Solutions in \(L^ p\) of abstract parabolic equations in Hilbert spaces (Q1321783)

Let \(H\) be a complex Hilbert space. The author considers the linear parabolic Cauchy problem \((C): u'(t)+ A(t)u(t)= f(t)\), \(t\in (0,T)\), \(u(0)= u_ 0\), where the operators \(-A(t)\) are infinitesimal generators of analytic semigroups in \(H\), with domains which may vary with \(t\). This problem has been considered by many authors, and major references and a unified approach to its studying can be found in the paper by \textit{P. Acquistapace} and \textit{B. Terreni} [Rend. Semin. Mat. Univ. Padova 78, 47-107 (1987; Zbl 0646.34006)]. In the present paper, the author uses this approach to study solutions of \((C)\) satisfying the following condition \(\int^ T_ 0 | u'(t)|^ p dt+ \int^ T_ 0 | A(t)u(t)|^ p dt<\infty\), \(1< p<\infty\). The author claims that ``the study of such solutions is used to obtain sharp regularity results in linear and nonlinear problems''.