Advanced Search
Article Contents
Article Contents

Pushed traveling fronts in monostable equations with monotone delayed reaction

Abstract Related Papers Cited by
  • We study the wavefront solutions of the scalar reaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone reaction term $g: \mathbb{R}_{+} → \mathbb{R}_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a similar result for $h>0$. To this end, we describe the asymptotics of all wavefronts (including critical and non-critical fronts) at $-\infty$. We also prove the uniqueness of wavefronts (up to a translation). In addition, a new uniqueness result for a class of nonlocal lattice equations is presented.
    Mathematics Subject Classification: Primary: 34K12, 35K31; Secondary: 92D25.


    \begin{equation} \\ \end{equation}
  • [1]

    M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts (Diekmann-Kaper theory of a nonlinear convolution equation re-visited), Math. Ann., 354 (2012), 73-109.


    R. D. Benguria and M. C. Depassier, Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation, Comm. Math. Phys., 175 (1996), 221-227.


    H. Berestycki and L. Nirenberg, Traveling waves in cylinders, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497-572.


    H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844 .


    A. Boumenir and V.M. Nguyen, Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations, J. Differential Equations, 244 (2008), 1551-1570.


    A. Calamai, C. Marcelli and F. Papalini, A general approach for front-propagation in functional reaction-diffusion equations, J. Dynam. Differential Equations, 21 (2009) 567-392.


    J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.


    X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.


    X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.


    J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.


    J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.


    O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.


    U. Ebert and W. van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts, Phys. D, 146 (2000), 1-99.


    J. Fang, J. Wei and X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations, Proc. Amer. Math. Soc., 139 (2011), 1361-1373.


    J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226.


    T. Faria and S. Trofimchuk, Non-monotone traveling waves in a single species reaction,-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376.


    B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction," Birkhauser, 2004.


    C. Gomez, H. Prado and S. TrofimchukSeparation dichotomy and wavefronts for a nonlinear convolution equation, preprint arXiv:1204.5760.


    K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.


    A. Kolmogorov, I. Petrovskii and N. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem, Byul. Mosk. Gos. Univ. Ser. A Mat. Mekh., 1 (1937) 1-26.


    X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. Erratum: 61(2008), 137-138.


    X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Anal., 259 (2010), 857-903.


    S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.


    S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277.


    S. Ma, X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.


    J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-48.


    M. Mei, Ch. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 233-258.


    G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 553-557.


    F. Rothe, Convergence to pushed fronts, Rocky Mountain J. Math. , 11 (1981), 617-633.


    K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.


    A.N. Stokes, On two types of moving front in quasilinear diffusion, Math. Biosciences, 31 (1976), 307-315.


    K. Schumacher, Travelling-front solutions for integro-differential equations. I , J. Reine Angew. Math., 316 (1980), 54-70.


    E. Trofimchuk, M. Pinto and S. TrofimchukTraveling wavefronts for a model of the Belousov-Zhabotinskii reaction, preprint arXiv:1103.0176v2.


    E. Trofimchuk, M. Pinto and S. TrofimchukPushed traveling fronts in monostable equations with monotone delayed reaction, preprint arXiv:1111.5161v1.


    E. Trofimchuk, V. Tkachenko and S. Trofimchuk, Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay, J. Differential Equations, 245 (2008), 2307-2332.


    E. Trofimchuk, P. Alvarado and S. Trofimchuk, On the geometry of wave solutions of a delayed reaction-diffusion equation, J. Differential Equations, 246 (2009), 1422-1444.


    E. Trofimchuk and S. Trofimchuk, Admissible wavefront speeds for a single species reaction-diffusion equation with delay, Discrete Contin. Dyn. Syst., 20 (2008), 407-423.


    Z.-C. Wang, W.T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607.


    H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.


    P. Weng, H. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439.


    J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.


    J. Xin}, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230.


    Z.-X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays, Proc. Amer. Math. Soc., 140 (2012), 3853-3859.


    B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation J. Differential Equations, 105 (1993), 46-62.

  • 加载中

Article Metrics

HTML views() PDF downloads(88) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint