We discuss and exploit the Carathéodory theorem on existence and uniqueness of an absolutely continuous solution x: ℐ (⊂ ℝ) → X of a general ODE for the right-hand side ℱ : dom ℱ ( ⊂ ℝ × X) → X taking values in an arbitrary Banach space X, and a related result concerning an extension of x. We propose a definition of solvability of (*) admitting all connected ℐ and unifying the cases “dom ℱ is open” and “dom ℱ = ℐ × Ω for some Ω ⊂ X”. We show how to use the theorems mentioned above to get approximate solutions of a nonlinear parabolic PDE and exact solutions of a linear evolution PDE with distribution data.
@article{bwmeta1.element.bwnjournal-article-apmv73z1p1bwm,
author = {Holly, Konstanty and Orewczyk, Joanna},
title = {Applications of the Carath\'eodory theorem to PDEs},
journal = {Annales Polonici Mathematici},
volume = {75},
year = {2000},
pages = {1-27},
zbl = {0968.34042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv73z1p1bwm}
}
Holly, Konstanty; Orewczyk, Joanna. Applications of the Carathéodory theorem to PDEs. Annales Polonici Mathematici, Tome 75 (2000) pp. 1-27. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv73z1p1bwm/
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