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

DOI:
https://doi.org/10.1090/S0002-9947-1989-0978369-9

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 is introduced by means of a lower semicontinuous convex functional 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 of the class to be the infinitesimal generator of a nonlinear semigroup on the domain of such that for the -valued function on provides a unique mild solution of the semilinear evolution equation satisfying a growth condition for the function . 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.

**[1]**J. Ball,*Strongly continuous semigroups, weak solutions, and the variation of constants formula*, Proc. Amer. Math. Soc.**63**(1977), 370-373. MR**0442748 (56:1128)****[2]**M. Crandall and T. Liggett,*Generation of semigroups of nonlinear transformations on general Banach spaces*, Amer. J. Math.**93**(1971), 265-298. MR**0287357 (44:4563)****[3]**E. Hille and R. Phillips,*Functional analysis and semi-groups*, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957. MR**0089373 (19:664d)****[4]**T. Iwamiya,*Global existence of mild solutions to semilinear differential equations in Banach spaces*, Hiroshima Math. J.**16**(1986), 499-530. MR**867577 (88a:34098)****[5]**T. Kato,*Perturbation theory for linear operators*, Springer-Verlag, New York, 1966. MR**0203473 (34:3324)****[6]**Y. Kobayashi,*Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups*, J. Math. Soc. Japan**27**(1975), 640-665. MR**0399974 (53:3812)****[7]**K. Kobayasi, Y. Kobayashi and S. Oharu,*Nonlinear evolution operators in Banach spaces*, Osaka J. Math.**21**(1984), 281-310. MR**752464 (85h:47073)****[8]**V. Lakshmikantham and S. Leela,*Differential and integral inequalities*, Academic Press, New York, 1969.**[9]**R. Martin, Jr.,*Invariant sets for perturbed semigroups of linear operators*, Ann. Mat. Pura Appl.**150**(1975), 221-239. MR**0390414 (52:11240)****[10]**-,*Nonlinear operators and differential equations in Banach spaces*, Wiley-Interscience, New York, 1976. MR**0492671 (58:11753)****[11]**S. Oharu and T. Takahashi,*Locally Lipschitz continuous perturbations of linear dissipative operators and nonlinear semigroups*, Proc. Amer. Math. Soc.**97**(1987), 139-145. MR**883426 (88d:47076)****[12]**N. Pavel,*Nonlinear evolution equations governed by*-*quasi-dissipative operators*, Nonlinear Anal.**5**(1981), 449-468. MR**613054 (82i:34073)****[13]**-,*Semilinear equations with dissipative*-*dependent domain perturbations*, Israel J. Math.**46**(1983), 103-122. MR**727025 (85b:47067)****[14]**-,*Differential equations, flow invariance and applications*, Pitman, London, 1984.**[15]**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Appl. Math. Sci., vol. 44, Springer-Verlag, New York, 1983. MR**710486 (85g:47061)****[16]**G. Webb,*Continuous nonlinear perturbations of linear accretive operators in Banach spaces*, J. Funct. Anal.**10**(1972), 191-203. MR**0361965 (50:14407)****[17]**K. Yosida,*Functional analysis*, Springer-Verlag, New York, 1968. MR**0239384 (39:741)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0978369-9

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