Abstract
In this article, we consider the dynamical behavior of the nonclassical diffusion equation in unbounded domain while the nonlinearity satisfy the arbitrary order polynomial growth conditions. Using the tail-estimated method and the asymptotic a priori estimate method, we obtain the existence of \({(H^1(\Omega)\cap L^{p}(\Omega),L^{2}(\Omega))}\)-global attractor, \({(H^1(\Omega)\cap L^{p}(\Omega),L^{p}(\Omega))}\)-global attractor, \({(H^1(\Omega)\cap L^{p}(\Omega),H^{1}(\Omega))}\)-global attractor and \({(H^1(\Omega)\cap L^{p}(\Omega),H^1(\Omega)\cap L^{p}(\Omega))}\)-global attractor.
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References
Aifantis E.C.: On the problem of diffusion in solids. Acta Mech. 37, 265–296 (1980)
Aifantis E.C.: Gradient nanomechanics: applications to deformation, fracture, and diffusion in nanopolycrystals. Metall. Mater. Trans. A 42, 2985–2998 (2011)
Anguiano, M., Caraballo, T., Real, J., et al.: Pullback attractors for reaction–diffusion equations in some unbounded domains with an H −1-valued non-autonomous forcing term and without uniqueness of solutions. Discret. Contin. Dyn. Syst. Ser. B 14, 307–326 (2010)
Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North-Holland, Amsterdam (1992)
Ball J.M.: Global attractors for damped semilinear wave equations. Discret. Contin. Dyn. Syst. 10, 31–52 (2004)
Cholewa J.W., Dlotko T.: Bi-spaces global attractors in abstract parabolic equations. Evol. Equ. Banach Cent. Publ. 60, 13–26 (2003)
Dlotko T., Kania M.B., Sun C.: Analysis of the viscous Cahn–Hilliard equation in \({\mathbb{R}^N}\). J. Differ. Equ. 252(3), 2771–2791 (2012)
Dlotko T., Kania M.B., Ma S.: Korteweg–de Vries–Burgers system in \({\mathbb{R}^N}\). J. Math. Anal. Appl. 411(2), 853–872 (2014)
Efendiev, M.A., Zelik, S.V.: The attractor for a nonlinear reaction diffusion system in an unbounded domain. Commun. Pure Appl. Math. LIV 54, 0625–0688 (2001)
Elliott C.M., Stuart A.M.: Viscous Cahn–Hilliard equation II. Analysis. J. Differ. Equ. 128(2), 387–414 (1996)
Kalantarov V.K.: Attractors for some non-linear problems of mathematical physics. Zap. Nauch. Sem. LOMI 152, 50–54 (1986)
Kuttler K., Aifantis E.C.: Existence and uniqueness in nonclassical diffusion. Q. Appl. Math. 45, 549–560 (1987)
Kuttler K., Aifantis E.C.: Quasilinear evolution equations in nonclassical diffusion. SIAM J. Math. Anal. 19, 110–120 (1988)
Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. Springer, Berlin (1972)
Liu Y.F., Ma Q.Z.: Exponential attractors for a nonclassical diffusion equation. Electron. J. Differ. Equ. 9, 1–9 (2009)
Ma Q.F., Wang S.H., Zhong C.K.: Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana Math. J. 6, 1542–1558 (2002)
Ma, Q.Z., Liu, Y.F., Zhang, F.H.: Global attractors in \({H^1(\mathbb{R}^N)}\) for nonclassical diffusion equations. Discret. Dyn. Nat. Soc. 2012, Article ID 672762. doi:10.1155/2012/672762
Pata V., Zelik S.: A remark on the damped wave equation. Commun. Pure Appl. Anal. 5(3), 611–616 (2006)
Robinson, C.: Infinite-Dimensional Dynamincal Systems: An Introduction to Disspative Parabolic PDEs and the Theory of Global attractors. Cambridge Universlty Press, Cambridge (2001)
Souplet P., Weissler F.: Poincare’s inequality and global solutions of a nonlinear parabolic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16(3), 335–371 (1999)
Souplet P.: Decay of heat semigroups in \({L^\infty}\) and applications to nonlinear parabolic problems in unbounded domains. J. Funct. Anal. 173(2), 343–360 (2000)
Stanoyevitch, A.: Products of Poincaré domains. Proc. Am. Math. Soc. 117(1), 79–87 (1993)
Sun C.Y., Wang S.Y., Zhong C.K.: Global attractors for a nonclassical diffusion equation. Acta Math. Sin. Engl. Ser. 23, 1271–1280 (2007)
Sun C.Y., Yang M.H.: Dynamics of the nonclassical diffusion equations. Asymptot. Anal. 59, 51–81 (2008)
Sun C.Y., Zhong C. K.: Attractors for the semilinear reaction–diffusion equation with distribution derivatives in unbounded domains. Nonlinear Anal. TMA 63(1), 49–65 (2005)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1997)
Wang S.Y., Li D.S., Zhong C.K.: On the dynamics of a class of nonclassical parabolic equations. J. Math. Anal. Appl. 317, 565–582 (2006)
Wang B.X.: Attractors for reaction diffusion equations in unbounded domains. Phys. D 128, 41–52 (1999)
Xiao Y.L.: Attractors for a nonclassical diffusion equation. Acta Math. Appl. Sin. 18, 273–276 (2002)
Zhang Y.H., Zhong C.K., Wang S.Y.: Attractors in \({{L}^{2}(\mathbb{R}^N)}\) for a class of reaction–diffusion equations. Nonlinear Anal. 71, 1901–1908 (2009)
Zhang F.H., Liu Y.F.: Pullback attractors in \({H^1(\mathbb{R}^N)}\) for non-autonomous nonclassical diffusion equations. Dyn. Syst. 29(1), 106–118 (2014)
Zhong C.K., Yang M.H., Sun C.Y.: The existence of global attractors for the norm-to-weak continuous semigroup and its application to the nonlinear reaction–diffusion equations. J. Differ. Equ. 223, 367–399 (2006)
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Li, Jl., Zhang, Fh. Bi-space Global Attractors for a Class of Nonclassical Parabolic Equations with Arbitrary Polynomial Growth in Unbounded Domain. Mediterr. J. Math. 13, 1807–1821 (2016). https://doi.org/10.1007/s00009-015-0617-0
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DOI: https://doi.org/10.1007/s00009-015-0617-0