Well-posedness of non-autonomous linear evolution equations in uniformly convex spaces

TL;DR

This paper establishes conditions for the well-posedness of non-autonomous linear evolution equations in uniformly convex Banach spaces, emphasizing the necessity of Lipschitz continuity of the generator mappings.

Contribution

It proves that Lipschitz continuity of the generator family is essential for well-posedness in uniformly convex spaces, extending previous results to this class of Banach spaces.

Findings

Lipschitz continuity of $A(t)$ ensures well-posedness.

H"older continuity with degree less than 1 is insufficient.

Uniform convexity of the space is crucial for the results.

Abstract

This paper addresses the problem of wellposedness of non-autonomous linear evolution equations $\dot x = A(t)x$ in uniformly convex Banach spaces. We assume that $A(t):D \subset X\to X$, for each $t$ is the generator of a quasi-contractive $C_0$-group where the domain $D$ and the growth exponent are independent of $t$. Well-posedness holds provided that $t\mapsto A(t)y$ is Lipschitz for all $y\in D$. H\"older continuity of degree $\alpha<1$ is not sufficient and the assumption of uniform convexity cannot be dropped.