Existence results for non-autonomous multiple-fragmentation models

Abstract

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)=Un(t,t0)f, t⩾t0, where f is the known data at some fixed time t0⩾0 and {Un(t,s)} italict 0⩽italics⩽italict⩽italicT is a uniformly continuous evolution system. A limit evolution system {U(t,s)} italict 0⩽italics⩽italict⩽italicT is shown to exist. Depending on the form of the known data f at time t0, the scalar‐valued function u, obtained from the limit evolution system via u(x, t)=[U(t, t0)f](x) for a.e. x>0, t⩾t0, 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.

Global Analysis of a Discrete Nonlocal and Nonautonomous Fragmentation Dynamics Occurring in a Moving Process

“…For this reason, the solution obtained in Theorem 10 can serve to approximate a solution for the Cauchy problem (3). The results obtained here, where we had to deal with a two parameter family of bounded linear operators, improve the preceding ones [2,3] where the two processes involved in the system, namely, transport and nonautonomous fragmentation, were treated separately. However, the problem of characterizing the full generator is still an open problem for this type of perturbed nonautonomous and transport model.…”

Discrete fragmentation equations with time-dependent coefficients

“…In contrast to the continuous fragmentation equation, where several investigations, such as [14] and [17], have dealt with time-dependent coefficients, previous investigations into (1.1) appear to have considered only the case when all fragmentation coefficients are time-independent. The approach used in [3,13,20] to analyse the constant-coefficient fragmentation system is to formulate the initial-value problem as an autonomous abstract Cauchy problem (ACP), posed in an appropriate Banach lattice.…”

“…In keeping with the semigroup approach used for the autonomous system, the strategy we adopt involves the application of the theory of evolution families, an account of which can be found in the seminal books on semigroups of operators by Goldstein [10] and Pazy [18]. Such families have been employed in the analysis of a variety of linear, non-autonomous evolution equations, such as the time-dependent coefficient versions of the continuous integro-differential fragmentation equation [14] and the Black-Scholes equation [19].…”

“…In [14], a slightly different, but equivalent, formulation of the initial-value problem (5.1) is posed as a non-autonomous ACP in the space L 1 (R + , x dx) (denoted by L 1,−1 in [14]). Only mass-conserving fragmentation is considered, and the fragmentation coefficients are assumed to satisfy the following conditions:…”

“…However, there is no corresponding result for a general ů ∈ L 1 (R + , x dx). Instead Some partial results on the uniqueness of the solution u(t) = U (t, s)ů, for the case when ů ∈ L 1 (R + , x dx) does not vanish on (n, ∞) for some n > 0, can be found in [15], where the notion of a weak solution is used. In particular, it is shown that u(t) = U (t, s)ů is the unique, non-negative, mass-conserving, weak solution of the non-autonomous ACP for any given non-negative initial data ů ∈ L 1 (R + , x dx), provided that the function b is independent of time, and a(x, t) = a 0 (x)α(t), with a 0 (x) ≤ C n on (0, n], and α a Lipschitz continuous function on [0, T ].…”