American Institute of Mathematical Sciences

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October  2019, 39(10): 5799-5823. doi: 10.3934/dcds.2019255

Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations

Abraham Solar 1,2,

1. 

Instituto de Física, Facultad de Física, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
2. 

Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile

Received  September 2018 Revised  April 2019 Published  July 2019

This paper deals with the stability of semi-wavefronts to the following delay non-local monostable equation: $\dot{v}(t,x) = \Delta v(t,x) - v(t,x) + \int_{\mathbb{R}^d}K(y)g(v(t-h,x-y))dy, x \in \mathbb{R}^d,\ t >0;$ where $h>0$ and $d\in\mathbb{Z}_+$. We give two general results for $d\geq1$: on the global stability of semi-wavefronts in $L^p$-spaces with unbounded weights and the local stability of planar wavefronts in $L^p$-spaces with bounded weights. We also give a global stability result for $d = 1$ which yields to the global stability in Sobolev spaces with bounded weights. Here $g$ is not assumed to be monotone and the kernel $K$ is not assumed to be symmetric, therefore non-monotone semi-wavefronts and backward semi-wavefronts appear for which we show their stability. In particular, the global stability of critical wavefronts is stated.

Keywords: Non-local equations, delay equations, semi-wavefronts, non-monotone wavefronts, global stability, local stability.
Mathematics Subject Classification: Primary: 35K57, 35R10; Secondary: 35B40, 92D25.
Citation: Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255
References:
[1]

M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts, Math. Ann., 354 (2012), 73-109.  doi: 10.1007/s00208-011-0722-8.  Google Scholar

[2]

M. Bani-Yaghoub, The traveling wavefront for a nonlocal delayed reaction-diffusion equation, J. Appl. Math. Comput., 53 (2017), 77-94.  doi: 10.1007/s12190-015-0958-7.  Google Scholar

[3]

R. Benguria and A. Solar, An estimation of level set for a non-local KPP equation with delay, Nonlinearity, 32 (2019), 777-799.  doi: 10.1088/1361-6544/aaedd7.  Google Scholar

[4]

R. Benguria and A. Solar, An iterative estimation for disturbances of semi-wavefronts to the delayed Fisher-KPP equation, Proc. Amer. Math. Soc., 147 (2019), 2495-2501.  Google Scholar

[5]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolutions equations, Advances in Differential Equations, 2 (1997), 125-160.   Google Scholar

[6]

I.-L. Chern, M. Mei, X. Yang and Q. Zhang, Stability of non-monotone critical traveling waves for reaction-diffusion equation with time-delay, J. Diff. Eqns, 259 (2015), 1503-1541.  doi: 10.1016/j.jde.2015.03.003.  Google Scholar

[7]

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.  doi: 10.1016/S0167-2789(00)00068-3.  Google Scholar

[8]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs (N.J., USA), 1964.  Google Scholar

[9]

T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity, 7 (1994), 741-764.   Google Scholar

[10]

A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts, J. Lond. Math. Soc., 89 (2014), 47-68.  doi: 10.1112/jlms/jdt050.  Google Scholar

[11]

C. Gomez, H. Prado and S. Trofimchuk, Separation dichotomy and wavefronts for a nonlinear convolution equation, J. Math. Anal. Appl., 420 (2014), 1-19.  doi: 10.1016/j.jmaa.2014.05.064.  Google Scholar

[12]

S. A. Gourley, J. So and J. Wu, Non-locality of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics, J. Math. Sci., 124 (2004), 5119-5153.   Google Scholar

[13] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York (N.Y., USA), 1966.   Google Scholar
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S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.  Google Scholar

[15]

R. Huang, M. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discret Contin. Dyn. Syst., 32 (2012), 3621-3649.  doi: 10.3934/dcds.2012.32.3621.  Google Scholar

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R. Huang, M. Mei, K. Zhang and Q. Zhang, Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations, Discrete Contin. Dyn. Syst., 36 (2016), 1331-1353.  doi: 10.3934/dcds.2016.36.1331.  Google Scholar

[17]

K. Kirchgassner, On the nonlinear dynamics of travelling fronts, J. Diff. Eqs., 96 (1992), 256-278.  doi: 10.1016/0022-0396(92)90153-E.  Google Scholar

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A. Kolmogorov, I. Petrovskii and N. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, (French) [Study of a Diffusion Equation That Is Related to the Growth of a Quality of Matter and Its Application to a Biological Problem], Moscow Uni. Bull. Math., 1 (1937), 1-25.   Google Scholar

[19]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[20]

C.-K. Lin, C.-T. Lin, Y. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084.  doi: 10.1137/120904391.  Google Scholar

[21]

G. Lv and M. Wang, Nonlinear stability of travelling wave fronts for delayed reaction diffusion equations, Nonlinearity, 23 (2010), 845-873. doi: 10.1088/0951-7715/23/4/005.  Google Scholar

[22]

S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J. Dyn. Diff. Eqns., 19 (2007), 391-436.  doi: 10.1007/s10884-006-9065-7.  Google Scholar

[23]

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), 2762-2790.  doi: 10.1137/090776342.  Google Scholar

[24]

M. Mei, K. Zhang and Q. Zhang, Global stability of critical traveling waves with oscillations for time-delayed reaction-diffusion equation, Int. J. Numer. Anal. Model., 16 (2019), 375-397.   Google Scholar

[25]

M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs (N.J., USA), 1967.  Google Scholar

[26]

D. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[27]

K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Am. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859.  Google Scholar

[28]

H. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence (R.I. USA), 1995.  Google Scholar

[29]

J. W.-H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure I. Travelling wavefronts on unbounded domains, Proc. R. Soc. A, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.  Google Scholar

[30]

A. Solar, Stability of semi-wavefronts for delayed reaction-diffusion equations, preprint, arXiv: 1704.03011. Google Scholar

[31]

A. Solar and S. Trofimchuk, Speed selection and stability of wavefronts for delayed monostable reaction-diffusion equations, J. Dyn. Diff. Eqns., 28 (2016), 1265-1292.  doi: 10.1007/s10884-015-9482-6.  Google Scholar

[32]

A. Stokes, On two types of moving front in quasilinear diffusion, Math.Biosciences, 31 (1976), 307-315.  doi: 10.1016/0025-5564(76)90087-0.  Google Scholar

[33]

H. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J.Diff. Eqns., 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar

[34]

E. Trofimchuk, P. Alvarado and S. Trofimchuk, On the geometry of wave solutions of a delayed reaction-diffusion equation, J. Diff. Eqns, 246 (2009), 1422-1444.  doi: 10.1016/j.jde.2008.10.023.  Google Scholar

[35]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Monotone waves for non-monotone and non-local monostable reaction-diffusion equations, J. Diff. Eqns, 261 (2016), 1203-1236.  doi: 10.1016/j.jde.2016.03.039.  Google Scholar

[36]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.  doi: 10.1215/kjm/1250522506.  Google Scholar

[37]

Z. Wang, W. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Diff. Eqns., 238 (2007), 153-200.  doi: 10.1016/j.jde.2007.03.025.  Google Scholar

[38]

Z. Wang, W. Li and S. Ruan, Travelling fronts in monostable equations with nonlocal delayed effects, J. Dyn. Differ. Equ., 20 (2008), 573-607.  doi: 10.1007/s10884-008-9103-8.  Google Scholar

[39]

H. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[40]

S.-L. Wu, W.-T. Li and S.-Y. Liu, Oscillatory waves in reaction-diffusion equations with nonlocal delay and crossing-monostability, Non. Lin. Anal.: Real World App., 10 (2009), 3141-3151.  doi: 10.1016/j.nonrwa.2008.10.012.  Google Scholar

[41]

T. Xu, S. Ji, M. Mei and J. Yin, Theoretical and numerical studies on global stability of traveling waves with oscillations for time-delayed nonlocal dispersion equations, preprint, arXiv: 1810.07484. Google Scholar

[42]

T. Yi and X. Zou, Asymptotic behavior, spreading speeds, and traveling waves of nonmonotone dynamical systems, SIAM J. Math.Anal., 47 (2015), 3005-3034.  doi: 10.1137/14095412X.  Google Scholar

[43]

T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition, Proc. R. Soc. A, 466 (2010), 2955-2973.  doi: 10.1098/rspa.2009.0650.  Google Scholar

[44]

T. Yi, Y. Chen and J. Wu, Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves, J. Diff. Eqns, 254 (2013), 3538-3572.  doi: 10.1016/j.jde.2013.01.031.  Google Scholar

show all references

References:
[1]

M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts, Math. Ann., 354 (2012), 73-109.  doi: 10.1007/s00208-011-0722-8.  Google Scholar

[2]

M. Bani-Yaghoub, The traveling wavefront for a nonlocal delayed reaction-diffusion equation, J. Appl. Math. Comput., 53 (2017), 77-94.  doi: 10.1007/s12190-015-0958-7.  Google Scholar

[3]

R. Benguria and A. Solar, An estimation of level set for a non-local KPP equation with delay, Nonlinearity, 32 (2019), 777-799.  doi: 10.1088/1361-6544/aaedd7.  Google Scholar

[4]

R. Benguria and A. Solar, An iterative estimation for disturbances of semi-wavefronts to the delayed Fisher-KPP equation, Proc. Amer. Math. Soc., 147 (2019), 2495-2501.  Google Scholar

[5]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolutions equations, Advances in Differential Equations, 2 (1997), 125-160.   Google Scholar

[6]

I.-L. Chern, M. Mei, X. Yang and Q. Zhang, Stability of non-monotone critical traveling waves for reaction-diffusion equation with time-delay, J. Diff. Eqns, 259 (2015), 1503-1541.  doi: 10.1016/j.jde.2015.03.003.  Google Scholar

[7]

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.  doi: 10.1016/S0167-2789(00)00068-3.  Google Scholar

[8]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs (N.J., USA), 1964.  Google Scholar

[9]

T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity, 7 (1994), 741-764.   Google Scholar

[10]

A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts, J. Lond. Math. Soc., 89 (2014), 47-68.  doi: 10.1112/jlms/jdt050.  Google Scholar

[11]

C. Gomez, H. Prado and S. Trofimchuk, Separation dichotomy and wavefronts for a nonlinear convolution equation, J. Math. Anal. Appl., 420 (2014), 1-19.  doi: 10.1016/j.jmaa.2014.05.064.  Google Scholar

[12]

S. A. Gourley, J. So and J. Wu, Non-locality of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics, J. Math. Sci., 124 (2004), 5119-5153.   Google Scholar

[13] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York (N.Y., USA), 1966.   Google Scholar
[14]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.  Google Scholar

[15]

R. Huang, M. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discret Contin. Dyn. Syst., 32 (2012), 3621-3649.  doi: 10.3934/dcds.2012.32.3621.  Google Scholar

[16]

R. Huang, M. Mei, K. Zhang and Q. Zhang, Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations, Discrete Contin. Dyn. Syst., 36 (2016), 1331-1353.  doi: 10.3934/dcds.2016.36.1331.  Google Scholar

[17]

K. Kirchgassner, On the nonlinear dynamics of travelling fronts, J. Diff. Eqs., 96 (1992), 256-278.  doi: 10.1016/0022-0396(92)90153-E.  Google Scholar

[18]

A. Kolmogorov, I. Petrovskii and N. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, (French) [Study of a Diffusion Equation That Is Related to the Growth of a Quality of Matter and Its Application to a Biological Problem], Moscow Uni. Bull. Math., 1 (1937), 1-25.   Google Scholar

[19]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[20]

C.-K. Lin, C.-T. Lin, Y. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084.  doi: 10.1137/120904391.  Google Scholar

[21]

G. Lv and M. Wang, Nonlinear stability of travelling wave fronts for delayed reaction diffusion equations, Nonlinearity, 23 (2010), 845-873. doi: 10.1088/0951-7715/23/4/005.  Google Scholar

[22]

S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J. Dyn. Diff. Eqns., 19 (2007), 391-436.  doi: 10.1007/s10884-006-9065-7.  Google Scholar

[23]

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), 2762-2790.  doi: 10.1137/090776342.  Google Scholar

[24]

M. Mei, K. Zhang and Q. Zhang, Global stability of critical traveling waves with oscillations for time-delayed reaction-diffusion equation, Int. J. Numer. Anal. Model., 16 (2019), 375-397.   Google Scholar

[25]

M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs (N.J., USA), 1967.  Google Scholar

[26]

D. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[27]

K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Am. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859.  Google Scholar

[28]

H. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence (R.I. USA), 1995.  Google Scholar

[29]

J. W.-H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure I. Travelling wavefronts on unbounded domains, Proc. R. Soc. A, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.  Google Scholar

[30]

A. Solar, Stability of semi-wavefronts for delayed reaction-diffusion equations, preprint, arXiv: 1704.03011. Google Scholar

[31]

A. Solar and S. Trofimchuk, Speed selection and stability of wavefronts for delayed monostable reaction-diffusion equations, J. Dyn. Diff. Eqns., 28 (2016), 1265-1292.  doi: 10.1007/s10884-015-9482-6.  Google Scholar

[32]

A. Stokes, On two types of moving front in quasilinear diffusion, Math.Biosciences, 31 (1976), 307-315.  doi: 10.1016/0025-5564(76)90087-0.  Google Scholar

[33]

H. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J.Diff. Eqns., 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar

[34]

E. Trofimchuk, P. Alvarado and S. Trofimchuk, On the geometry of wave solutions of a delayed reaction-diffusion equation, J. Diff. Eqns, 246 (2009), 1422-1444.  doi: 10.1016/j.jde.2008.10.023.  Google Scholar

[35]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Monotone waves for non-monotone and non-local monostable reaction-diffusion equations, J. Diff. Eqns, 261 (2016), 1203-1236.  doi: 10.1016/j.jde.2016.03.039.  Google Scholar

[36]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.  doi: 10.1215/kjm/1250522506.  Google Scholar

[37]

Z. Wang, W. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Diff. Eqns., 238 (2007), 153-200.  doi: 10.1016/j.jde.2007.03.025.  Google Scholar

[38]

Z. Wang, W. Li and S. Ruan, Travelling fronts in monostable equations with nonlocal delayed effects, J. Dyn. Differ. Equ., 20 (2008), 573-607.  doi: 10.1007/s10884-008-9103-8.  Google Scholar

[39]

H. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[40]

S.-L. Wu, W.-T. Li and S.-Y. Liu, Oscillatory waves in reaction-diffusion equations with nonlocal delay and crossing-monostability, Non. Lin. Anal.: Real World App., 10 (2009), 3141-3151.  doi: 10.1016/j.nonrwa.2008.10.012.  Google Scholar

[41]

T. Xu, S. Ji, M. Mei and J. Yin, Theoretical and numerical studies on global stability of traveling waves with oscillations for time-delayed nonlocal dispersion equations, preprint, arXiv: 1810.07484. Google Scholar

[42]

T. Yi and X. Zou, Asymptotic behavior, spreading speeds, and traveling waves of nonmonotone dynamical systems, SIAM J. Math.Anal., 47 (2015), 3005-3034.  doi: 10.1137/14095412X.  Google Scholar

[43]

T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition, Proc. R. Soc. A, 466 (2010), 2955-2973.  doi: 10.1098/rspa.2009.0650.  Google Scholar

[44]

T. Yi, Y. Chen and J. Wu, Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves, J. Diff. Eqns, 254 (2013), 3538-3572.  doi: 10.1016/j.jde.2013.01.031.  Google Scholar

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