Applications of the Carathéodory theorem to PDEs
Volume 73 / 2000
Annales Polonici Mathematici 73 (2000), 1-27
DOI: 10.4064/ap-73-1-1-27
Abstract
We discuss and exploit the Carathéodory theorem on existence and uniqueness of an absolutely continuous solution x: ℐ (⊂ ℝ) → X of a general ODE $ẋ {(*)\over=} ℱ(t,x)$ 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.
