Maximal L^p-regularity for evolution equations governed by non-autonomous forms in weighted spaces

Abstract

We consider the maximal regularity for non-autonomous Cauchy problem $u^{\prime }(t)+A(t)u(t)=f(t),t\text{-a.e.},u(0)=u_0.$Each operator $A(t)$ arises from a time depending on a sesquilinear form $\mathfrak{a}(t)$ on a Hilbert space $\mathcal{H}$ with a constant domain $\mathcal{V}$. We show maximal $L^p-$regularity result in temporally weighted $L^p$-spaces for $2<p<\mathrm{\infty }$ and other regularity properties for the solution of the previous problem under sufficient regularity assumption on the forms and the initial value $u_0,$ which is significantly weaker than those from previous contributions. Our results are motivated by boundary value problems.

Keywords:

• Besov spaces
• maximal regularity
• non-autonomous evolution equations
• sesquilinear forms
• weighted spaces

2020 Mathematics Subject Classifications:

• 35K90
• 35K45
• 47D06

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

All data generated or analyzed during this study are included in this published article.