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Zbl 1123.34044
Kobayashi, Yoshikazu; Matsumoto, Toshitaka; Tanaka, Naoki
Semigroups of locally Lipschitz operators associated with semilinear evolution equations.
(English)
[J] J. Math. Anal. Appl. 330, No. 2, 1042-1067 (2007). ISSN 0022-247X

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.
[Jin Liang (Hefei)]
MSC 2000:
*34G20 Nonlinear ODE in abstract spaces
47H20 Semigroups of nonlinear operators

Keywords: Semigroup of locally Lipschitz operators; semilinear evolution equation; semilinear stability condition; sub-tangential condition; growth condition

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