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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 ϕ, which presents a mild solution to the Cauchy problem for the semilinear evolution equation

u ' (t)=(A+B)u(t)(t0),u(0)=u 0 (u 0 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×X. An application to the complex Ginzburg-Landau equation is given.

34G20Nonlinear ODE in abstract spaces
47H20Semigroups of nonlinear operators