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

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May 2014, 19(3): 801-815. doi: 10.3934/dcdsb.2014.19.801

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

Lina Wang ^1, , Xueli Bai ^1, and Yang Cao ^2,

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

Received July 2013 Revised November 2013 Published February 2014

Abstract
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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.

Keywords: exponential stability., spectral analysis, Traveling fronts, Evans function, singular perturbation method.

Mathematics Subject Classification: Primary: 35A18, 35B35; Secondary: 35B4.

Citation: Lina Wang, Xueli Bai, Yang Cao. Exponential stability of the traveling fronts for a viscous Fisher-KPP equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 801-815. doi: 10.3934/dcdsb.2014.19.801

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.

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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.

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