# American Institute of Mathematical Sciences

2013, 33(5): 2169-2187. doi: 10.3934/dcds.2013.33.2169

## Pushed traveling fronts in monostable equations with monotone delayed reaction

 1 Department of Differential Equations, National Technical University, Kyiv, Ukraine 2 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile 3 Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca

Received  November 2011 Revised  August 2012 Published  December 2012

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.
Citation: Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. Pushed traveling fronts in monostable equations with monotone delayed reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2169-2187. doi: 10.3934/dcds.2013.33.2169
##### References:
 [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. [2] 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. [3] H. Berestycki and L. Nirenberg, Traveling waves in cylinders,, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497. [4] H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: travelling waves and steady states,, Nonlinearity, 22 (2009), 2813. [5] 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. [6] 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. [7] J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. [8] X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics,, Math. Ann., 326 (2003), 123. [9] 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. [10] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation,, Ann. Mat. Pura Appl., 185 (2006), 461. [11] J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Differential Equations, 244 (2008), 3080. [12] O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation,, Nonlinear Anal., 2 (1978), 721. [13] U. Ebert and W. van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts,, Phys. D, 146 (2000), 1. [14] J. Fang, J. Wei and X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations,, Proc. Amer. Math. Soc., 139 (2011), 1361. [15] 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. [16] T. Faria and S. Trofimchuk, Non-monotone traveling waves in a single species reaction,-diffusion equation with delay,, J. Differential Equations, 228 (2006), 357. [17] B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction,", Birkhauser, (2004). [18] C. Gomez, H. Prado and S. Trofimchuk, Separation dichotomy and wavefronts for a nonlinear convolution equation,, preprint , (). [19] K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations,, J. Math. Biol., 2 (1975), 251. [20] 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. [21] 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. [22] X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Functional Anal., 259 (2010), 857. [23] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294. [24] S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations,, J. Differential Equations, 237 (2007), 259. [25] 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. [26] J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type,, J. Dynam. Differential Equations, 11 (1999), 1. [27] 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. [28] 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, 349 (2011), 553. [29] F. Rothe, Convergence to pushed fronts, , Rocky Mountain J. Math., 11 (1981), 617. [30] K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations,, Trans. Amer. Math. Soc., 302 (1987), 587. [31] A.N. Stokes, On two types of moving front in quasilinear diffusion,, Math. Biosciences, 31 (1976), 307. [32] K. Schumacher, Travelling-front solutions for integro-differential equations. I ,, J. Reine Angew. Math., 316 (1980), 54. [33] E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling wavefronts for a model of the Belousov-Zhabotinskii reaction,, preprint , (). [34] E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction,, preprint , (). [35] 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. [36] 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. [37] E. Trofimchuk and S. Trofimchuk, Admissible wavefront speeds for a single species reaction-diffusion equation with delay,, Discrete Contin. Dyn. Syst., 20 (2008), 407. [38] 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. [39] H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353. [40] 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. [41] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dynam. Differential Equations, 13 (2001), 651. [42] J. Xin}, Front propagation in heterogeneous media,, SIAM Review, 42 (2000), 161. [43] Z.-X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays,, Proc. Amer. Math. Soc., 140 (2012), 3853. [44] B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46.

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##### References:
 [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. [2] 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. [3] H. Berestycki and L. Nirenberg, Traveling waves in cylinders,, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497. [4] H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: travelling waves and steady states,, Nonlinearity, 22 (2009), 2813. [5] 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. [6] 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. [7] J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. [8] X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics,, Math. Ann., 326 (2003), 123. [9] 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. [10] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation,, Ann. Mat. Pura Appl., 185 (2006), 461. [11] J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Differential Equations, 244 (2008), 3080. [12] O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation,, Nonlinear Anal., 2 (1978), 721. [13] U. Ebert and W. van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts,, Phys. D, 146 (2000), 1. [14] J. Fang, J. Wei and X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations,, Proc. Amer. Math. Soc., 139 (2011), 1361. [15] 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. [16] T. Faria and S. Trofimchuk, Non-monotone traveling waves in a single species reaction,-diffusion equation with delay,, J. Differential Equations, 228 (2006), 357. [17] B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction,", Birkhauser, (2004). [18] C. Gomez, H. Prado and S. Trofimchuk, Separation dichotomy and wavefronts for a nonlinear convolution equation,, preprint , (). [19] K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations,, J. Math. Biol., 2 (1975), 251. [20] 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. [21] 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. [22] X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Functional Anal., 259 (2010), 857. [23] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294. [24] S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations,, J. Differential Equations, 237 (2007), 259. [25] 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. [26] J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type,, J. Dynam. Differential Equations, 11 (1999), 1. [27] 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. [28] 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, 349 (2011), 553. [29] F. Rothe, Convergence to pushed fronts, , Rocky Mountain J. Math., 11 (1981), 617. [30] K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations,, Trans. Amer. Math. Soc., 302 (1987), 587. [31] A.N. Stokes, On two types of moving front in quasilinear diffusion,, Math. Biosciences, 31 (1976), 307. [32] K. Schumacher, Travelling-front solutions for integro-differential equations. I ,, J. Reine Angew. Math., 316 (1980), 54. [33] E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling wavefronts for a model of the Belousov-Zhabotinskii reaction,, preprint , (). [34] E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction,, preprint , (). [35] 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. [36] 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. [37] E. Trofimchuk and S. Trofimchuk, Admissible wavefront speeds for a single species reaction-diffusion equation with delay,, Discrete Contin. Dyn. Syst., 20 (2008), 407. [38] 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. [39] H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353. [40] 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. [41] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dynam. Differential Equations, 13 (2001), 651. [42] J. Xin}, Front propagation in heterogeneous media,, SIAM Review, 42 (2000), 161. [43] Z.-X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays,, Proc. Amer. Math. Soc., 140 (2012), 3853. [44] B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46.
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