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.

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.

[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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