"The solution of an evolution equation describing certain types of mech" by Kenneth L. Kuttler, John W. Hilgers et al.
 

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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