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May  2012, 11(3): 1231-1252. doi: 10.3934/cpaa.2012.11.1231

Dynamics of non-autonomous nonclassical diffusion equations on $R^n$

Cung The Anh 1, and  Tang Quoc Bao 2,

1. 

Department of Mathematics, Hanoi National University of Education, 36 Xuan Thuy, Cau Giay, Hanoi, Vietnam
2. 

Faculty of Applied Mathematics and Informatics, Hanoi University of Science and Technology, No 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam

Received  December 2010 Revised  May 2011 Published  December 2011

We consider the Cauchy problem for a non-autonomous nonclassical diffusion equation of the form $u_t-\varepsilon\Delta u_t - \Delta u+f(u)+\lambda u=g(t)$ on $R^n$. Under an arbitrary polynomial growth order of the nonlinearity $f$ and a suitable exponent growth of the external force $g$, using the method of tail-estimates and the asymptotic a priori estimate method, we prove the existence of an $(H^{1}(R^n) L^{p}(R^n), H^{1}(R^n) L^{p}(R^n))$ - pullback attractor $\hat{A}_{\varepsilon}$ for the process associated to the problem. We also prove the upper semicontinuity of $\{\hat{A}_{\varepsilon}: \varepsilon\in [0,1]\}$ at $\varepsilon = 0$.
Keywords: method of tail estimates, weak solution, Non-autonomous nonclassical diffusion equation, unbounded domain., pullback attractor, upper semicontinuity, a priori.
Mathematics Subject Classification: 35B41, 35K57, 35D05, 35B30.
Citation: Cung The Anh, Tang Quoc Bao. Dynamics of non-autonomous nonclassical diffusion equations on $R^n$. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1231-1252. doi: 10.3934/cpaa.2012.11.1231
References:
[1]

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

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

[3]

A. N. Carvalho, J. A. Langa and J. C. Robinson, On the continuity of pullback attractors for evolution processes, Nonlinear Anal., 71 (2009), 1812-1824. doi: 10.1016/j.na.2009.01.016.  Google Scholar

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Y. Li and C. K. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comp., 190 (2007), 1020-1029. doi: 10.1016/j.amc.2006.11.187.  Google Scholar

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Y. Li, S. Wang and H. Wu, Pullback attractors for non-autonomous reaction-diffusion equations in $L^p$, Appl. Math. Comp., 207 (2009), 373-379. doi: 10.1016/j.amc.2008.10.065.  Google Scholar

[6]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Paris, 1969.  Google Scholar

[7]

Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroups and applications, Indian University Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.  Google Scholar

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J. C. Peter and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627. doi: 10.1007/BF01594969.  Google Scholar

[9]

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

[10]

C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymp. Anal., 59 (2009), 51-81. doi: 10.3233/ASY-2008-0886.  Google Scholar

[11]

C. Truesdell and W. Noll, "The Nonlinear Field Theories of Mechanics," Encyclomedia of Physics, Springer, Berlin, 1995.  Google Scholar

[12]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 179 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[13]

B. Wang, Pullback attractors for non-autonomous reaction-diffusion equations on $\mathbb R^n$, Front. Math. China, 4 (2009), 563-583. doi: 10.1007/s11464-009-0033-5.  Google Scholar

[14]

S. Wang, D. Li and C. 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

[15]

Y. Xiao, Attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin., Engl. Ser, 18 (2002), 273-276. doi: 10.1007/s102550200026.  Google Scholar

[16]

C. K. Zhong, M. H. Yang and C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 15 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.  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 Anal., 73 (2010), 399-412. doi: 10.1016/j.na.2010.03.031.  Google Scholar

[3]

A. N. Carvalho, J. A. Langa and J. C. Robinson, On the continuity of pullback attractors for evolution processes, Nonlinear Anal., 71 (2009), 1812-1824. doi: 10.1016/j.na.2009.01.016.  Google Scholar

[4]

Y. Li and C. K. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comp., 190 (2007), 1020-1029. doi: 10.1016/j.amc.2006.11.187.  Google Scholar

[5]

Y. Li, S. Wang and H. Wu, Pullback attractors for non-autonomous reaction-diffusion equations in $L^p$, Appl. Math. Comp., 207 (2009), 373-379. doi: 10.1016/j.amc.2008.10.065.  Google Scholar

[6]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Paris, 1969.  Google Scholar

[7]

Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroups and applications, Indian University Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.  Google Scholar

[8]

J. C. Peter and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627. doi: 10.1007/BF01594969.  Google Scholar

[9]

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

[10]

C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymp. Anal., 59 (2009), 51-81. doi: 10.3233/ASY-2008-0886.  Google Scholar

[11]

C. Truesdell and W. Noll, "The Nonlinear Field Theories of Mechanics," Encyclomedia of Physics, Springer, Berlin, 1995.  Google Scholar

[12]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 179 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[13]

B. Wang, Pullback attractors for non-autonomous reaction-diffusion equations on $\mathbb R^n$, Front. Math. China, 4 (2009), 563-583. doi: 10.1007/s11464-009-0033-5.  Google Scholar

[14]

S. Wang, D. Li and C. 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

[15]

Y. Xiao, Attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin., Engl. Ser, 18 (2002), 273-276. doi: 10.1007/s102550200026.  Google Scholar

[16]

C. K. Zhong, M. H. Yang and C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 15 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.  Google Scholar

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