Maximal regularity for non-autonomous evolution equations governed by forms having less regularity

Abstract

We consider the maximal regularity problem for non-autonomous evolution equations

$$\begin{array}{l}{u'(t) + A(t)\,u(t) = f(t), \quad t \in (0, \tau]}\\ {u(0) = u_0.}\end{array}$$

(0.1)

Each operator A(t) is associated with a sesquilinear form a(t) on a Hilbert space H. We assume that these forms all have the same domain V. It is proved in Haak and Ouhabaz (Math Ann, doi:10.1007/s00208-015-1199-7, 2015) that if the forms have some regularity with respect to t (e.g., piecewise α-Hölder continuous for some α > ½) then the above problem has maximal L _p -regularity for all u ₀ in the real-interpolation space $(H, \fancyscript{D}(A(0)))_{1-{1}/{p},p}$. In this paper we prove that the regularity required there can be improved for a class of sesquilinear forms. The forms considered here are such that the difference a(t;.,.) − a(s;.,.) is continuous on a larger space than the common domain V. We give three examples which illustrate our results.