Construction of the evolution operator of parabolic type (Q5939266)

The authors show the existence of an evolution family for the linear differential equation NEWLINE\[NEWLINE{du(t)\over dt}+A(t)u(t) =f(t)\quad (a<t<b)NEWLINE\]NEWLINE of parabolic type in a Banach space \(X\). The operators \(A(t)\) have \((t\)-independent) domain \(Y=D(A(t))\), a Banach space. Besides more traditional assumptions, the new hypothesis is that for NEWLINE\[NEWLINE\omega(h): =\sup\biggl\{\bigl \|A(t+h)- A(t)\bigr\|_{Y\to X}; \;a\leq t\leq b-h\biggr\},NEWLINE\]NEWLINE the function \(\omega (h)/h\) is integrable at 0. This generalises previous results where \(t\mapsto A(t)\in {\mathcal L}(Y,X)\) is Hölder continuous, or where \(\omega(h) |\log h|/h\) is integrable at 0.