Interplay between degenerate convergence of semigroups and asymptotic analysis: A study of a singularly perturbed abstract telegraph system

Journal of Evolution Equations

Volumn 9, Issue 2, 2009, Pages 293-314

  Banasiak, Jacek ^a,b   Bobrowski, Adam ^c  

^a UNIVERSITY OF KWAZULU NATAL   (South Africa)

^b LODZ UNIVERSITY OF TECHNOLOGY   (Poland)

^c INSTITUTE OF MATHEMATICS   (Poland)

Author keywords

Asymptotic analysis; Extended limits of generators; Hydrodynamic approximation; Linear evolution systems; Semigroups; Trotter Kato theorem

Indexed keywords

EID: 67349102044     PISSN: 14243199     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00028-009-0009-7     Document Type: Article

References (29)

1. Arendt, W., Batty, C.J.K., Hieber, M. and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser, 2001.
2. J. Banasiak 1997 Singularly perturbed linear and semilinear hyperbolic systems: kinetic theory approach to some folk's theorems Acta Applicandae Mathematicae 49 199 228 (Pubitemid 127714693)
3. J. Banasiak J.R. Mika 1998 Singularly perturbed telegraph equations with applications in the random walk theory J. Appl. Math. Stochastic Anal. 11 1 9 28
4. J. Banasiak 1996 Asymptotic analysis of abstract linear kinetic equation Math. Methods Appl. Sci. 19 6 481 505 (Pubitemid 126577076)
5. J. Banasiak 1997 Diffusion approximation of the linear Boltzmann equationwith and without an external field: Analysis of initial layer J. Math. Anal. Appl. 205 1 216 238 (Pubitemid 127171314)
6. J. Banasiak G. Frosali G. Spiga 2002 An interplay between elastic and inelastic scattering in models of extended kinetic theory and their hydrodynamic limits-Reference manual Transport Theory Statist. Phys. 31 3 187 248
7. A. Bobrowski 1994 Degenerate Convergence of Semigroups Semigroup Forum 49 303 327
8. Bobrowski, A., ANote onConvergence of Semigroups, Ann. Polon. Math. LXIX.2 (1998), 107-127.
9. Bobrowski, A., Functional Analysis forProbability and Stochastic Processes, Cambridge University Press 2005.
10. Bobrowski, A., Degenerate convergence of semigroups related to amodel of stochastic gene expression, Semigroup Forum 73 (2006), 345-366. (With correction in vol. 77.3)
11. A. Bobrowski 2007 On limitations and insufficiency of the Trotter-Kato theorem Semigroup Forum 75 317 336
12. A. Bobrowski 2007 On a semigroup generated by a convex combination of two Feller generators J. Evol. Equ. 7 555 565 (Pubitemid 47246700)
13. Engel, K.-J. and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer, 2000.
14. Ethier, S.N. and T.G. Kurtz, Markov Processes. Characterization and Convergence, Wiley, 1986.
15. Goldstein, J.A., Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs 1985.
16. Janssen, A., The distance between the Kac process and the Wiener process with applications to generalized telegraph equations, J. Theoret. Probab. 3(2), 1990, 349-360.
17. D.D. Joseph L. Preziosi 1989 Heat waves Rev. Modern Phys. 61 1 41 73
18. Kac, M., Some Stochastic Problems in Physics and Mechanics, Magnolia Petrolum Co. Colloq. Lect. 2, 1956.
19. Kato, T., Singular perturbation and semigroup theory. In: Turbululence and Navier-Stokes Equations Theory, Proc. Conf. Univ. Paris-Sud, Orsay 1975, Lect. Notes Math., Springer-Verlag, Berlin 565 (1976), 104-112.
20. Kato T.: Perturbation Theory for Linear Operators 2nd ed. Springer, Berlin (1984)
21. T.G. Kurtz 1969 Extensions of Trotter's operator semigroup approximation theorems J. Funct. Anal. 3 354 375
22. J.R. Mika 1977 Singularly perturbed evolution equations in Banach space J. Math. Anal. Apl. 58 1 189 201
23. J.R. Mika 1981 New asymptotic expansion algorithm for singularly perturbed evolution equations Math. Methods Appl. Sci. 3 2 172 188
24. Mika, J. R., Banasiak, J., Singularly Perturbed Evolution Equations with Applications to Kinetic Theory, Series on Advances in Mathematics for Applied Sciences, vol. 34,World Scientific, Singapore 1995.
25. Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer 1983.
26. Pinsky, M., Lectures on Random Evolutions, World Scientific 1991.
27. Salvarani, F., Diffusion limits for the initial-boundary value problem for the Goldstein-Taylormodel, Rend. Sem. Mat. Univ. Pol. Torino Vol., 57(3), (1999), 211-222.
28. M. Sova 1967 Convergence d'opérations lineaires non bornées Rev. Roum. Math. Pures et App. 12 373 389
29. Zauderer, E., Partial Differential Equations of Applied Mathematics, 2nd ed., Wiley, 1988.