Arlotti, L.; Banasiak, J. Nonautonomous fragmentation equation via evolution semigroups. (English) Zbl 1213.47045 Math. Methods Appl. Sci. 33, No. 10, 1201-1210 (2010). The following Cauchy problem for a nonautonomous multiple-fragmentation equation is treated in this paper:\[ u_t(t,x)=-a(t,x)u(t,x)+\int_x^\infty a(t,y)b(t,x,y)u(t,y)\,dy, \quad t>s, \text{ a.e. } x>0, \]with the initial value \(u(s,x)=f(x)\), a.e. \(x>0\). The main tool is the evolution semigroup. Reviewer: Jin Liang (Shanghai) Cited in 4 Documents MSC: 47D06 One-parameter semigroups and linear evolution equations 34G10 Linear differential equations in abstract spaces 35R09 Integro-partial differential equations 47A55 Perturbation theory of linear operators Keywords:evolution family; substochastic semigroup; perturbation family; integro-partial differential equation; nonautonomous multiple-fragmentation equation PDFBibTeX XMLCite \textit{L. Arlotti} and \textit{J. Banasiak}, Math. Methods Appl. Sci. 33, No. 10, 1201--1210 (2010; Zbl 1213.47045) Full Text: DOI References: [1] McLaughlin, Existence results for non-autonomous multiple-fragmentation models, Mathematical Methods in the Applied Sciences 20 pp 1313– (1997) · Zbl 0907.45008 [2] Arlotti L Lods B Mokhtar-Kharroubi M Honesty results for positive evolution families in abstract state spaces · Zbl 1296.47036 [3] Liskevich, Gaussian bounds for propagators perturbed by potentials, Journal of Functional Analysis 238 (1) pp 245– (2006) · Zbl 1104.47043 [4] Räbiger, Non-autonomous Miyadera perturbations, Differential Integral Equations 13 (1-3) pp 341– (2000) · Zbl 0980.34056 [5] Chicone, Evolution Semigroups in Dynamical Systems and Differential Equations (1999) · Zbl 0970.47027 · doi:10.1090/surv/070 [6] Banasiak, Perturbations of Positive Semigroups with Applications (2006) [7] Webb, Theory of Nonlinear Age-dependent Population Dynamics (1985) · Zbl 0555.92014 [8] Royden, Real Analysis (1968) [9] Arendt, Vector-Valued Laplace Transforms and Cauchy Problems (2001) · Zbl 0978.34001 · doi:10.1007/978-3-0348-5075-9 [10] Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (1983) · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1 [11] Banasiak J Goswami A Shindin S Aggregation in age and space structured population models-asymptotic analysis approach · Zbl 1241.35206 [12] Hille, Colloquium Publications, in: Functional Analysis and Semi-groups (1957) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.