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On the application of substochastic semigroup theory to fragmentation models with mass loss. (English) Zbl 1059.47046

In this paper, the theory of semigroups and evolution equations is applied to the integro-differential initial value problem \[ \partial_t u(x,t) = -a(x)u(x,t) + \int_x^\infty a(y) b(x| y) u(y,t)\,dy + \partial_x [r(x)u(x,t)], \qquad u(x,0) = u_0(x), \] modeling fragmentation with mass loss, where \(u(x,t)\) is the time-dependent number density of particles with mass \(x\). The above i.v.p. is recast into an abstract Cauchy problem \[ \dot u(t) = (A+B) u(t), \qquad u(0) = u_0, \] in the space \(L^1([0,\infty), x\,dx)\), where \(Au\) is formally given by the loss term \(-au + \partial_x [ru]\) and \(Bu\) is formally given by the integral gain term. Since the non-negative functions \(a\), \(b\) and \(r\) are not assumed to be bounded, all the operators involved are unbounded.
The authors prove that \(A\), defined on a suitable domain \({\mathcal D}(A)\) which is explicitly described, generates a substochastic semigroup (i.e., a positive \(C_0\)-semigroup). Then, generalizing previous results of J. Voigt [Transp. Theory Stat. Phys. 16, 453–466 (1987; Zbl 0634.47040)], L. Arlotti [Acta Appl. Math. 23, 129–144 (1991; Zbl 0734.45005)] and J. Banasiak [Taiwanese J. Math. 5, 169–191 (2001; Zbl 1002.47021)], they are able to prove that the closure \(\overline{A+B}\) of \(A+B\) still generates a substochastic semigroup and, moreover, that the formal mass-rate equation \[ {d \over dt} \int_0^\infty u(x,t)x\,dx = -\int_0^\infty a(x)\lambda(x) u(x,t) x\,dx - \int_0^\infty r(x) u(x,t)\,dx, \] where \(\lambda\) is defined by \(\int_0^y x b(x| y)\,dx = y - \lambda(y)y\), is rigorously satisfied if \(u_0 \in {\mathcal D}(\overline{A+B})^+\).
Dealing with the subtle question of unbounded perturbations, this very interesting paper may be recommended (in spite of its “applied” motivations) also to theoretically-oriented semigroup people.

MSC:

47D06 One-parameter semigroups and linear evolution equations
45K05 Integro-partial differential equations
47A55 Perturbation theory of linear operators
45M05 Asymptotics of solutions to integral equations
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