Exponential stability of the traveling fronts fora viscous Fisher-KPP equation

Lina Wang; Xueli Bai; Yang Cao

doi:10.3934/dcdsb.2014.19.801

Article Contents

2014, Volume 19, Issue 3: 801-815. Doi: 10.3934/dcdsb.2014.19.801

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Exponential stability of the traveling fronts for a viscous Fisher-KPP equation

1.
Center for PDE, East China Normal University, Shanghai, 200241
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024

Received: June 30, 2013

Revised: October 31, 2013

Published: April 2014

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Abstract

This paper is concerned with the stability of traveling front solutions for a viscous Fisher-KPP equation. By applying geometric singular perturbation method, special Evans function estimates, detailed spectral analysis and $C_0$ semigroup theories, each traveling front solution with wave speed $c<-2\sqrt{f^\prime(0)}$ is proved to be locally exponentially stable in some appropriate exponentially weighted spaces.

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Mathematics Subject Classification: Primary: 35A18, 35B35; Secondary: 35B40.

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References

[1]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. in Math., 30 (1978), 33-76.doi: 10.1016/0001-8708(78)90130-5.
[2]	G. Barenblatt, I. Zheltov and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303.doi: 10.1016/0021-8928(60)90107-6.
[3]	M. Bramson, Convergence of Solutions of the Kolmogorov Equation to Traveling Waves, Mem. Amer. Math. Soc., 44 (1983), No. 285, iv+190.doi: 10.1090/memo/0285.
[4]	C. M. Cuesta, Linear stability analysis of travelling waves for a pseudo-parabolic Burgers' equation, Dyn. Partial Differ. Equ, 7 (2010), 77-105.doi: 10.4310/DPDE.2010.v7.n1.a5.
[5]	C. J. van Duijn, L. A. Peletier and I. S. Pop, A new class of entropy solutions of the Buckley-Leverett equation, SIAM J. Math. Anal., 39 (2007), 507-536.doi: 10.1137/05064518X.
[6]	C. J. van Duijn, L. A. Peletier and I. S. Pop, Travelling wave solutions for degenerate pseudo-parabolic equations modelling two-phase flow in porous media, Nonlinear Anal. Real World Appl., 14 (2013), 1361-1383.doi: 10.1016/j.nonrwa.2012.10.002.
[7]	N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.doi: 10.1016/0022-0396(79)90152-9.
[8]	R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369.doi: 10.1111/j.1469-1809.1937.tb02153.x.
[9]	R. Gardner and C. Jones, Traveling waves of a perturbed diffusion equation arising in a phase field model, Indiana U. Math. J., 39 (1990), 1197-1222.doi: 10.1512/iumj.1990.39.39054.
[10]	F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions, J. Differential Equations, 249 (2010), 1726-1745.doi: 10.1016/j.jde.2010.06.025.
[11]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
[12]	F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
[13]	C. K. R. T. Jones, Geometric singular perturbation theories, in Dynamical systems Lecture Notes in Mathematics 1609, Springer-Verlag, Berlin, (1995), 44-118.doi: 10.1007/BFb0095239.
[14]	A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou Sér. Inter. A, 1 (1937), 1-26.
[15]	P. Lafitte and C. Mascia, Numerical exploration of a forward-backward diffusion equation, Math. Models Methods Appl. Sci., 22 (2012), 1250004, 33 pages.doi: 10.1142/S0218202512500042.
[16]	K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov, J. Differential Equations, 59 (1985), 44-70.doi: 10.1016/0022-0396(85)90137-8.
[17]	C. Mascia, A. Terracina and A. Tesei, Two-phase entropy solutions of a forward-backward parabolic equation, Arch. Ration. Mech. Anal., 194 (2009), 887-925.doi: 10.1007/s00205-008-0185-6.
[18]	V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756.doi: 10.1090/S0002-9947-03-03340-3.
[19]	R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349.doi: 10.1007/BF02101705.
[20]	D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.doi: 10.1016/0001-8708(76)90098-0.
[21]	R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math.Anal., 1 (1970), 1-26.doi: 10.1137/0501001.
[22]	A. Terracina, Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem, SIAM J. Math.Anal., 43 (2011), 228-252.doi: 10.1137/090778833.
[23]	K. Uchiyama, The behaviour of solutions of some non-linear diffusion equation for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.
[24]	L. N. Wang, Y. P. Wu and T. Li, Exponential stability of large-amplitude traveling fronts for quasi-linear relaxation systems with diffusion, Physica D., 240 (2011), 971-983.doi: 10.1016/j.physd.2011.02.003.