Article
MSC:
20M10,
35B40,
35F10,
35R20,
47D06,
82D75 | MR 3164576 | Zbl 06346372 | DOI: 10.1007/s10492-014-0041-y
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Keywords:
evolution equation; semi-group; transport equation; perturbed semigroups; spectral theory
evolution equation; semi-group; transport equation; perturbed semigroups; spectral theory
Summary:
In this paper, we study the time asymptotic behavior of the solution to an abstract Cauchy problem on Banach spaces without restriction on the initial data. The abstract results are then applied to the study of the time asymptotic behavior of solutions of an one-dimensional transport equation with boundary conditions in $L_1$-space arising in growing cell populations and originally introduced by M. Rotenberg, J. Theoret. Biol. 103 (1983), 181–199.
References:
[1] Abdelmoumen, B., Jeribi, A., Mnif, M.: Time asymptotic description of the solution to an abstract Cauchy problem and application to transport equation. Math. Z. 268 (2011), 837-869. DOI 10.1007/s00209-010-0698-1 | MR 2818733 | Zbl 1231.47010
[2] Amar, A. Ben, Jeribi, A., Mnif, M.: Some applications of the regularity and irreducibility on transport theory. Acta Appl. Math. 110 (2010), 431-448. DOI 10.1007/s10440-009-9430-8 | MR 2601665
[3] Boulanouar, M.: A Rotenberg's model with perfect memory rule. C. R. Acad. Sci., Paris, Sér. I, Math. 327 (1998), 965-968 French. MR 1659158 | Zbl 0914.92014
[4] Dunford, N., Pettis, B. J.: Linear operations on summable functions. Trans. Am. Math. Soc. 47 (1940), 323-392. DOI 10.1090/S0002-9947-1940-0002020-4 | MR 0002020 | Zbl 0023.32902
[5] Dunford, N., Schwartz, J. T.: Linear Operators I. General Theory. Pure and Applied Mathematics 6 Interscience Publishers, New York (1958). MR 0117523 | Zbl 0084.10402
[6] Greenberg, W., Mee, C. van der, Protopopescu, V.: Boundary Value Problems in Abstract Kinetic Theory. Operator Theory: Advances and Applications 23 Birkhäuser, Basel (1987). MR 0896904
[7] Hille, E., Phillips, R. S.: Functional Analysis and Semi-Groups. rev. ed. Colloquium Publications 31 American Mathematical Society, Providence (1957); Functional Analysis and Semi-Groups. 3rd printing of rev. ed. of 1957 Colloquium Publications 31 American Mathematical Society, Providence (1974). MR 0089373
[8] Jeribi, A.: A characterization of the essential spectrum and applications. Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. 8 (2002), 805-825. MR 1934382 | Zbl 1099.47501
[9] Jeribi, A.: Time asymptotic behaviour for unbounded linear operators arising in growing cell populations. Nonlinear Anal., Real World Appl. 4 (2003), 667-688. MR 1978556 | Zbl 1039.92033
[10] Kato, T.: Perturbation Theory for Linear Operators. 2nd ed. Grundlehren der mathematischen Wissenschaften 132 Springer, Berlin (1976). MR 0407617 | Zbl 0342.47009
[11] Lods, B.: On linear kinetic equations involving unbounded cross-sections. Math. Methods Appl. Sci. 27 (2004), 1049-1075. DOI 10.1002/mma.485 | MR 2063095 | Zbl 1072.37062
[12] Mokhtar-Kharroubi, M.: Time asymptotic behaviour and compactness in transport theory. European J. Mech. B Fluids 11 (1992), 39-68. MR 1151539
[13] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44 Springer, New York (1983). MR 0710486 | Zbl 0516.47023
[14] Rotenberg, M.: Transport theory for growing cell populations. J. Theor. Biol. 103 (1983), 181-199. DOI 10.1016/0022-5193(83)90024-3 | MR 0713945
[15] Smul'yan, Y. L.: Completely continuous perturbations of operators. Dokl. Akad. Nauk SSSR (N.S.) 101 (1955), 35-38 Russian; Completely continuous perturbations of operators. Translat. by M. G. Arsove Am. Math. Soc., Transl., II. Ser. 10 (1958), 341-344. Zbl 0080.10603
[16] Vidav, I.: Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator. J. Math. Anal. Appl. 22 (1968), 144-155. DOI 10.1016/0022-247X(68)90166-2 | MR 0230531 | Zbl 0155.19203
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