The Solution of an Evolution Equation Describing Certain Types of Mechanical and Chemical Interaction

Abstract

AMS (MOS): 45K05, 45G10

Consider the initial value problem (∗) where A is a certain quadratic integral operator which does not depend on t explicitly. The equation describes the evolution in time of the volume distribution, u , of an ensemble of particles undergoing concurrent coalescence and fracture. It is shown (∗) has a unique solution valid for all t⩾0 in the Banach spaces and X , the space of bounded Lebesgue measurable functions on [0, V₀] V₀ is the total ensemble volume. The solution satisfies u ⩾0 for all (or almost all) conserves total volume and depends continuously on u₀. While in general equations like (∗) do not possess solutions valid for all t ⩾ 0 , (∗) does precisely because of the non-negativity and volume conservation. The proof exploits an interesting inter-play between the two spaces. Both spaces must be considered to get the solution in either one.

Additional information

Notes on contributors

W. Walter

Kenneth L. Kuttler

John W Hilgers

Thomas H Courtney