doi: 10.3934/dcdsb.2021103

Asymptotic behavior of time periodic solutions for extended Fisher-Kolmogorov equations with delays

1. 

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

2. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

3. 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: Pengyu Chen

Received  September 2020 Revised  February 2021 Published  March 2021

Fund Project: Research supported by National Natural Science Foundations of China (No. 12061063, No. 12061062, No. 11771428), Natural Science Foundation of Gansu Province (No. 20JR5RA522), Science Research Project for Colleges and Universities of Gansu Province (No. 2019B-047), Project of NWNU-LKQN2019-3 and Project of NWNU-LKQN2019-13

In this paper, we investigate the global existence, uniqueness and asymptotic stability of time periodic classical solution for a class of extended Fisher-Kolmogorov equations with delays and general nonlinear term. We establish a general framework to investigate the asymptotic behavior of time periodic solutions for nonlinear extended Fisher-Kolmogorov equations with delays and general nonlinear function, which will provide an effective way to deal with such kinds of problems. The discussion is based on the theory of compact and analytic operator semigroups and maximal regularization method.

Citation: Pengyu Chen, Xuping Zhang, Zhitao Zhang. Asymptotic behavior of time periodic solutions for extended Fisher-Kolmogorov equations with delays. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021103
References:
[1]

H. Amann, Periodic solutions of semilinear parabolic equations, in: L. Cesari, R. Kannan, R. Weinberger (Eds.), Nonlinear Analysis: Collection of Papers in Honor of Erich H. Rothe, Academic Press, New York, 1978, 1–29.  Google Scholar

[2]

A. L. A. de Araujo, Periodic solutions for extended Fisher-Kolmogorov and Swift-Hohenberg equations obtained using a continuation theorem, Nonlinear Anal., 94 (2014), 100-106.  doi: 10.1016/j.na.2013.08.007.  Google Scholar

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T. A. Burton, Stability and Periodic Solutions of Ordinary Differential Equations and Functional Differential Equations, Academic Press, Orlando, FL, 1985.  Google Scholar

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A. CaicedoC. CuevasG. M. Mophou and G. M. N'Guérékata, Asymptotic behavior of solutions of some semilinear functional differential and integro-differential equations with infinite delay in Banach spaces, J. Franklin Inst., 349 (2012), 1-24.  doi: 10.1016/j.jfranklin.2011.02.001.  Google Scholar

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P. Danumjaya and A. K. Pani, Orthogonal cubic spline collocation method for the extended Fisher-Kolmogorov equation, J. Comput. Appl. Math., 174 (2005), 101-117.  doi: 10.1016/j.cam.2004.04.002.  Google Scholar

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P. Danumjaya and A. K. Pani, Numerical methods for the extended Fisher-Kolmogorov equation, Int. J. Numer. Anal. Model., 3 (2006), 186-210.   Google Scholar

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G. T. Dee and W. van Saarloos, Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett., 60 (1988), 2641-2644.  doi: 10.1103/PhysRevLett.60.2641.  Google Scholar

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M. do Rosário GrossinhoL. Sanchez and S. A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation, Appl. Math. Lett., 18 (2005), 439-444.  doi: 10.1016/j.aml.2004.03.011.  Google Scholar

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J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981.  Google Scholar

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N. Khiari and K. Omrani, Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions, Comput. Math. Appl., 62 (2011), 4151-4160.  doi: 10.1016/j.camwa.2011.09.065.  Google Scholar

[15]

D. Li and Y. Wang, Asymptotic behavior of gradient systems with small time delays, Nonlinear Anal. Real World Appl., 11 (2010), 1627-1633.  doi: 10.1016/j.nonrwa.2009.03.015.  Google Scholar

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Y. Li, Positive solutions of fourth-order boundary value problems with two parameters, J. Math. Anal. Appl., 281 (2003), 477-484.  doi: 10.1016/S0022-247X(03)00131-8.  Google Scholar

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Y. Li, Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Funct. Anal., 261 (2011), 1309-1324.  doi: 10.1016/j.jfa.2011.05.001.  Google Scholar

[18]

Y. Liu and Z. Li, Schaefer type theorem and periodic solutions of evolution equations, J. Math. Anal. Appl., 316 (2006), 237-255.  doi: 10.1016/j.jmaa.2005.04.045.  Google Scholar

[19]

Z. Ouyang, Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Comput. Math. Appl., 61 (2011), 860-870.  doi: 10.1016/j.camwa.2010.12.034.  Google Scholar

[20]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[21]

L. A. Peletier and W. C. Troy, Spatial patterns described by the extended Fisher-Kolmogorov equation: Periodic solutions, SIAM J. Math. Anal., 28 (1997), 1317-1353.  doi: 10.1137/S0036141095280955.  Google Scholar

[22]

D. Smets and J. B. van den Berg, Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations, J. Differential Equations, 184 (2002), 78-96.  doi: 10.1006/jdeq.2001.4135.  Google Scholar

[23]

R. Teman, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, second ed., Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[24]

Z. WangY. Liu and X. Liu, On global asymptotic stability of neural networks with discrete and distributed delays, Physics Lett. A, 345 (2005), 299-308.  doi: 10.1016/j.physleta.2005.07.025.  Google Scholar

[25]

J. Wu, Theory and Application of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[26]

X. Xiang and N. U. Ahmed, Existence of periodic solutions of semilinear evolution equations with time lags, Nonlinear Anal., 18 (1992), 1063-1070.  doi: 10.1016/0362-546X(92)90195-K.  Google Scholar

[27]

J. ZhuY. Liu and Z. Li, The existence and attractivity of time periodic solutions for evolution equations with delays, Nonlinear Anal. Real World Appl., 9 (2008), 842-851.  doi: 10.1016/j.nonrwa.2007.01.004.  Google Scholar

[28]

B. ZhuL. Liu and Y. Wu, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl. Math. Lett., 61 (2016), 73-79.  doi: 10.1016/j.aml.2016.05.010.  Google Scholar

show all references

References:
[1]

H. Amann, Periodic solutions of semilinear parabolic equations, in: L. Cesari, R. Kannan, R. Weinberger (Eds.), Nonlinear Analysis: Collection of Papers in Honor of Erich H. Rothe, Academic Press, New York, 1978, 1–29.  Google Scholar

[2]

A. L. A. de Araujo, Periodic solutions for extended Fisher-Kolmogorov and Swift-Hohenberg equations obtained using a continuation theorem, Nonlinear Anal., 94 (2014), 100-106.  doi: 10.1016/j.na.2013.08.007.  Google Scholar

[3]

T. A. Burton and B. Zhang, Periodic solutions of abstract differential equations with infinite delay, J. Differential Equations, 90 (1991), 357-396.  doi: 10.1016/0022-0396(91)90153-Z.  Google Scholar

[4]

T. A. Burton, Stability and Periodic Solutions of Ordinary Differential Equations and Functional Differential Equations, Academic Press, Orlando, FL, 1985.  Google Scholar

[5]

A. CaicedoC. CuevasG. M. Mophou and G. M. N'Guérékata, Asymptotic behavior of solutions of some semilinear functional differential and integro-differential equations with infinite delay in Banach spaces, J. Franklin Inst., 349 (2012), 1-24.  doi: 10.1016/j.jfranklin.2011.02.001.  Google Scholar

[6]

X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153.  Google Scholar

[7]

P. CoulletC. Elphick and D. Repaux, Nature of spatial chaos, Phys. Rev. Lett., 58 (1987), 431-434.  doi: 10.1103/PhysRevLett.58.431.  Google Scholar

[8]

P. Danumjaya and A. K. Pani, Orthogonal cubic spline collocation method for the extended Fisher-Kolmogorov equation, J. Comput. Appl. Math., 174 (2005), 101-117.  doi: 10.1016/j.cam.2004.04.002.  Google Scholar

[9]

P. Danumjaya and A. K. Pani, Numerical methods for the extended Fisher-Kolmogorov equation, Int. J. Numer. Anal. Model., 3 (2006), 186-210.   Google Scholar

[10]

G. T. Dee and W. van Saarloos, Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett., 60 (1988), 2641-2644.  doi: 10.1103/PhysRevLett.60.2641.  Google Scholar

[11]

M. do Rosário GrossinhoL. Sanchez and S. A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation, Appl. Math. Lett., 18 (2005), 439-444.  doi: 10.1016/j.aml.2004.03.011.  Google Scholar

[12]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981.  Google Scholar

[14]

N. Khiari and K. Omrani, Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions, Comput. Math. Appl., 62 (2011), 4151-4160.  doi: 10.1016/j.camwa.2011.09.065.  Google Scholar

[15]

D. Li and Y. Wang, Asymptotic behavior of gradient systems with small time delays, Nonlinear Anal. Real World Appl., 11 (2010), 1627-1633.  doi: 10.1016/j.nonrwa.2009.03.015.  Google Scholar

[16]

Y. Li, Positive solutions of fourth-order boundary value problems with two parameters, J. Math. Anal. Appl., 281 (2003), 477-484.  doi: 10.1016/S0022-247X(03)00131-8.  Google Scholar

[17]

Y. Li, Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Funct. Anal., 261 (2011), 1309-1324.  doi: 10.1016/j.jfa.2011.05.001.  Google Scholar

[18]

Y. Liu and Z. Li, Schaefer type theorem and periodic solutions of evolution equations, J. Math. Anal. Appl., 316 (2006), 237-255.  doi: 10.1016/j.jmaa.2005.04.045.  Google Scholar

[19]

Z. Ouyang, Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Comput. Math. Appl., 61 (2011), 860-870.  doi: 10.1016/j.camwa.2010.12.034.  Google Scholar

[20]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[21]

L. A. Peletier and W. C. Troy, Spatial patterns described by the extended Fisher-Kolmogorov equation: Periodic solutions, SIAM J. Math. Anal., 28 (1997), 1317-1353.  doi: 10.1137/S0036141095280955.  Google Scholar

[22]

D. Smets and J. B. van den Berg, Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations, J. Differential Equations, 184 (2002), 78-96.  doi: 10.1006/jdeq.2001.4135.  Google Scholar

[23]

R. Teman, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, second ed., Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[24]

Z. WangY. Liu and X. Liu, On global asymptotic stability of neural networks with discrete and distributed delays, Physics Lett. A, 345 (2005), 299-308.  doi: 10.1016/j.physleta.2005.07.025.  Google Scholar

[25]

J. Wu, Theory and Application of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[26]

X. Xiang and N. U. Ahmed, Existence of periodic solutions of semilinear evolution equations with time lags, Nonlinear Anal., 18 (1992), 1063-1070.  doi: 10.1016/0362-546X(92)90195-K.  Google Scholar

[27]

J. ZhuY. Liu and Z. Li, The existence and attractivity of time periodic solutions for evolution equations with delays, Nonlinear Anal. Real World Appl., 9 (2008), 842-851.  doi: 10.1016/j.nonrwa.2007.01.004.  Google Scholar

[28]

B. ZhuL. Liu and Y. Wu, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl. Math. Lett., 61 (2016), 73-79.  doi: 10.1016/j.aml.2016.05.010.  Google Scholar

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