Regularity properties for evolution family governed by non-autonomous forms

TL;DR

This paper investigates the regularity, norm continuity, compactness, and Gibbs properties of evolution families generated by non-autonomous forms, with applications to time-dependent Robin boundary conditions for the Laplacian.

Contribution

It provides new regularity results and properties of evolution families from non-autonomous forms, including the Gibbs property, with concrete application to Laplacian operators.

Findings

Established norm continuity and compactness of evolution families.

Proved Gibbs property for the evolution family.

Applied abstract results to Laplacian with time-dependent Robin boundary conditions.

Abstract

This paper gives further regularity properties of the evolution family associated with a non-autonomous evolution equation \begin{equation*}\label{Abstract equation} \dot u(t)+A(t)u(t)=f(t),\ \ t\in[0,T],\ \ u(0)=u_0, \end{equation*} where $A(t),\ t\in [0,T],$ arise from non-autonomous sesquilinear forms $\mathfrak a(t,\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ Results on norm continuity, compactness and results on the \textit{Gibbs} character of the evolution family are established. The abstract results are applied to the Laplacian operator with time dependent Robin boundary conditions.