Maximal regularity for semilinear non-autonomous evolution equations in temporally weighted spaces

Abstract

We consider the problem of maximal regularity for the semilinear non-autonomous evolution equations u′(t)+A(t)u(t)=F(t,u),t-a.e.,u(0)=u0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u'(t)+A(t)u(t)=F(t,u),\\, t \ ext {-a.e.}, \\, u(0)=u_0. \\end{aligned}$$\\end{document}Here, the time-dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space mathcal {H}. We prove the maximal regularity result in temporally weighted L^2-spaces and other regularity properties for the solution of the previous problem under minimal regularity assumptions on the forms, the initial value u_0 and the inhomogeneous term F. Our results are motivated by boundary value problems.