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Asymptotic profiles for a traveling front solution of a biological equation. (English) Zbl 1252.35061
The existence of depolarization waves in the human brain is considered. For the modeling of the studied biological process, the equation \[ \frac{{\partial u}} {{\partial \tau}} = \Delta u + f(u)I_\Omega - \alpha uI_{\mathbb{R}^N \backslash \Omega } ,\quad \tau \in \mathbb{R},\;x \in \mathbb{R}^N \] is used, where \(f(u) = \lambda u(u - a)(1 - u)\) , \(a \in (0,1/2)\) and \(\lambda > 0\), \(\alpha > 0\) are numerical parameters; \(I_\Omega \) is the characteristic function of the domain \(\Omega \). It is assumed that \(\Omega \) is a straight cylinder of radius r. The influence of \(\Omega \) on the geometry of the propagation of the waves, the existence, the stability and the energy of nontrivial asymptotic profiles of the possible traveling fronts is studied. Applying dynamical systems’ techniques and graphic criteria based on Sturm-Liouville, trhee different types of behavior for the solution \(u(t, x)\) are established depending on the thickness of the grey matter. Some biological effects are demonstrated as results of numerical experiments.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K58 Semilinear parabolic equations
35C07 Traveling wave solutions
35B35 Stability in context of PDEs
35K57 Reaction-diffusion equations
92C20 Neural biology
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References:
[1] DOI: 10.1090/S0002-9904-1974-13349-5 · Zbl 0281.35010
[2] DOI: 10.1016/j.jde.2007.01.021 · Zbl 1129.35042
[3] DOI: 10.1080/03605300500300006 · Zbl 1087.35052
[4] DOI: 10.1006/jdeq.1997.3255 · Zbl 0994.34047
[5] DOI: 10.1006/jdeq.1996.0031 · Zbl 0849.35056
[6] DOI: 10.1007/978-3-662-05281-5_2
[7] DOI: 10.1016/0022-0396(91)90134-U · Zbl 0768.35048
[8] Gallay Th., Ann. Scient. l’Ecole Normale Supérieure 42 pp 103–
[9] Gallay Th., Differential Integral Equations 20 pp 901–
[10] DOI: 10.1016/S0165-0173(01)00081-9
[11] DOI: 10.1016/j.pbiomolbio.2007.10.002
[12] Hale J. K., Mathematical Survey, in: Asymptotic Behavior of Dissipative Systems (1988)
[13] Henry D., Lecture Notes in Mathematics, in: Geometric Theory of Semilinear Parabolic Equations (1981)
[14] DOI: 10.1016/S0166-2236(00)01793-8
[15] Leão A. A. P., J. Neurophysiol. 10 pp 359–
[16] DOI: 10.1016/0006-8993(94)90295-X
[17] DOI: 10.1016/S0006-8993(96)00874-8
[18] DOI: 10.1097/00001756-199306000-00027
[19] Obeidat A. S., J. Cereb. Blood Flow Metab. 20 pp 412–
[20] DOI: 10.1016/0010-4825(95)00051-8
[21] Revett K., J. Cereb. Blood Flow Metab. 18 pp 998–
[22] DOI: 10.1016/j.anihpc.2006.12.005 · Zbl 1152.35047
[23] DOI: 10.1016/j.jde.2008.02.046 · Zbl 1162.35044
[24] DOI: 10.1023/A:1008924227961 · Zbl 05967436
[25] DOI: 10.1007/978-1-4612-0873-0
[26] Somjen G., Ions in the Brain: Normal Function, Seizures, and Stroke (2004)
[27] Sramka M., Appl. Neurophysiol. 40 pp 48–
[28] DOI: 10.1016/S0006-3495(78)85447-2
[29] DOI: 10.1103/PhysRevE.74.021909
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