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Traveling wave solutions for some density dependent diffusion equations. (English) Zbl 0612.35069

Existence and stability of monotone traveling wave solutions of the equation \[ u_ t=(u^ m)_{xx}+f(u),\quad m>1 \] and f(u) vanishes at the three points \(0<\alpha <1\). The asymptotic behaviour of the solutions of the initial value problem is also investigated.
Reviewer: G.Boillat

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
92D25 Population dynamics (general)
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[1] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems”. Israel Program of Scientific Translation, Jerusalem, 1973. · Zbl 0282.34022
[2] D. G. Aronson, Density dependent interaction-diffusion systems. Dynamics and Modeling of Reactive Systems, Academic Press, New York, 1980, 161–176.
[3] D. G. Aronson, M. G. Crandall and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal.,6 (1982), 1001–1022. · Zbl 0518.35050 · doi:10.1016/0362-546X(82)90072-4
[4] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation. Partial Differential Equations and Related Topics, Lecture Notes in Math., vol. 446, Springer, 1975, 5–49. · Zbl 0325.35050
[5] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics. Advances in Math.,30 (1978), 33–76. · Zbl 0407.92014 · doi:10.1016/0001-8708(78)90130-5
[6] C. Atkinson, G. E. H. Reuter and C. J. Ridler-Rowe, Traveling wave solutions for some nonlinear diffusion equations. SIAM J. Math. Anal.,12 (1981), 880–892. · Zbl 0471.35042 · doi:10.1137/0512074
[7] M. Bertsch, R. Kersner and L. A. Peletier, Positivity versus localization in degenerate diffusion equations. Math. Inst. Univ. Leiden, The Netherlands, No. 3, 1983. · Zbl 0596.35073
[8] H. Engler, Relations between traveling wave solutions of quasilinear parabolic equations. Proc. Amer. Math. Soc.,93 (1985), 297–302. · Zbl 0535.35042 · doi:10.1090/S0002-9939-1985-0770540-6
[9] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear equations to travelling front solutions. Arch. Rational Mech. Anal.,65 (1977), 335–361. · Zbl 0361.35035 · doi:10.1007/BF00250432
[10] W. S. C. Gurney and R. M. Nisbet, The regulation of inhomogeneous populations. J. Theoret. Biol.,52 (1975), 441–457. · doi:10.1016/0022-5193(75)90011-9
[11] M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations. Math. Biosci.,33 (1979), 35–49. · Zbl 0362.92007 · doi:10.1016/0025-5564(77)90062-1
[12] P. S. Hagan, Traveling wave and multiple traveling wave solutions of parabolic equations. SIAM J. Math. Anal.,13 (1982), 717–738. · Zbl 0504.35050 · doi:10.1137/0513049
[13] P. Hartman, Ordinary Differential Equations. Wiley, New York, 1973. · Zbl 0281.34001
[14] Y. Hosono, Traveling wave front solutions for some competitive systems with density dependent diffusion. Computational and Asymptotic Methods for Boundary and Interior Layers, Boole Press, Dublin, 1982, 285–290.
[15] Ya. I. Kanel’, On the stabilization of solutions of Cauchy problem for the equations arising in the theory of combustion. Mat. Sb.,59 (1962), 245–288.
[16] R. Kersner, Nonlinear heat conduction with absorption: space localization and extinction in finite time. SIAM J. Appl. Math.,43 (1983), 1275–1285. · Zbl 0536.35039 · doi:10.1137/0143085
[17] T. Nagai and M. Mimura, Asymptotic behavior for a nonlinear degenerate diffusion equation in population dynamics. SIAM J. Appl. Math.,43 (1983), 449–464. · Zbl 0554.35060 · doi:10.1137/0143029
[18] W. I. Newman, Some exact solutions to a non-linear diffusion problem in population genetics and combustion. J. Theoret. Biol.,85 (1980), 325–334. · doi:10.1016/0022-5193(80)90024-7
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