Strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss

Journal of Mathematical Analysis and Applications

Volumn 293, Issue 2, 2004, Pages 693-720

  Arlotti, Luisa ^a   Banasiak, Jacek ^b  

^a UNIVERSITY OF UDINE   (Italy)

^b UNIVERSITY OF KWAZULU NATAL   (South Africa)

Author keywords

Fragmentation models; Shattering transition; Substochastic semigroups

Indexed keywords

EID: 2442487952     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jmaa.2004.01.028     Document Type: Article

References (24)

1. M. Aizenman, T. Bak, Convergence to equilibrium in a system of reacting polymers, Comm. Math. Phys. 65 (1979) 203-230.
2. W.J. Anderson, Continuous-Time Markov Chains. An Applications-Oriented Approach, Springer-Verlag, New York, 1991.
3. L. Arlotti, The Cauchy problem for the linear Maxwell-Boltzmann equation, J. Differential Equations 69 (1987) 166-184.
4. 1-spaces with applications to transport equations and Kolmogorov's differential equations, Acta Appl. Math. 23 (1991) 129-144.
5. J. Banasiak, On a diffusion-kinetic equation arising in extended kinetic theory, Math. Methods Appl. Sci. 23 (2000) 1237-1256.
6. J. Banasiak, On an extension of Kato-Voigt perturbation theorem for substochastic semigroups and its applications, Taiwanese J. Math. 5 (2001) 169-191.
7. J. Banasiak, On a non-uniqueness in fragmentation models, Math. Methods Appl. Sci. 25 (2002) 541-556.
8. J. Banasiak, W. Lamb, On the application of substochastic semigroup theory to fragmentation models with mass loss, J. Math. Anal. Appl. 284 (2003) 9-30.
9. J. Banasiak, A complete description of dynamics generated by birth-and-death problems: a semigroup approach, in: R. Rudnicki (Ed.), Populations Dynamics. Proc. Conf. on Mathematical Modelling of Population +Dynamics, Bedlewo, Poland, June 24-28, 2002, Banach Center Publ. 63 (2004), in press.
10. J. Banasiak, Conservative and shattering solutions for some classes of fragmentation equations, Math. Models Methods Appl. Sci. 14 (2004), in press.
11. M. Cai, B.F. Edwards, H. Han, Exact and asymptotic scaling solutions for fragmentation with mass loss, Phys. Rev. A 43 (1991) 656-662.
12. B.F. Edwards, M. Cai, H. Han, Rate equation and scaling for fragmentation with mass loss, Phys. Rev. A 41 (1990) 5755-5757.
13. K.-J. Engel, R. Nagel, One-parameter Semigroups for Linear Evolution Equations, in: Grad. Texts in Math., Springer-Verlag, Berlin, 1999.
14. I. Filippov, On the distribution of the sizes of particles which undergo splitting, Theory Probab. Appl. 6 (1961) 275-293.
15. G. Frosali, C.V.M. van der Mee, S.L. Paveri-Fontana, Conditions for runaway phenomena in the kinetic theory of particle swarms, J. Math. Phys. 30 (1989) 1177-1186.
16. G. Frosali, C. van der Mee, F. Mugelli, A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators, Math. Methods Appl. Sci. 27 (2004) 669-686.
17. J. Huang, B.E. Edwards, A.D. Levine, General solutions and scaling violation for fragmentation with mass loss, J. Phys. A 24 (1991) 3967-3977.
18. I. Jeon, Stochastic fragmentation and some sufficient conditions for shattering transition, J. Korean Math. Soc. 39 (2002) 543-558.
19. T. Kato, On the semi-groups generated by Kolmogoroff's differential equations, J. Math. Soc. Japan 6 (1954) 1-15.
20. D.J. McLaughlin, W. Lamb, A.C. McBride, A semigroup approach to fragmentation models, SIAM J. Math. Anal. 28 (1997) 1158-1172.
21. E.D. McGrady, R.M. Ziff, "Shattering" transition in fragmentation, Phys. Rev. Lett. 58 (1987) 892-895.
22. J.R. Norris, Markov Chains, Cambridge Univ. Press, Cambridge, UK, 1998.
23. 0-semigroups and their generators, Transport Theory Statist. Phys. 16 (1987) 453-466.
24. R.M. Ziff, E.D. McGrady, Kinetics of polymer degradation, Macromolecules 19 (1986) 2513-2519.