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

TL;DR

This paper establishes necessary and sufficient conditions for weak solutions of an abstract evolution equation with a spectral operator to be Gevrey ultradifferentiable on the semi-axis, revealing an inherent smoothness enhancement.

Contribution

It provides a complete characterization of Gevrey ultradifferentiability for weak solutions of spectral operator evolution equations, including smoothness improvement effects.

Findings

Conditions for Gevrey ultradifferentiability of solutions are derived.

Weak solutions can be inherently smoother than initially assumed.

Results include cases of analytic and entire solutions.

Abstract

Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\ge 0, \] 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 $\beta\ge 1$, in particular analytic or entire, on the open semi-axis $(0,\infty)$. Also, revealed is a certain interesting inherent smoothness improvement effect.