# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 105-136. doi: 10.1007/s002050050037. [2] M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190 pp. [3] E. Chasseigne, M. Chaves and J. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pure Appl. (9), 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. [4] F. Chen, Almost periodic traveling waves of nonlocal evolution equations, Nonlinear Anal., 50 (2002), 807-838. 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), 125-160. [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), 137-156. 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), 461-485. 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), 3080-3118. 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), 727-755. 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), 797-819. doi: 10.1016/j.na.2003.10.030. [11] P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 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 (1980/81), 281-314. doi: 10.1007/BF00256381. [13] T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity, 7 (1994), 741-764. 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), 147-161. doi: 10.1007/s00028-009-0043-5. [15] S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284. doi: 10.1007/s002850000047. [16] S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in "Nonlinear Dynamics and Evolution Equations" (eds. H. Brunner, X.-Q. Zhao and X. Zou), Fields Institute Communications, 48, Amer. Math. Soc., Providence, RI, (2006), 137-200. [17] F. Hamel and L. Roques, Uniqueness and stability of properties of monostable pulsating fronts, J. European Math. Soc., 13 (2011), 345-390. 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), 599-622. 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), 163-187. 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), 399-437. 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), 260-282. doi: 10.1007/s11072-009-0075-3. [22] K. Kirchgassner, On the nonlinear dynamics of travelling fronts, J. Differential Equations, 96 (1992), 256-278. 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, Série Internationale Sect. A, 1 (1937), 1-26. [24] 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. [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), 85-100. [26] J.-F. Mallordy and J.-M. Roquejoffre, A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal., 26 (1995), 1-20. 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), 495-510. 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), 511-529. 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), 579-594. 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), 551-568. 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), 2762-2790; Erratum, SIAM J. Math. Anal., 44 (2012), 538-540. 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), 379-401. [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), 743-752. 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), 728-732. 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), 3150-3158. 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), 312-355. 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), 1841-1853. 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), 453-508. [39] J. Wu, D. Wei and M. Mei, Analysis on the critical speed of traveling waves, Appl. Math. Lett., 20 (2007), 712-718. 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), 925-953. doi: 10.2977/prims/1260476648.

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##### 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), 105-136. doi: 10.1007/s002050050037. [2] M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190 pp. [3] E. Chasseigne, M. Chaves and J. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pure Appl. (9), 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. [4] F. Chen, Almost periodic traveling waves of nonlocal evolution equations, Nonlinear Anal., 50 (2002), 807-838. 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), 125-160. [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), 137-156. 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), 461-485. 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), 3080-3118. 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), 727-755. 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), 797-819. doi: 10.1016/j.na.2003.10.030. [11] P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 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 (1980/81), 281-314. doi: 10.1007/BF00256381. [13] T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity, 7 (1994), 741-764. 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), 147-161. doi: 10.1007/s00028-009-0043-5. [15] S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284. doi: 10.1007/s002850000047. [16] S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in "Nonlinear Dynamics and Evolution Equations" (eds. H. Brunner, X.-Q. Zhao and X. Zou), Fields Institute Communications, 48, Amer. Math. Soc., Providence, RI, (2006), 137-200. [17] F. Hamel and L. Roques, Uniqueness and stability of properties of monostable pulsating fronts, J. European Math. Soc., 13 (2011), 345-390. 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), 599-622. 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), 163-187. 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), 399-437. 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), 260-282. doi: 10.1007/s11072-009-0075-3. [22] K. Kirchgassner, On the nonlinear dynamics of travelling fronts, J. Differential Equations, 96 (1992), 256-278. 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, Série Internationale Sect. A, 1 (1937), 1-26. [24] 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. [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), 85-100. [26] J.-F. Mallordy and J.-M. Roquejoffre, A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal., 26 (1995), 1-20. 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), 495-510. 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), 511-529. 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), 579-594. 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), 551-568. 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), 2762-2790; Erratum, SIAM J. Math. Anal., 44 (2012), 538-540. 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), 379-401. [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), 743-752. 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), 728-732. 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), 3150-3158. 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), 312-355. 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), 1841-1853. 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), 453-508. [39] J. Wu, D. Wei and M. Mei, Analysis on the critical speed of traveling waves, Appl. Math. Lett., 20 (2007), 712-718. 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), 925-953. doi: 10.2977/prims/1260476648.
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