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August  2019, 24(8): 3947-3970. doi: 10.3934/dcdsb.2018338

Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay

Rafael Obaya 1,, and  Ana M. Sanz 2,

1. 

Departamento de Matemática Aplicada, E. Ingenierías Industriales, Universidad de Valladolid, 47011 Valladolid, Spain, IMUVA, Instituto de Investigación en Matemáticas, Universidad de Valladolid
2. 

Departamento de Didáctica de las Ciencias Experimentales, Sociales y de la Matemática. Facultad de Educación de Palencia, Universidad de Valladolid, 34004 Palencia, Spain, IMUVA, Instituto de Investigación en Matemáticas, Universidad de Valladolid

* Corresponding author

Dedicated to Peter E. Kloeden on the occasion of his 70th birthday

Received  June 2018 Revised  August 2018 Published  August 2019 Early access  January 2019

Fund Project: The authors were partly supported by MINECO/FEDER grant MTM2015-66330-P, and the European Commission under project H2020-MSCA-ITN-2014 643073 CRITICS.

This paper provides a dynamical frame to study non-autonomous parabolic partial differential equations with finite delay. Assuming monotonicity of the linearized semiflow, conditions for the existence of a continuous separation of type Ⅱ over a minimal set are given. Then, practical criteria for the uniform or strict persistence of the systems above a minimal set are obtained.

Keywords: Topological dynamics, skew-product semiflows, continuous separation, partial functional differential equations with finite delay, uniform and strict persistence.
Mathematics Subject Classification: 35K58, 37B55, 37C60, 37C65.
Citation: Rafael Obaya, Ana M. Sanz. Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3947-3970. doi: 10.3934/dcdsb.2018338
References:
[1]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.

[2]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964.

[3]

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

[4]

R. Johnson, R. Obaya, S. Novo, C. Núñez and R. Fabbri, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control, Developments in Mathematics 36, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-29025-6.

[5]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moskow, 1967 (Russian). English transl.: Transl. Math. Monographs, AMS, Providence, 1968.

[6]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications Vol. 16, Birkhäuser, Basel, Boston, Berlin, 1995. doi: 10.1007/978-3-0348-9234-6.

[7]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc. 321 (1990), 1-44. doi: 10.2307/2001590.

[8]

R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35. 

[9]

J. Mierczyński and W. Shen, Lyapunov exponents and asymptotic dynamics in random Kolmogorov models, J. Evol. Equ., 4 (2004), 371-390. doi: 10.1007/s00028-004-0160-0.

[10]

S. Novo, C. Núñez, R. Obaya and A. M. Sanz, Skew-product semiflows for non-autonomous partial functional differential equations with delay, Discrete Continuous Dynam. Systems - A, 34 (2014), 4291-4321.  doi: 10.3934/dcds.2014.34.4291.

[11]

S. Novo, R. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dynamics Differential Equations, 25 (2013), 1201-1231. doi: 10.1007/s10884-013-9337-y.

[12]

S. Novo, R. Obaya and A. M. Sanz, Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows, Nonlinearity, 26 (2013), 2409-2440.  doi: 10.1088/0951-7715/26/9/2409.

[13]

C. Núñez, R. Obaya and A. M. Sanz, Minimal sets in monotone and sublinear skew-product semiflows Ⅱ: Two-dimensional systems of differential equations, J. Differential Equations, 248 (2010), 1899-1925. doi: 10.1016/j.jde.2009.12.006.

[14]

R. Obaya and A. M. Sanz, Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems, J. Differential Equations, 261 (2016), 4135-4163. doi: 10.1016/j.jde.2016.06.019.

[15]

R. Obaya and A. M. Sanz, Is uniform persistence a robust property in almost periodic models? A well-behaved family: almost periodic Nicholson systems, Nonlinearity, 31 (2018), 388-413. doi: 10.1088/1361-6544/aa92e7.

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[17]

P. Poláčik and I. Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynamics Differential Equations, 5 (1993), 279-303. doi: 10.1007/BF01053163.

[18]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.

[19]

R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations, 113 (1994), 17-67.  doi: 10.1006/jdeq.1994.1113.

[20]

W. Shen and Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows, Mem. Amer. Math. Soc., 647, Amer. Math. Soc., Providence, 1998. doi: 10.1090/memo/0647.

[21]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, 1995.

[22]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.

[23]

C. C. Travis and G. F. Webb, Existence, stability, and compactness in the $\alpha$-norm for partial functional differential equations, Trans. Amer. Math. Soc., 240 (1978), 129-143. doi: 10.2307/1998809.

[24]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

show all references

References:
[1]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.

[2]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964.

[3]

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

[4]

R. Johnson, R. Obaya, S. Novo, C. Núñez and R. Fabbri, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control, Developments in Mathematics 36, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-29025-6.

[5]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moskow, 1967 (Russian). English transl.: Transl. Math. Monographs, AMS, Providence, 1968.

[6]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications Vol. 16, Birkhäuser, Basel, Boston, Berlin, 1995. doi: 10.1007/978-3-0348-9234-6.

[7]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc. 321 (1990), 1-44. doi: 10.2307/2001590.

[8]

R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35. 

[9]

J. Mierczyński and W. Shen, Lyapunov exponents and asymptotic dynamics in random Kolmogorov models, J. Evol. Equ., 4 (2004), 371-390. doi: 10.1007/s00028-004-0160-0.

[10]

S. Novo, C. Núñez, R. Obaya and A. M. Sanz, Skew-product semiflows for non-autonomous partial functional differential equations with delay, Discrete Continuous Dynam. Systems - A, 34 (2014), 4291-4321.  doi: 10.3934/dcds.2014.34.4291.

[11]

S. Novo, R. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dynamics Differential Equations, 25 (2013), 1201-1231. doi: 10.1007/s10884-013-9337-y.

[12]

S. Novo, R. Obaya and A. M. Sanz, Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows, Nonlinearity, 26 (2013), 2409-2440.  doi: 10.1088/0951-7715/26/9/2409.

[13]

C. Núñez, R. Obaya and A. M. Sanz, Minimal sets in monotone and sublinear skew-product semiflows Ⅱ: Two-dimensional systems of differential equations, J. Differential Equations, 248 (2010), 1899-1925. doi: 10.1016/j.jde.2009.12.006.

[14]

R. Obaya and A. M. Sanz, Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems, J. Differential Equations, 261 (2016), 4135-4163. doi: 10.1016/j.jde.2016.06.019.

[15]

R. Obaya and A. M. Sanz, Is uniform persistence a robust property in almost periodic models? A well-behaved family: almost periodic Nicholson systems, Nonlinearity, 31 (2018), 388-413. doi: 10.1088/1361-6544/aa92e7.

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[17]

P. Poláčik and I. Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynamics Differential Equations, 5 (1993), 279-303. doi: 10.1007/BF01053163.

[18]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.

[19]

R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations, 113 (1994), 17-67.  doi: 10.1006/jdeq.1994.1113.

[20]

W. Shen and Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows, Mem. Amer. Math. Soc., 647, Amer. Math. Soc., Providence, 1998. doi: 10.1090/memo/0647.

[21]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, 1995.

[22]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.

[23]

C. C. Travis and G. F. Webb, Existence, stability, and compactness in the $\alpha$-norm for partial functional differential equations, Trans. Amer. Math. Soc., 240 (1978), 129-143. doi: 10.2307/1998809.

[24]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

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