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Mild well-posedness of abstract differential equations. (English) Zbl 1170.47026
Amann, Herbert (ed.) et al., Functional analysis and evolution equations. The Günter Lumer volume. Basel: Birkhäuser (ISBN 978-3-7643-7793-9/hbk). 371-387 (2008).
In [Math. Z. 240, No. 2, 311--343 (2002; Zbl 1018.47008)], {\it W. Arendt} and {\it S.--Q.\thinspace Bu} established an elegant and simple characterization for strong periodic solutions of abstract differential operators. In the present paper, the authors generalize this result by giving spectral conditions that characterize mild well-posed differential equations in a general Banach space $X$. This approach gives a unified framework for several notions of strong and mild solutions and it is expressed in terms of operator-valued Fourier multipliers. Specificially, the authors study periodic solutions of first and second order equations in $L^p$ spaces and give some applications of this new notion in the setting of semilinear equations of second order in Hilbert spaces. For the entire collection see [Zbl 1131.46001].

47D06One-parameter semigroups and linear evolution equations
47D09Operator sine and cosine functions and higher-order Cauchy problems
34G10Linear ODE in abstract spaces