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

Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity

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.
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:
[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.

[2]

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

[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.

[4]

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

[5]

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

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[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.

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.

[2]

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

[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.

[4]

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

[5]

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

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[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.

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