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Semigroups of locally Lipschitz operators associated with semilinear evolution equations. (English) Zbl 1123.34044

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

${u}^{\text{'}}\left(t\right)=\left(A+B\right)u\left(t\right)\phantom{\rule{1.em}{0ex}}\left(t\ge 0\right),\phantom{\rule{1.em}{0ex}}u\left(0\right)={u}_{0}\phantom{\rule{1.em}{0ex}}\left({u}_{0}\in D\right)·$

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 $X×X$. An application to the complex Ginzburg-Landau equation is given.

MSC:
 34G20 Nonlinear ODE in abstract spaces 47H20 Semigroups of nonlinear operators