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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Quasilinear evolution equations in Banach spaces
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by Michael G. Murphy PDF
Trans. Amer. Math. Soc. 259 (1980), 547-557 Request permission

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.
References
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 259 (1980), 547-557
  • MSC: Primary 34G20; Secondary 47D05
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0567096-X
  • MathSciNet review: 567096