On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis

Marat V Markin

doi:10.1515/math-2019-0083

Abstract

Abstract Given the abstract evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array}$$ with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic or entire, on the open semi-axis (0, ∞). Also, revealed is a certain interesting inherent smoothness improvement effect.

Highlights

Given the abstract evolution equation y (t) = Ay(t), t ≥, with scalar type spectral operator A in a complex Banach space, found are conditions necessary and su cient for all weak solutions of the equation, which a priori need not be strongly di erentiable, to be strongly Gevrey ultradi erentiable of order β ≥, in particular analytic or entire, on the open semi-axis (, ∞)
We nd conditions on a scalar type spectral operator A in a complex Banach space necessary and su cient for all weak solutions of the evolution equation y (t) = Ay(t), t ≥, (1.1)
The found results generalize the corresponding ones of paper [1], where similar consideration is given to equation (1.1) with a normal operator A in a complex Hilbert space, and the characterizations of the generation of Gevrey ultradi erentiable C -semigroups of Roumieu and Beurling types by scalar type spectral operators found in papers [2, 4]

Summary

Introduction

We need the following characterization of a particular weak solution’s of equation (1.1) with a scalar type spectral operator A in a complex Banach space being strongly Gevrey ultradi erentiable on a subinterval I of [ , ∞). The case of the strong Gevrey ultradi erentiability of the weak solutions of equation (1.1) with a scalar type spectral operator in a complex Banach space on the open semi-axis ( , ∞), to the analogous setup with a normal operator A in a complex Hilbert space [1], signi cantly di ers from its counterpart over the closed semi-axis [ , ∞) studied in [6].

Results

Conclusion