American Institute of Mathematical Sciences

Advanced
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

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 & 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.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[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.  doi: 10.1016/0021-8928(60)90107-6.  Google Scholar

[3]

M. Bramson, Convergence of Solutions of the Kolmogorov Equation to Traveling Waves,, Mem. Amer. Math. Soc., 44 (1983).  doi: 10.1090/memo/0285.  Google Scholar

[4]

C. M. Cuesta, Linear stability analysis of travelling waves for a pseudo-parabolic Burgers' equation,, Dyn. Partial Differ. Equ, 7 (2010), 77.  doi: 10.4310/DPDE.2010.v7.n1.a5.  Google Scholar

[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.  doi: 10.1137/05064518X.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2012.10.002.  Google Scholar

[7]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, J. Differential Equations, 31 (1979), 53.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[8]

R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 335.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[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.  doi: 10.1512/iumj.1990.39.39054.  Google Scholar

[10]

F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions,, J. Differential Equations, 249 (2010), 1726.  doi: 10.1016/j.jde.2010.06.025.  Google Scholar

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).   Google Scholar

[12]

F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43.   Google Scholar

[13]

C. K. R. T. Jones, Geometric singular perturbation theories,, in Dynamical systems Lecture Notes in Mathematics 1609, 1609 (1995), 44.  doi: 10.1007/BFb0095239.  Google Scholar

[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.   Google Scholar

[15]

P. Lafitte and C. Mascia, Numerical exploration of a forward-backward diffusion equation,, Math. Models Methods Appl. Sci., 22 (2012).  doi: 10.1142/S0218202512500042.  Google Scholar

[16]

K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov,, J. Differential Equations, 59 (1985), 44.  doi: 10.1016/0022-0396(85)90137-8.  Google Scholar

[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.  doi: 10.1007/s00205-008-0185-6.  Google Scholar

[18]

V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation,, Trans. Amer. Math. Soc., 356 (2004), 2739.  doi: 10.1090/S0002-9947-03-03340-3.  Google Scholar

[19]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves,, Comm. Math. Phys., 164 (1994), 305.  doi: 10.1007/BF02101705.  Google Scholar

[20]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems,, Adv. Math., 22 (1976), 312.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[21]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations,, SIAM J. Math.Anal., 1 (1970), 1.  doi: 10.1137/0501001.  Google Scholar

[22]

A. Terracina, Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem,, SIAM J. Math.Anal., 43 (2011), 228.  doi: 10.1137/090778833.  Google Scholar

[23]

K. Uchiyama, The behaviour of solutions of some non-linear diffusion equation for large time,, J. Math. Kyoto Univ., 18 (1978), 453.   Google Scholar

[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.  doi: 10.1016/j.physd.2011.02.003.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. in Math., 30 (1978), 33.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[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.  doi: 10.1016/0021-8928(60)90107-6.  Google Scholar

[3]

M. Bramson, Convergence of Solutions of the Kolmogorov Equation to Traveling Waves,, Mem. Amer. Math. Soc., 44 (1983).  doi: 10.1090/memo/0285.  Google Scholar

[4]

C. M. Cuesta, Linear stability analysis of travelling waves for a pseudo-parabolic Burgers' equation,, Dyn. Partial Differ. Equ, 7 (2010), 77.  doi: 10.4310/DPDE.2010.v7.n1.a5.  Google Scholar

[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.  doi: 10.1137/05064518X.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2012.10.002.  Google Scholar

[7]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, J. Differential Equations, 31 (1979), 53.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[8]

R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 335.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[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.  doi: 10.1512/iumj.1990.39.39054.  Google Scholar

[10]

F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions,, J. Differential Equations, 249 (2010), 1726.  doi: 10.1016/j.jde.2010.06.025.  Google Scholar

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).   Google Scholar

[12]

F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43.   Google Scholar

[13]

C. K. R. T. Jones, Geometric singular perturbation theories,, in Dynamical systems Lecture Notes in Mathematics 1609, 1609 (1995), 44.  doi: 10.1007/BFb0095239.  Google Scholar

[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.   Google Scholar

[15]

P. Lafitte and C. Mascia, Numerical exploration of a forward-backward diffusion equation,, Math. Models Methods Appl. Sci., 22 (2012).  doi: 10.1142/S0218202512500042.  Google Scholar

[16]

K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov,, J. Differential Equations, 59 (1985), 44.  doi: 10.1016/0022-0396(85)90137-8.  Google Scholar

[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.  doi: 10.1007/s00205-008-0185-6.  Google Scholar

[18]

V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation,, Trans. Amer. Math. Soc., 356 (2004), 2739.  doi: 10.1090/S0002-9947-03-03340-3.  Google Scholar

[19]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves,, Comm. Math. Phys., 164 (1994), 305.  doi: 10.1007/BF02101705.  Google Scholar

[20]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems,, Adv. Math., 22 (1976), 312.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[21]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations,, SIAM J. Math.Anal., 1 (1970), 1.  doi: 10.1137/0501001.  Google Scholar

[22]

A. Terracina, Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem,, SIAM J. Math.Anal., 43 (2011), 228.  doi: 10.1137/090778833.  Google Scholar

[23]

K. Uchiyama, The behaviour of solutions of some non-linear diffusion equation for large time,, J. Math. Kyoto Univ., 18 (1978), 453.   Google Scholar

[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.  doi: 10.1016/j.physd.2011.02.003.  Google Scholar

[1]

Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885
[2]

Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations & Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1
[3]

Hua Chen, Ling-Jun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic & Related Models, 2012, 5 (2) : 261-281. doi: 10.3934/krm.2012.5.261
[4]

Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347
[5]

Peter Howard, K. Zumbrun. The Evans function and stability criteria for degenerate viscous shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 837-855. doi: 10.3934/dcds.2004.10.837
[6]

Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857
[7]

Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783
[8]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939
[9]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
[10]

Hongmei Cheng, Rong Yuan. Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 3007-3022. doi: 10.3934/dcdsb.2017160
[11]

Zhen-Hui Bu, Zhi-Cheng Wang. Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations Ⅰ. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2395-2430. doi: 10.3934/dcds.2017104
[12]

Hirokazu Ninomiya, Masaharu Taniguchi. Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 819-832. doi: 10.3934/dcds.2006.15.819
[13]

Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241
[14]

Roberto Triggiani, Jing Zhang. Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay. Evolution Equations & Control Theory, 2018, 7 (1) : 153-182. doi: 10.3934/eect.2018008
[15]

Je-Chiang Tsai. Global exponential stability of traveling waves in monotone bistable systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601
[16]

Weiran Sun, Min Tang. A relaxation method for one dimensional traveling waves of singular and nonlocal equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1459-1491. doi: 10.3934/dcdsb.2013.18.1459
[17]

Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405
[18]

Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941
[19]

Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115
[20]

Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences