American Institute of Mathematical Sciences

Advanced
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

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 & 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. Google Scholar

[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.  Google Scholar

[3]

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

[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 .  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

C. Gomez, H. Prado and S. Trofimchuk, Separation dichotomy and wavefronts for a nonlinear convolution equation,, preprint , ().   Google Scholar

[19]

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

[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. Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling wavefronts for a model of the Belousov-Zhabotinskii reaction,, preprint , ().   Google Scholar

[34]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction,, preprint , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

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. Google Scholar

[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.  Google Scholar

[3]

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

[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 .  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

C. Gomez, H. Prado and S. Trofimchuk, Separation dichotomy and wavefronts for a nonlinear convolution equation,, preprint , ().   Google Scholar

[19]

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

[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. Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling wavefronts for a model of the Belousov-Zhabotinskii reaction,, preprint , ().   Google Scholar

[34]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction,, preprint , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

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