Semigroups of locally Lipschitz operators associated with semilinear evolution equations (Q879047)

Let \(A\) be the generator of a \(C_0\) semigroup on a Banach space \(X\) and \(B\) a nonlinear operator from a subset \(D\) of \(X\) into \(X\). This paper concerns the semigroup of locally Lipschitz operators on \(D\) with respect to a given vector-valued functional \(\varphi\), which presents a mild solution to the Cauchy problem for the semilinear evolution equation \[ u'(t)= (A+B)u(t)\quad (t\geq 0),\quad u(0)=u_0\quad (u_0\in D). \] Under some assumptions, the authors obtain a characterization of such a semigroup in terms of a sub-tangential condition, a growth condition and a semilinear stability condition indicated by a family of metric-like functionals on \(X\times X\). An application to the complex Ginzburg-Landau equation is given.