The solution of an evolution equation describing certain types of mechanical and chemical interaction

Document Type

Article

Publication Date

5-1-1985

Abstract

Consider the initial value problem (*) ϑu/ϑt = Au, u(0) = uo 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 L1[0, Vo] and X, the space of bounded Lebesgue measurable functions on [0, Vo]. Vo is the total ensemble volume. The solution satisfies u 0 for all (or almost all) x ∊ [0, Vo], conserves total volume and depends continuously on uo. 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 interplay between the two spaces. Both spaces must be considered to get the solution in either one. © 1985, Taylor & Francis Group, LLC. All rights reserved.

Publication Title

Applicable Analysis

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