Pushed traveling fronts in monostable  equations with
monotone delayed reaction

Elena Trofimchuk; Manuel Pinto; Sergei Trofimchuk

doi:10.3934/dcds.2013.33.2169

Previous Article
Application of the subharmonic Melnikov method to piecewise-smooth systems
DCDS Home
This Issue
Next Article
Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations

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
Related Papers

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 - 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. 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. Google Scholar
[3]	H. Berestycki and L. Nirenberg, Traveling waves in cylinders,, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497. 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. 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. 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. Google Scholar
[7]	J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. 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. 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. Google Scholar
[10]	J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation,, Ann. Mat. Pura Appl., 185 (2006), 461. Google Scholar
[11]	J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Differential Equations, 244 (2008), 3080. Google Scholar
[12]	O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation,, Nonlinear Anal., 2 (1978), 721. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[23]	S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294. Google Scholar
[24]	S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations,, J. Differential Equations, 237 (2007), 259. 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. Google Scholar
[26]	J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type,, J. Dynam. Differential Equations, 11 (1999), 1. 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. 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, 349 (2011), 553. Google Scholar
[29]	F. Rothe, Convergence to pushed fronts, , Rocky Mountain J. Math., 11 (1981), 617. Google Scholar
[30]	K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations,, Trans. Amer. Math. Soc., 302 (1987), 587. Google Scholar
[31]	A.N. Stokes, On two types of moving front in quasilinear diffusion,, Math. Biosciences, 31 (1976), 307. Google Scholar
[32]	K. Schumacher, Travelling-front solutions for integro-differential equations. I ,, J. Reine Angew. Math., 316 (1980), 54. 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. 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. 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. 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. Google Scholar
[39]	H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353. 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. Google Scholar
[41]	J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dynam. Differential Equations, 13 (2001), 651. Google Scholar
[42]	J. Xin}, Front propagation in heterogeneous media,, SIAM Review, 42 (2000), 161. Google Scholar
[43]	Z.-X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays,, Proc. Amer. Math. Soc., 140 (2012), 3853. Google Scholar
[44]	B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46. 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. 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. Google Scholar
[3]	H. Berestycki and L. Nirenberg, Traveling waves in cylinders,, Ann. Inst. H. Poincare Anal. Non. Lineaire, 9 (1992), 497. 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. 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. 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. Google Scholar
[7]	J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. 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. 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. Google Scholar
[10]	J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation,, Ann. Mat. Pura Appl., 185 (2006), 461. Google Scholar
[11]	J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Differential Equations, 244 (2008), 3080. Google Scholar
[12]	O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation,, Nonlinear Anal., 2 (1978), 721. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[23]	S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294. Google Scholar
[24]	S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations,, J. Differential Equations, 237 (2007), 259. 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. Google Scholar
[26]	J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type,, J. Dynam. Differential Equations, 11 (1999), 1. 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. 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, 349 (2011), 553. Google Scholar
[29]	F. Rothe, Convergence to pushed fronts, , Rocky Mountain J. Math., 11 (1981), 617. Google Scholar
[30]	K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations,, Trans. Amer. Math. Soc., 302 (1987), 587. Google Scholar
[31]	A.N. Stokes, On two types of moving front in quasilinear diffusion,, Math. Biosciences, 31 (1976), 307. Google Scholar
[32]	K. Schumacher, Travelling-front solutions for integro-differential equations. I ,, J. Reine Angew. Math., 316 (1980), 54. 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. 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. 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. 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. Google Scholar
[39]	H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353. 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. Google Scholar
[41]	J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dynam. Differential Equations, 13 (2001), 651. Google Scholar
[42]	J. Xin}, Front propagation in heterogeneous media,, SIAM Review, 42 (2000), 161. Google Scholar
[43]	Z.-X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays,, Proc. Amer. Math. Soc., 140 (2012), 3853. Google Scholar
[44]	B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46. Google Scholar

[1]	Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147
[2]	Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043
[3]	Zhi-Cheng Wang. Traveling curved fronts in monotone bistable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2339-2374. doi: 10.3934/dcds.2012.32.2339
[4]	Armengol Gasull, Hector Giacomini, Joan Torregrosa. Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3567-3582. doi: 10.3934/dcds.2013.33.3567
[5]	Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331
[6]	Thuc Manh Le, Nguyen Van Minh. Monotone traveling waves in a general discrete model for populations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3221-3234. doi: 10.3934/dcdsb.2017171
[7]	João Fialho, Feliz Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Conference Publications, 2013, 2013 (special) : 217-226. doi: 10.3934/proc.2013.2013.217
[8]	Massimo Tarallo, Zhe Zhou. Limit periodic upper and lower solutions in a generic sense. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 293-309. doi: 10.3934/dcds.2018014
[9]	Luisa Malaguti, Cristina Marcelli. Existence of bounded trajectories via upper and lower solutions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 575-590. doi: 10.3934/dcds.2000.6.575
[10]	Shiwang Ma, Xiao-Qiang Zhao. Global asymptotic stability of minimal fronts in monostable lattice equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 259-275. doi: 10.3934/dcds.2008.21.259
[11]	Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89
[12]	Chufen Wu, Peixuan Weng. Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 867-892. doi: 10.3934/dcdsb.2011.15.867
[13]	Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405
[14]	Hongmei Cheng, Rong Yuan. Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 3007-3022. doi: 10.3934/dcdsb.2017160
[15]	Jian Fang, Jianhong Wu. Monotone traveling waves for delayed Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3043-3058. doi: 10.3934/dcds.2012.32.3043
[16]	Je-Chiang Tsai. Global exponential stability of traveling waves in monotone bistable systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601
[17]	Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 315-328. doi: 10.3934/dcds.2000.6.315
[18]	Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441
[19]	Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126
[20]	Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5695-5707. doi: 10.3934/dcdsb.2019102

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences

Pushed traveling fronts in monostable equations with monotone delayed reaction

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Pushed traveling fronts in monostable equations with monotone delayed reaction

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors