A note on semigroups of locally Lipschitz operators associated with semilinear evolution equations (Q2887963)

Regarding the semilinear Cauchy problem \(u'(t) = Au(t) + Bu(t)\) on the Banach space \(X\) in which \(A\) is the infinitesimal generator of an analytic \(C_0\) semigroup \(T(t)\), \textit{T. Matsumoto} and \textit{N. Tanaka} [J. Approx. Theory 163, No. 9, 1217--1237 (2011; Zbl 1229.47133)] proved the existence of a semigroup \(S(t)\) of Lipschitz operators on a closed subset \(D\) of \(X\) such that \(S(t)x\) is in fact the mild solution of the above Cauchy problem on \(D\). The authors' aim is to give a simple proof for a part of this theorem. First, they consider the continuous dependence of the mild solution on initial data and then characterize semigroups of locally Lipschitz operators associated with semilinear equations of parabolic type. At the end, they prove the above mentioned main result.