Time dependent perturbation in a non-autonomous non-classical parabolic equation

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January 2013, 18(1): 209-221. doi: 10.3934/dcdsb.2013.18.209

Time dependent perturbation in a non-autonomous non-classical parabolic equation

Felipe Rivero ^1,

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

Received January 2012 Revised June 2012 Published September 2012

Abstract
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n this paper we study the existence and characterization of a pullback attractor for a non-autonomous non-classical parabolic equation of the form \begin{equation}\label{EQnoncla} \left\{ \begin{split} &u_t-\gamma(t)\Delta u_t-\Delta u=f(u) \mbox{ in }\Omega,\\ &u=0 \mbox{ on }\partial\Omega \end{split} \right. (1) \end{equation} in a sufficiently smooth bounded domain $\Omega\subset\mathbb R^n$ with $f$ and $\gamma$ satisfying some suitable natural conditions. We prove the well posedness of this model and the existence of a pullback attractor. We show that this pullback attractor is characterized as the union of unstable sets of the associated equilibria and that this characterization is stable under time dependent perturbation of the nonlinearity.

Keywords: Evolution processes, non-autonomous parabolic equations, attractors structure, time dependent perturbation, pullback attractors, continuity of attractors., non-autonomous dynamical systems.

Mathematics Subject Classification: 35B20, 35B50, 35K55, 37B55, 37C7.

Citation: Felipe Rivero. Time dependent perturbation in a non-autonomous non-classical parabolic equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 209-221. doi: 10.3934/dcdsb.2013.18.209

References:

[1]	A. B. Babin and M. I. Vishik, "Attractors of Evolution Equations,'' North-Holland, Amsterdam, 1992.
[2]	T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Analysis, 72 (2010), 1967-1976. doi: 10.1016/j.na.2009.09.037.
[3]	T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A gradient-like non-autonomous evolution processes, Int. Journal of Bifurcation and Chaos, 20 (2010), 2751-2760. doi: 10.1142/S0218127410027337.
[4]	T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Analysis, 74 (2011), 2272-2283. doi: 10.1016/j.na.2010.11.032.
[5]	A. N. Carvallo and J. W. Cholewa, Local well possed, asymptotic behaviour and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time, Trans. Amer. Math. Soc., 361 (2009), 2567-2586. doi: 10.1090/S0002-9947-08-04789-2.
[6]	A. N. Carvalho and J. A. Langa, An extension of the concept of gradient systems which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668. doi: 10.1016/j.jde.2009.01.007.
[7]	A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds, J. Differential Equations, 233 (2007), 622-653. doi: 10.1016/j.jde.2006.08.009.
[8]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Lower semicontinuity of attractors for non-autonomous dynamical systems, Ergod. Th. & Dynam. Systems, 29 (2009), 1765-1780. doi: 10.1017/S0143385708000850.
[9]	A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed gradient system, J. Differential Equations, 236 (2007), 570-603. doi: 10.1016/j.jde.2007.01.017.
[10]	V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,'' Colloquium Publications 49. American Mathematical Society, 2002.
[11]	J. K. Hale, "Asymptotic Behavior of Dissipative System,'' Mathematical Surveys and Monographs vol. 25, 1989.
[12]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'' Springer-Verlag, Berlin-New York, 1981.
[13]	P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical System,'' Mathematical Surveys and Monographs vol. 176, 2011.
[14]	O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,'' Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991.
[15]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Springer-Verlag, New York, 1983.
[16]	J. C. Robinson, "Infinite-Dimensional Dynamical System. An introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,'' Cambridge Text in Applied Mathematics, Cambridge, 2001.
[17]	G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,'' Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002.
[18]	Ch. Sun, S. Wang and Ch. Zhong, Global attractors for a nonclassical diffusion equation, Acta Math. Sin. (Engl. Ser.), 23 (2007), 1271-1280.
[19]	P. E. Sobolevskiĭ, Equations of parabolic type in a Banach space, Trudy Moskov. Mat. Obšč, 10 (1961), 297-350.
[20]	R. Temam, "Infinite-Dimensional Dynamical System in Mechanics and Physics,'' Applied Mathematical Sciences 68, Springer-Verlag, New-York, 1988.
[21]	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.

show all references

References:

[1]	A. B. Babin and M. I. Vishik, "Attractors of Evolution Equations,'' North-Holland, Amsterdam, 1992.
[2]	T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Analysis, 72 (2010), 1967-1976. doi: 10.1016/j.na.2009.09.037.
[3]	T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A gradient-like non-autonomous evolution processes, Int. Journal of Bifurcation and Chaos, 20 (2010), 2751-2760. doi: 10.1142/S0218127410027337.
[4]	T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Analysis, 74 (2011), 2272-2283. doi: 10.1016/j.na.2010.11.032.
[5]	A. N. Carvallo and J. W. Cholewa, Local well possed, asymptotic behaviour and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time, Trans. Amer. Math. Soc., 361 (2009), 2567-2586. doi: 10.1090/S0002-9947-08-04789-2.
[6]	A. N. Carvalho and J. A. Langa, An extension of the concept of gradient systems which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668. doi: 10.1016/j.jde.2009.01.007.
[7]	A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds, J. Differential Equations, 233 (2007), 622-653. doi: 10.1016/j.jde.2006.08.009.
[8]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Lower semicontinuity of attractors for non-autonomous dynamical systems, Ergod. Th. & Dynam. Systems, 29 (2009), 1765-1780. doi: 10.1017/S0143385708000850.
[9]	A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed gradient system, J. Differential Equations, 236 (2007), 570-603. doi: 10.1016/j.jde.2007.01.017.
[10]	V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,'' Colloquium Publications 49. American Mathematical Society, 2002.
[11]	J. K. Hale, "Asymptotic Behavior of Dissipative System,'' Mathematical Surveys and Monographs vol. 25, 1989.
[12]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'' Springer-Verlag, Berlin-New York, 1981.
[13]	P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical System,'' Mathematical Surveys and Monographs vol. 176, 2011.
[14]	O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,'' Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991.
[15]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Springer-Verlag, New York, 1983.
[16]	J. C. Robinson, "Infinite-Dimensional Dynamical System. An introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,'' Cambridge Text in Applied Mathematics, Cambridge, 2001.
[17]	G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,'' Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002.
[18]	Ch. Sun, S. Wang and Ch. Zhong, Global attractors for a nonclassical diffusion equation, Acta Math. Sin. (Engl. Ser.), 23 (2007), 1271-1280.
[19]	P. E. Sobolevskiĭ, Equations of parabolic type in a Banach space, Trudy Moskov. Mat. Obšč, 10 (1961), 297-350.
[20]	R. Temam, "Infinite-Dimensional Dynamical System in Mechanics and Physics,'' Applied Mathematical Sciences 68, Springer-Verlag, New-York, 1988.
[21]	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.

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