American Institute of Mathematical Sciences

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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

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

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