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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
$u'(t)= (A+B)u(t)\quad (t\geq 0),\quad u(0)=u_0\quad (u_0\in D).$
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.
Reviewer: Jin Liang (Hefei)

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 47H20 Semigroups of nonlinear operators
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