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Existence results for non‐autonomous multiple‐fragmentation models
Author(s) -
McLaughlin D. J.,
Lamb W.,
McBride A. C.
Publication year - 1997
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(199710)20:15<1313::aid-mma915>3.0.co;2-q
Subject(s) - mathematics , initial value problem , limit (mathematics) , scalar (mathematics) , cauchy problem , cauchy distribution , mathematical physics , function (biology) , fragmentation (computing) , mathematical analysis , combinatorics , pure mathematics , geometry , computer science , operating system , evolutionary biology , biology
We investigate an initial‐value problem modelling fragmentation processes where particles split into two or more pieces at a rate, γ, that not only depends on the sizes of the particles involved but also on time. The existence of non‐negative, mass‐conserving solutions is established by considering a truncated version of an associated non‐autonomous abstract Cauchy problem. The latter has solutions of the form u ( t )= U n ( t , t 0 ) f , t ⩾ t 0 , where f is the known data at some fixed time t 0 ⩾0 and { U n ( t , s )} t 0 ⩽ s ⩽ t ⩽ Tis a uniformly continuous evolution system. A limit evolution system { U ( t , s )} t 0 ⩽ s ⩽ t ⩽ Tis shown to exist. Depending on the form of the known data f at time t 0 , the scalar‐valued function u , obtained from the limit evolution system via u ( x , t )=[ U ( t , t 0 ) f ]( x ) for a.e. x >0, t ⩾ t 0 , is a solution of either the original initial‐value problem or an integral version of this problem. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.