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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,'' Colloquium Publications 49. American Mathematical Society, 2002.  Google Scholar

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative System,'' Mathematical Surveys and Monographs vol. 25, 1989.  Google Scholar

[12]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'' Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[13]

P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical System,'' Mathematical Surveys and Monographs vol. 176, 2011.  Google Scholar

[14]

O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,'' Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991.  Google Scholar

[15]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Springer-Verlag, New York, 1983.  Google Scholar

[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.  Google Scholar

[17]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,'' Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002.  Google Scholar

[18]

Ch. Sun, S. Wang and Ch. Zhong, Global attractors for a nonclassical diffusion equation, Acta Math. Sin. (Engl. Ser.), 23 (2007), 1271-1280.  Google Scholar

[19]

P. E. Sobolevskiĭ, Equations of parabolic type in a Banach space, Trudy Moskov. Mat. Obšč, 10 (1961), 297-350.  Google Scholar

[20]

R. Temam, "Infinite-Dimensional Dynamical System in Mechanics and Physics,'' Applied Mathematical Sciences 68, Springer-Verlag, New-York, 1988.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. B. Babin and M. I. Vishik, "Attractors of Evolution Equations,'' North-Holland, Amsterdam, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,'' Colloquium Publications 49. American Mathematical Society, 2002.  Google Scholar

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative System,'' Mathematical Surveys and Monographs vol. 25, 1989.  Google Scholar

[12]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'' Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[13]

P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical System,'' Mathematical Surveys and Monographs vol. 176, 2011.  Google Scholar

[14]

O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,'' Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991.  Google Scholar

[15]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Springer-Verlag, New York, 1983.  Google Scholar

[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.  Google Scholar

[17]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,'' Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002.  Google Scholar

[18]

Ch. Sun, S. Wang and Ch. Zhong, Global attractors for a nonclassical diffusion equation, Acta Math. Sin. (Engl. Ser.), 23 (2007), 1271-1280.  Google Scholar

[19]

P. E. Sobolevskiĭ, Equations of parabolic type in a Banach space, Trudy Moskov. Mat. Obšč, 10 (1961), 297-350.  Google Scholar

[20]

R. Temam, "Infinite-Dimensional Dynamical System in Mechanics and Physics,'' Applied Mathematical Sciences 68, Springer-Verlag, New-York, 1988.  Google Scholar

[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.  Google Scholar

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