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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quasilinear evolution equations in Banach spaces


Author: Michael G. Murphy
Journal: Trans. Amer. Math. Soc. 259 (1980), 547-557
MSC: Primary 34G20; Secondary 47D05
MathSciNet review: 567096
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Abstract: This paper is concerned with the quasi-linear evolution equation $ u'(t)\, + \,A(t,\,u(t))u(t)\, = \,0$ in $ [0,\,T],\,u(0)\, = \,{x_0}$ in a Banach space setting. The spirit of this inquiry follows that of T. Kato and his fundamental results concerning linear evolution equations. We assume that we have a family of semigroup generators that satisfies continuity and stability conditions. A family of approximate solutions to the quasi-linear problem is constructed that converges to a ``limit solution.'' The limit solution must be the strong solution if one exists. It is enough that a related linear problem has a solution in order that the limit solution be the unique solution of the quasi-linear problem. We show that the limit solution depends on the initial value in a strong way. An application and the existence aspect are also addressed.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0567096-X
PII: S 0002-9947(1980)0567096-X
Keywords: Quasi-linear evolution equations, Banach space, evolution operator, strongly continuous semigroup
Article copyright: © Copyright 1980 American Mathematical Society