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Kinetic-type models with diffusion: conservative and nonconservative solutions. (English) Zbl 1147.82344

Summary: In some cases mathematical models of physical or biological phenomena do not return the laws of nature used to build them. Well-known examples of this type appear in fragmentation-coagulation theory or in birth-and-death processes, as well as in some branches of transport theory. In these examples models based on the principle of conservation of mass (individuals, or particles) have solutions that are not conservative. In this paper we consider such models, augmented by diffusion in the physical space, and show that the diffusive part does not affect the breach of the conservation laws.

MSC:

82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
35K57 Reaction-diffusion equations
82C40 Kinetic theory of gases in time-dependent statistical mechanics
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References:

[1] DOI: 10.1007/BF02774019 · Zbl 0535.35017
[2] DOI: 10.1007/BF00048802 · Zbl 0734.45005
[3] DOI: 10.1016/j.jmaa.2004.01.028 · Zbl 1075.47023
[4] DOI: 10.1142/S0218202506001595 · Zbl 1139.47312
[5] Banasiak J., Taiwanese J. Math. 5 pp 169– (2001)
[6] Banasiak J., Mathematical Modelling of Population Dynamics 63 pp 165– (2004)
[7] DOI: 10.1142/S0218202504003325 · Zbl 1090.47026
[8] Banasiak J., Perturbations of Positive Semigroups with Applications (2006) · Zbl 1097.47038
[9] DOI: 10.1081/TT-120014800 · Zbl 1029.76049
[10] DOI: 10.1016/S0034-4877(03)90015-2 · Zbl 1068.82019
[11] DOI: 10.1016/S0022-247X(03)00154-9 · Zbl 1059.47046
[12] DOI: 10.2969/jmsj/02540565 · Zbl 0278.35041
[13] DOI: 10.1017/CBO9780511566158
[14] DOI: 10.1088/0305-4470/27/8/009 · Zbl 0834.45011
[15] DOI: 10.1016/S0304-4149(03)00045-0 · Zbl 1075.60553
[16] Henry D., Geometric Theory of Semilinear Equations (1981) · Zbl 0456.35001
[17] DOI: 10.1007/s002050100186 · Zbl 0997.45005
[18] DOI: 10.1006/jmaa.2001.7444 · Zbl 0986.35112
[19] Markowich P., Semiconductor Equations (1990)
[20] DOI: 10.1103/PhysRevLett.58.892
[21] DOI: 10.1137/S0036141095291701 · Zbl 0892.47044
[22] DOI: 10.1214/105051605000000386 · Zbl 1082.60075
[23] Walker C., Adv. Differential Equations 10 pp 121– (2005)
[24] DOI: 10.1088/0305-4470/18/15/026
[25] DOI: 10.1021/ma00164a010
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