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Abstract Cauchy problems for quasi-linear evolution equations in the sense of Hadamard

Published online by Cambridge University Press:  30 June 2004

Naoki Tanaka
Naoki Tanaka
Affiliation:
Department of Mathematics, Faculty of Science, Okayama University, Okayama 700-8530, Japan. E-mail: tanaka@math.okayama-u.ac.jp
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Abstract

This paper is devoted to the well-posedness of abstract Cauchy problems for quasi-linear evolution equations. The notion of Hadamard well-posedness is considered, and a new type of stability condition is introduced from the viewpoint of the theory of finite difference approximations. The result obtained here generalizes not only some results on abstract Cauchy problems closely related with the theory of integrated semigroups or regularized semigroups but also the Kato theorem on quasi-linear evolution equations. An application to some quasi-linear partial differential equation of weakly hyperbolic type is also given.

Keywords

abstract Cauchy problem in the sense of Hadamardregularized semigroupabstract quasi-linear evolution equationstability conditionfinite difference approximation34G2047J25 (primary)47D6047D62 (secondary)
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 89 , Issue 1 , July 2004 , pp. 123 - 160
Copyright
2004 London Mathematical Society

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