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

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

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 $\varepsilon = 0$.
References:
[1]

E. C. Aifantis, On the problem of diffusion in solids,, Acta Mech., 37 (1980), 265.  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.  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.  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.  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.  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, (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.  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.  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., 23 (2007), 1271.  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.  doi: 10.3233/ASY-2008-0886.  Google Scholar

[11]

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

[12]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains,, Physica D, 179 (1999), 41.  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.  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.  doi: 10.1016/j.jmaa.2005.06.094.  Google Scholar

[15]

Y. Xiao, Attractors for a nonclassical diffusion equation,, Acta Math. Appl. Sin., 18 (2002), 273.  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.  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.  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.  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.  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.  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.  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, (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.  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.  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., 23 (2007), 1271.  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.  doi: 10.3233/ASY-2008-0886.  Google Scholar

[11]

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

[12]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains,, Physica D, 179 (1999), 41.  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.  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.  doi: 10.1016/j.jmaa.2005.06.094.  Google Scholar

[15]

Y. Xiao, Attractors for a nonclassical diffusion equation,, Acta Math. Appl. Sin., 18 (2002), 273.  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.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

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