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

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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.