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Abstract Cauchy Problems for Quasi‐Linear Evolution Equations in the Sense of Hadamard
Author(s) -
Tanaka Naoki
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611503014643
Subject(s) - mathematics , hadamard transform , cauchy problem , cauchy distribution , initial value problem , hyperbolic partial differential equation , type (biology) , stability (learning theory) , mathematics subject classification , pure mathematics , cauchy's integral theorem , partial differential equation , mathematical analysis , cauchy's integral formula , ecology , machine learning , computer science , biology
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. 2000 Mathematics Subject Classification 34G20, 47J25 (primary), 47D60, 47D62 (secondary).