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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.

MSC:
34G20Nonlinear ODE in abstract spaces
47H20Semigroups of nonlinear operators
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References:
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