf [5], see [6])
In [1–3], found are characterizations of the strong di erentiability and Gevrey ultradi erentiability of order β ≥, in particular analyticity and entireness, on [, ∞) and (, ∞) of all weak solutions of the evolution equation y (t) = Ay(t), t ≥, (1.1)with a scalar type spectral operator A in a complex Banach space
As is shown by [2, Theorem . ], all weak solutions of equation (1.1) can be entire vector functions, i.e., belong to the rst-order Beurling type Gevrey class E ( )([, ∞), X), while the operator A is unbounded, e.g., when A is a semibounded below self-adjoint operator in a complex Hilbert space

Summary

Introduction

This remarkable fact contrasts the situation when, in (1.1), a closed densely de ned linear operator A generates a C -semigroup, in which case the strong di erentiability of all weak solutions of (1.1) at alone immediately implies boundedness for A (cf [5], see [6]) It remains to examine whether all weak solutions of equation (1.1) with a scalar type spectral operator A in a complex Banach space can belong to the Gevrey classes of orders less than one (not necessarily to the same one) with A remaining unbounded. In this paper, developing the results of [1–3], we show that an unbounded scalar type spectral operator A in a complex Banach space cannot sustain the strong Gevrey ultradi erentiability of all weak solutions of equation (1.1) for orders less than one, i.e., that imposing on all the weak solutions along with the entireness requirement certain growth at in nity conditions (see Preliminaries) necessarily makes the operator A bounded. For the reader’s convenience, we outline certain essential preliminaries

On the Gevrey ultradi erentiability of weak solutions

One Lemma

Main Result