American Institute of Mathematical Sciences

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October  2012, 32(10): 3621-3649. doi: 10.3934/dcds.2012.32.3621

Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity

Rui Huang 1, , Ming Mei 2, and  Yong Wang 3,

1. 

School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong, 510631, China
2. 

Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2
3. 

Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, 100190

Received  May 2011 Revised  December 2011 Published  May 2012

In this paper, we study a class of nonlocal dispersion equation with monostable nonlinearity in $n$-dimensional space \[ \begin{cases} u_t - J\ast u +u+d(u(t,x))= \displaystyle \int_{\mathbb{R}^n} f_\beta (y) b(u(t-\tau,x-y)) dy, \\ u(s,x)=u_0(s,x), \ \ s\in[-\tau,0], \ x\in \mathbb{R}^n, \end{cases} \] where the nonlinear functions $d(u)$ and $b(u)$ possess the monostable characters like Fisher-KPP type, $f_\beta(x)$ is the heat kernel, and the kernel $J(x)$ satisfies ${\hat J}(\xi)=1-\mathcal{K}|\xi|^\alpha+o(|\xi|^\alpha)$ for $0<\alpha\le 2$ and $\mathcal{K}>0$. After establishing the existence for both the planar traveling waves $\phi(x\cdot{\bf e}+ct)$ for $c\ge c_*$ ($c_*$ is the critical wave speed) and the solution $u(t,x)$ for the Cauchy problem, as well as the comparison principles, we prove that, all noncritical planar wavefronts $\phi(x\cdot{\bf e}+ct)$ are globally stable with the exponential convergence rate $t^{-n/\alpha}e^{-\mu_\tau t}$ for $\mu_\tau>0$, and the critical wavefronts $\phi(x\cdot{\bf e}+c_*t)$ are globally stable in the algebraic form $t^{-n/\alpha}$, and these rates are optimal. As application,we also automatically obtain the stability of traveling wavefronts to the classical Fisher-KPP dispersion equations. The adopted approach is Fourier transform and the weighted energy method with a suitably selected weight function.
Keywords: Fisher-KPP equation, time-delays, Nonlocal dispersion equations, global stability, Fourier transform., traveling waves, weighted energy.
Mathematics Subject Classification: Primary: 35K57, 34K20; Secondary: 92D2.
Citation: Rui Huang, Ming Mei, Yong Wang. Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3621-3649. doi: 10.3934/dcds.2012.32.3621
References:
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[2]

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F. Chen, Almost periodic traveling waves of nonlocal evolution equations,, Nonlinear Anal., 50 (2002), 807.  doi: 10.1016/S0362-546X(01)00787-8.  Google Scholar

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X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125.   Google Scholar

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F. Hamel and L. Roques, Uniqueness and stability of properties of monostable pulsating fronts,, J. European Math. Soc., 13 (2011), 345.  doi: 10.4171/JEMS/256.  Google Scholar

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L. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods,, J. Math. Pure Appl. (9), 92 (2009), 163.  doi: 10.1016/j.matpur.2009.04.009.  Google Scholar

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L. Ignat and J. D. Rossi, A nonlocal convolution-diffusion equation,, J. Func. Anal., 251 (2007), 399.  doi: 10.1016/j.jfa.2007.07.013.  Google Scholar

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D. Ya. Khusainov, A. F. Ivanov and I. V. Kovarzh, Solution of one heat equation with delay,, Nonlinear Oscillasions (N. Y.), 12 (2009), 260.  doi: 10.1007/s11072-009-0075-3.  Google Scholar

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K. Kirchgassner, On the nonlinear dynamics of travelling fronts,, J. Differential Equations, 96 (1992), 256.  doi: 10.1016/0022-0396(92)90153-E.  Google Scholar

[23]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l' équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bulletin Université d'Etat à Moscou, 1 (1937), 1.   Google Scholar

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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

[25]

G. Li, M. Mei and Y. S. Wong, Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model,, Math. Biosci. Engin., 5 (2008), 85.   Google Scholar

[26]

J.-F. Mallordy and J.-M. Roquejoffre, A parabolic equation of the KPP type in higher dimensions,, SIAM J. Math. Anal., 26 (1995), 1.  doi: 10.1137/S0036141093246105.  Google Scholar

[27]

M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. I. Local nonlinearity,, J. Differential Equations, 247 (2009), 495.  doi: 10.1016/j.jde.2008.12.026.  Google Scholar

[28]

M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. II. Nonlocal nonlinearity,, J. Differential Equations, 247 (2009), 511.  doi: 10.1016/j.jde.2008.12.020.  Google Scholar

[29]

M. Mei, J. W.-H. So, M. Li and S. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion,, Proc. Roy. Soc. Edinburgh Sec. A, 134 (2004), 579.  doi: 10.1017/S0308210500003358.  Google Scholar

[30]

M. Mei and J. W.-H. So, Stability of strong travelling waves for a non-local time-delayed reaction-diffusion equation,, Proc. Roy. Soc. Edinburgh Sec. A, 138 (2008), 551.  doi: 10.1017/S0308210506000333.  Google Scholar

[31]

M. Mei, C. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations,, SIAM J. Math. Anal., 42 (2010), 2762.  doi: 10.1137/090776342.  Google Scholar

[32]

M. Mei and Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations,, Int. J. Numer. Anal. Model. Seris B, 2 (2011), 379.   Google Scholar

[33]

M. Mei and Y. S. Wong, Novel stability results for traveling wavefronts in an age-structured reaction-diffusion equations,, Math. Biosci. Engin., 6 (2009), 743.  doi: 10.3934/mbe.2009.6.743.  Google Scholar

[34]

H. J. K. Moet, A note on the asymptotic behavior of solutions of the KPP equation,, SIAM J. Math. Anal., 10 (1979), 728.  doi: 10.1137/0510067.  Google Scholar

[35]

S. Pan, W.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay,, Nonlinear Anal., 72 (2010), 3150.  doi: 10.1016/j.na.2009.12.008.  Google Scholar

[36]

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

[37]

J. W.-H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure: I. Traveling wavefronts on unbounded domains,, Roy. Soc. London Proc. Series A Math. Phys. Eng. Sci., 457 (2001), 1841.  doi: 10.1098/rspa.2001.0789.  Google Scholar

[38]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time,, J. Math. Kyoto Univ., 18 (1978), 453.   Google Scholar

[39]

J. Wu, D. Wei and M. Mei, Analysis on the critical speed of traveling waves,, Appl. Math. Lett., 20 (2007), 712.  doi: 10.1016/j.aml.2006.08.006.  Google Scholar

[40]

H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation,, Publ. Res. Inst. Math. Sci., 45 (2009), 925.  doi: 10.2977/prims/1260476648.  Google Scholar

show all references

References:
[1]

P. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rational Mech. Anal., 138 (1997), 105.  doi: 10.1007/s002050050037.  Google Scholar

[2]

M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves,, Mem. Amer. Math. Soc., 44 (1983).   Google Scholar

[3]

E. Chasseigne, M. Chaves and J. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pure Appl. (9), 86 (2006), 271.  doi: 10.1016/j.matpur.2006.04.005.  Google Scholar

[4]

F. Chen, Almost periodic traveling waves of nonlocal evolution equations,, Nonlinear Anal., 50 (2002), 807.  doi: 10.1016/S0362-546X(01)00787-8.  Google Scholar

[5]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations,, Adv. Differential Equations, 2 (1997), 125.   Google Scholar

[6]

C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems,, Arch. Rational Mech. Anal., 187 (2008), 137.  doi: 10.1007/s00205-007-0062-8.  Google Scholar

[7]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction-diffusion equation,, Annali. di Matematica Pura Appl. (4), 185 (2006), 461.  doi: 10.1007/s10231-005-0163-7.  Google Scholar

[8]

J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Differential Equations, 244 (2008), 3080.  doi: 10.1016/j.jde.2007.11.002.  Google Scholar

[9]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics,, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727.  doi: 10.1017/S0308210504000721.  Google Scholar

[10]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations,, Nonlinear Anal., 60 (2005), 797.  doi: 10.1016/j.na.2003.10.030.  Google Scholar

[11]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems,", Lecture Notes in Biomathematics, 28 (1979).   Google Scholar

[12]

P. C. Fife and J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusion,, Arch. Rational Mech. Anal., 75 (): 281.  doi: 10.1007/BF00256381.  Google Scholar

[13]

T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations,, Nonlinearity, 7 (1994), 741.  doi: 10.1088/0951-7715/7/3/003.  Google Scholar

[14]

J. García-Melián and F. Quirós, Fujita exponents for evolution problems with nonlocal diffusion,, J. Evolution Equations, 10 (2010), 147.  doi: 10.1007/s00028-009-0043-5.  Google Scholar

[15]

S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation,, J. Math. Biol., 41 (2000), 272.  doi: 10.1007/s002850000047.  Google Scholar

[16]

S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread,, in, 48 (2006), 137.   Google Scholar

[17]

F. Hamel and L. Roques, Uniqueness and stability of properties of monostable pulsating fronts,, J. European Math. Soc., 13 (2011), 345.  doi: 10.4171/JEMS/256.  Google Scholar

[18]

R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $\mathbb R^N$,, Nonlinear Differential Equations Appl., 15 (2008), 599.  doi: 10.1007/s00030-008-7041-0.  Google Scholar

[19]

L. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods,, J. Math. Pure Appl. (9), 92 (2009), 163.  doi: 10.1016/j.matpur.2009.04.009.  Google Scholar

[20]

L. Ignat and J. D. Rossi, A nonlocal convolution-diffusion equation,, J. Func. Anal., 251 (2007), 399.  doi: 10.1016/j.jfa.2007.07.013.  Google Scholar

[21]

D. Ya. Khusainov, A. F. Ivanov and I. V. Kovarzh, Solution of one heat equation with delay,, Nonlinear Oscillasions (N. Y.), 12 (2009), 260.  doi: 10.1007/s11072-009-0075-3.  Google Scholar

[22]

K. Kirchgassner, On the nonlinear dynamics of travelling fronts,, J. Differential Equations, 96 (1992), 256.  doi: 10.1016/0022-0396(92)90153-E.  Google Scholar

[23]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l' équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bulletin Université d'Etat à Moscou, 1 (1937), 1.   Google Scholar

[24]

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

[25]

G. Li, M. Mei and Y. S. Wong, Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model,, Math. Biosci. Engin., 5 (2008), 85.   Google Scholar

[26]

J.-F. Mallordy and J.-M. Roquejoffre, A parabolic equation of the KPP type in higher dimensions,, SIAM J. Math. Anal., 26 (1995), 1.  doi: 10.1137/S0036141093246105.  Google Scholar

[27]

M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. I. Local nonlinearity,, J. Differential Equations, 247 (2009), 495.  doi: 10.1016/j.jde.2008.12.026.  Google Scholar

[28]

M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. II. Nonlocal nonlinearity,, J. Differential Equations, 247 (2009), 511.  doi: 10.1016/j.jde.2008.12.020.  Google Scholar

[29]

M. Mei, J. W.-H. So, M. Li and S. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion,, Proc. Roy. Soc. Edinburgh Sec. A, 134 (2004), 579.  doi: 10.1017/S0308210500003358.  Google Scholar

[30]

M. Mei and J. W.-H. So, Stability of strong travelling waves for a non-local time-delayed reaction-diffusion equation,, Proc. Roy. Soc. Edinburgh Sec. A, 138 (2008), 551.  doi: 10.1017/S0308210506000333.  Google Scholar

[31]

M. Mei, C. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations,, SIAM J. Math. Anal., 42 (2010), 2762.  doi: 10.1137/090776342.  Google Scholar

[32]

M. Mei and Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations,, Int. J. Numer. Anal. Model. Seris B, 2 (2011), 379.   Google Scholar

[33]

M. Mei and Y. S. Wong, Novel stability results for traveling wavefronts in an age-structured reaction-diffusion equations,, Math. Biosci. Engin., 6 (2009), 743.  doi: 10.3934/mbe.2009.6.743.  Google Scholar

[34]

H. J. K. Moet, A note on the asymptotic behavior of solutions of the KPP equation,, SIAM J. Math. Anal., 10 (1979), 728.  doi: 10.1137/0510067.  Google Scholar

[35]

S. Pan, W.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay,, Nonlinear Anal., 72 (2010), 3150.  doi: 10.1016/j.na.2009.12.008.  Google Scholar

[36]

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

[37]

J. W.-H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure: I. Traveling wavefronts on unbounded domains,, Roy. Soc. London Proc. Series A Math. Phys. Eng. Sci., 457 (2001), 1841.  doi: 10.1098/rspa.2001.0789.  Google Scholar

[38]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time,, J. Math. Kyoto Univ., 18 (1978), 453.   Google Scholar

[39]

J. Wu, D. Wei and M. Mei, Analysis on the critical speed of traveling waves,, Appl. Math. Lett., 20 (2007), 712.  doi: 10.1016/j.aml.2006.08.006.  Google Scholar

[40]

H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation,, Publ. Res. Inst. Math. Sci., 45 (2009), 925.  doi: 10.2977/prims/1260476648.  Google Scholar

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