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Characterization of nonlinear semigroups associated with semilinear evolution equations

Authors: Shinnosuke Oharu and Tadayasu Takahashi
Journal: Trans. Amer. Math. Soc. 311 (1989), 593-619
MSC: Primary 47H20; Secondary 58D25
MathSciNet review: 978369
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Abstract: Nonlinear continuous perturbations of linear dissipative operators are considered from the point of view of the nonlinear semigroup theory. A general class of nonlinear perturbations of linear contraction semigroups in a Banach space $ X$ is introduced by means of a lower semicontinuous convex functional $ [{\text{unk}}]:X \to [0,\infty ]$ and two notions of semilinear infinitesimal generators of the associated nonlinear semigroups are formulated. Four types of necessary and sufficient conditions are given for a semilinear operator $ A + B$ of the class to be the infinitesimal generator of a nonlinear semigroup $ \{ S(t):t \geqslant 0\} $ on the domain $ C$ of $ B$ such that for $ x \in C$ the $ C$-valued function $ S( \cdot )x$ on $ [0,\infty )$ provides a unique mild solution of the semilinear evolution equation $ u'(t) = (A + B)u(t)$ satisfying a growth condition for the function $ [{\text{unk]}}(u( \cdot ))$. It turns out that various types of characterizations of nonlinear semigroups associated with semilinear evolution equations are obtained and, in particular, a semilinear version of the Hille-Yosida theorem is established in a considerably general form.

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Keywords: Nonlinear perturbations of linear operators, semilinear evolution equation, mild solution, nonlinear semigroup, full infinitesimal generator, range condition, local quasi-dissipativity
Article copyright: © Copyright 1989 American Mathematical Society

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