Quasilinear evolution equations in Banach spaces

Murphy, Michael G.

doi:10.1090/S0002-9947-1980-0567096-X

Quasilinear evolution equations in Banach spaces
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by Michael G. Murphy

Trans. Amer. Math. Soc. 259 (1980), 547-557

DOI: https://doi.org/10.1090/S0002-9947-1980-0567096-X

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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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Functional analysis