Pushed traveling fronts in monostable  equations with
monotone delayed reaction

Elena Trofimchuk; Manuel Pinto; Sergei Trofimchuk

doi:10.3934/dcds.2013.33.2169

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

Pushed traveling fronts in monostable equations with monotone delayed reaction

Elena Trofimchuk ^1, , Manuel Pinto ^2, and Sergei Trofimchuk ^3,

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

Abstract
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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.

Keywords: monotone traveling waves, minimal speed., pushed fronts, Upper and lower solutions, asymptotic integration.

Mathematics Subject Classification: Primary: 34K12, 35K31; Secondary: 92D2.

Citation: Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. Pushed traveling fronts in monostable equations with monotone delayed reaction. Discrete and Continuous Dynamical Systems, 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), 73-109.
[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), 221-227.
[3]	H. Berestycki and L. Nirenberg, Traveling waves in cylinders, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497-572.
[4]	H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844 .
[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), 1551-1570.
[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) 567-392.
[7]	J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.
[8]	X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.
[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), 233-258.
[10]	J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.
[11]	J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.
[12]	O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.
[13]	U. Ebert and W. van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts, Phys. D, 146 (2000), 1-99.
[14]	J. Fang, J. Wei and X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations, Proc. Amer. Math. Soc., 139 (2011), 1361-1373.
[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), 2199-2226.
[16]	T. Faria and S. Trofimchuk, Non-monotone traveling waves in a single species reaction,-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376.
[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 arXiv:1204.5760.
[19]	K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.
[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) 1-26.
[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), 1-40. Erratum: 61(2008), 137-138.
[22]	X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Anal., 259 (2010), 857-903.
[23]	S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.
[24]	S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277.
[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), 54-87.
[26]	J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-48.
[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), 233-258.
[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, Ser. I, 349 (2011), 553-557.
[29]	F. Rothe, Convergence to pushed fronts, Rocky Mountain J. Math. , 11 (1981), 617-633.
[30]	K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.
[31]	A.N. Stokes, On two types of moving front in quasilinear diffusion, Math. Biosciences, 31 (1976), 307-315.
[32]	K. Schumacher, Travelling-front solutions for integro-differential equations. I , J. Reine Angew. Math., 316 (1980), 54-70.
[33]	E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling wavefronts for a model of the Belousov-Zhabotinskii reaction, preprint arXiv:1103.0176v2.
[34]	E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction, preprint arXiv:1111.5161v1.
[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), 2307-2332.
[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), 1422-1444.
[37]	E. Trofimchuk and S. Trofimchuk, Admissible wavefront speeds for a single species reaction-diffusion equation with delay, Discrete Contin. Dyn. Syst., 20 (2008), 407-423.
[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), 573-607.
[39]	H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
[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), 409-439.
[41]	J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.
[42]	J. Xin}, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230.
[43]	Z.-X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays, Proc. Amer. Math. Soc., 140 (2012), 3853-3859.
[44]	B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation J. Differential Equations, 105 (1993), 46-62.

show all references

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), 73-109.
[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), 221-227.
[3]	H. Berestycki and L. Nirenberg, Traveling waves in cylinders, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497-572.
[4]	H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844 .
[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), 1551-1570.
[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) 567-392.
[7]	J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.
[8]	X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.
[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), 233-258.
[10]	J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.
[11]	J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.
[12]	O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.
[13]	U. Ebert and W. van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts, Phys. D, 146 (2000), 1-99.
[14]	J. Fang, J. Wei and X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations, Proc. Amer. Math. Soc., 139 (2011), 1361-1373.
[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), 2199-2226.
[16]	T. Faria and S. Trofimchuk, Non-monotone traveling waves in a single species reaction,-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376.
[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 arXiv:1204.5760.
[19]	K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.
[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) 1-26.
[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), 1-40. Erratum: 61(2008), 137-138.
[22]	X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Anal., 259 (2010), 857-903.
[23]	S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.
[24]	S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277.
[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), 54-87.
[26]	J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-48.
[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), 233-258.
[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, Ser. I, 349 (2011), 553-557.
[29]	F. Rothe, Convergence to pushed fronts, Rocky Mountain J. Math. , 11 (1981), 617-633.
[30]	K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.
[31]	A.N. Stokes, On two types of moving front in quasilinear diffusion, Math. Biosciences, 31 (1976), 307-315.
[32]	K. Schumacher, Travelling-front solutions for integro-differential equations. I , J. Reine Angew. Math., 316 (1980), 54-70.
[33]	E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling wavefronts for a model of the Belousov-Zhabotinskii reaction, preprint arXiv:1103.0176v2.
[34]	E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction, preprint arXiv:1111.5161v1.
[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), 2307-2332.
[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), 1422-1444.
[37]	E. Trofimchuk and S. Trofimchuk, Admissible wavefront speeds for a single species reaction-diffusion equation with delay, Discrete Contin. Dyn. Syst., 20 (2008), 407-423.
[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), 573-607.
[39]	H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
[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), 409-439.
[41]	J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.
[42]	J. Xin}, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230.
[43]	Z.-X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays, Proc. Amer. Math. Soc., 140 (2012), 3853-3859.
[44]	B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation J. Differential Equations, 105 (1993), 46-62.

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