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July  2017, 22(5): 1817-1833. doi: 10.3934/dcdsb.2017108

Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic

Tomás Caraballo 1, , Antonio M. Márquez-Durán 2, and  Rivero Felipe 3,

1. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160,41080-Sevilla, Spain
2. 

Departamento de Economía, Métodos Cuantitativos e Historia Económica, Universidad Pablo de Olavide, Ctra. de Utrera, Km. 1,41013-Sevilla, Spain
3. 

Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Mário Santos Braga s/n, Centro, CEP 24020140 -Niterói, RJ, Brazil

Received  January 2016 Revised  July 2016 Published  March 2017

Fund Project: Partially supported by FEDER and Ministerio de Economía y Competitividad grant # MTM2015-63723-P and Junta de Andalucía under Proyecto de Excelencia P12-FQM-1492 and # FQM314 (Spain).

Our aim in this work is the study of the existence and uniqueness of solutions for a non-classical and non-autonomous diffusion equation containing infinite delay terms. We also analyze the asymptotic behaviour of the system in the pullback sense and, under suitable additional conditions, we obtain global exponential decay of the solutions of the evolutionary problem to stationary solutions.

Keywords: Delay equation, pullback attractor, nonautonomous problem, evolution process, nonclassical diffusion equation.
Mathematics Subject Classification: Primary: 35R15, 35B41, 37B55; Secondary: 47J3.
Citation: Tomás Caraballo, Antonio M. Márquez-Durán, Rivero Felipe. Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1817-1833. doi: 10.3934/dcdsb.2017108
References:
[1]

E. C. Aifantis, On the problem of diffusion in solids, Acta Mech., 37 (1980), 265-296.  doi: 10.1007/BF01202949.  Google Scholar

[2]

C. T. Anh and T. Q. Bao, Pullback attractors for a class of non-autonomous nonclassical diffusion equations, Nonlinear Analysis, 73 (2010), 399-412.  doi: 10.1016/j.na.2010.03.031.  Google Scholar

[3]

F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2591-2636.  doi: 10.1142/S0218127410027246.  Google Scholar

[4]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1665.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[5]

T. Caraballo, M. J. Garrido-Atienza, B Schmalfußand and J. Valero, non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.  Google Scholar

[6]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal, 72 (2010), 1967-1976.  doi: 10.1016/j.na.2009.09.037.  Google Scholar

[7]

T. Caraballo and A. M. Márquez-Durán, Existence, uniqueness and asymptotic behavior of solutions for a nonclassical difusion equation with delay, Dynamics of Partial Differential Equations, 10 (2013), 267-281.  doi: 10.4310/DPDE.2013.v10.n3.a3.  Google Scholar

[8]

T. Caraballo, A. M. Márquez-Durán and F. Rivero, ell-posedness and asymptotic behaviour for a non-classical and non-autonomous diffusion equation with delay, nternational J. Bifurcation and Chaos, 25 (2015), 1540021, 11pp.  doi: 10.1142/S0218127415400210.  Google Scholar

[9]

T. Caraballo, A. M. Márquez-Durán and F. Rivero, A Nonclassical and Nonautonomous Diffusion Equation Containing Infinite Delays, Differential and Difference Equations with Applications. ICDDEA 2015. Springer Proceedings in Mathematics & Statistics, vol. 164 (2016) 385-399. Google Scholar

[10]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of non-autonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[11]

J. K. Hale, Asymptotic Behavior of Dissipative System American Mathematical Society, 1988. Google Scholar

[12]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.   Google Scholar

[13]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay Lecture Notes in Mathematics, 1473, Berlin: Springer-Verlag, 1991. Google Scholar

[14]

Z. Hu and Y. Wang, Pullback attractors for a non-autonomous nonclassical diffusion equation with variable delay, J. Math. Phys., 53 (2012), 072702, 17pp.  doi: 10.1063/1.4736847.  Google Scholar

[15]

K. Kuttlerand and E. C. Aifantis, Existence and uniqueness in non classical diffusion, Quarterly of Applied Mathematics, 45 (1987), 549-560.  doi: 10.1090/qam/910461.  Google Scholar

[16]

K. Kuttlerand and E. C. Aifantis, Quasilinear evolution equations in nonclassical diffusion, SIAM Journal on Mathematical Analysis, 19 (1988), 110-120.  doi: 10.1137/0519008.  Google Scholar

[17]

J. L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969. Google Scholar

[18]

Q. Ma, Y. Liu and F. Zhang, Global attractors in $H^1(\mathbb R^N)$ for nonclassical diffusion equations, Discrete Dyn. Nat. Soc. 2012, Art. ID 672762, 16 pp. Google Scholar

[19]

P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a 2D-Navier-Stokes model in an infinite delay case, submitted. Google Scholar

[20]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[21]

J. C. Peter and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627.   Google Scholar

[22]

F. Rivero, Pullback attractor for non-autonomous non-classical parabolic equation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 209-221.  doi: 10.3934/dcdsb.2013.18.209.  Google Scholar

[23]

C. Sun, S. Wang and C. Zhong, Global attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin. Engl., 23 (2007), 1271-1280.  doi: 10.1007/s10114-005-0909-6.  Google Scholar

[24]

C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymptotic Analysis, 59 (2008), 51-81.   Google Scholar

[25]

S. Wang, D. Li and Ch. Zhong, On the dynamics of a class of nonclassical parabolic equations, J. Math. Anal. Appl., 317 (2006), 565-582.  doi: 10.1016/j.jmaa.2005.06.094.  Google Scholar

show all references

References:
[1]

E. C. Aifantis, On the problem of diffusion in solids, Acta Mech., 37 (1980), 265-296.  doi: 10.1007/BF01202949.  Google Scholar

[2]

C. T. Anh and T. Q. Bao, Pullback attractors for a class of non-autonomous nonclassical diffusion equations, Nonlinear Analysis, 73 (2010), 399-412.  doi: 10.1016/j.na.2010.03.031.  Google Scholar

[3]

F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2591-2636.  doi: 10.1142/S0218127410027246.  Google Scholar

[4]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1665.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[5]

T. Caraballo, M. J. Garrido-Atienza, B Schmalfußand and J. Valero, non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.  Google Scholar

[6]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal, 72 (2010), 1967-1976.  doi: 10.1016/j.na.2009.09.037.  Google Scholar

[7]

T. Caraballo and A. M. Márquez-Durán, Existence, uniqueness and asymptotic behavior of solutions for a nonclassical difusion equation with delay, Dynamics of Partial Differential Equations, 10 (2013), 267-281.  doi: 10.4310/DPDE.2013.v10.n3.a3.  Google Scholar

[8]

T. Caraballo, A. M. Márquez-Durán and F. Rivero, ell-posedness and asymptotic behaviour for a non-classical and non-autonomous diffusion equation with delay, nternational J. Bifurcation and Chaos, 25 (2015), 1540021, 11pp.  doi: 10.1142/S0218127415400210.  Google Scholar

[9]

T. Caraballo, A. M. Márquez-Durán and F. Rivero, A Nonclassical and Nonautonomous Diffusion Equation Containing Infinite Delays, Differential and Difference Equations with Applications. ICDDEA 2015. Springer Proceedings in Mathematics & Statistics, vol. 164 (2016) 385-399. Google Scholar

[10]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of non-autonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[11]

J. K. Hale, Asymptotic Behavior of Dissipative System American Mathematical Society, 1988. Google Scholar

[12]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.   Google Scholar

[13]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay Lecture Notes in Mathematics, 1473, Berlin: Springer-Verlag, 1991. Google Scholar

[14]

Z. Hu and Y. Wang, Pullback attractors for a non-autonomous nonclassical diffusion equation with variable delay, J. Math. Phys., 53 (2012), 072702, 17pp.  doi: 10.1063/1.4736847.  Google Scholar

[15]

K. Kuttlerand and E. C. Aifantis, Existence and uniqueness in non classical diffusion, Quarterly of Applied Mathematics, 45 (1987), 549-560.  doi: 10.1090/qam/910461.  Google Scholar

[16]

K. Kuttlerand and E. C. Aifantis, Quasilinear evolution equations in nonclassical diffusion, SIAM Journal on Mathematical Analysis, 19 (1988), 110-120.  doi: 10.1137/0519008.  Google Scholar

[17]

J. L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969. Google Scholar

[18]

Q. Ma, Y. Liu and F. Zhang, Global attractors in $H^1(\mathbb R^N)$ for nonclassical diffusion equations, Discrete Dyn. Nat. Soc. 2012, Art. ID 672762, 16 pp. Google Scholar

[19]

P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a 2D-Navier-Stokes model in an infinite delay case, submitted. Google Scholar

[20]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[21]

J. C. Peter and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627.   Google Scholar

[22]

F. Rivero, Pullback attractor for non-autonomous non-classical parabolic equation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 209-221.  doi: 10.3934/dcdsb.2013.18.209.  Google Scholar

[23]

C. Sun, S. Wang and C. Zhong, Global attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin. Engl., 23 (2007), 1271-1280.  doi: 10.1007/s10114-005-0909-6.  Google Scholar

[24]

C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymptotic Analysis, 59 (2008), 51-81.   Google Scholar

[25]

S. Wang, D. Li and Ch. Zhong, On the dynamics of a class of nonclassical parabolic equations, J. Math. Anal. Appl., 317 (2006), 565-582.  doi: 10.1016/j.jmaa.2005.06.094.  Google Scholar

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