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July  2019, 24(7): 3211-3226. doi: 10.3934/dcdsb.2018316

Rate of attraction for a semilinear thermoelastic system with variable coefficients

Fágner D. Araruna , Flank D. M. Bezerra and  Milton L. Oliveira ,

Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa, PB, 58051-900, Brasil

* Corresponding author: Milton L. Oliveira

Received  July 2017 Revised  September 2018 Published  July 2019 Early access  January 2019

The present paper is concerned with the problem of determining the rate of convergence of global attractors of the family of dissipative semilinear thermoelastic systems with variable coefficients
$ \begin{cases} \partial_t^2u-\partial_x(a_\varepsilon(x) \partial_xu)+\partial_x(m(x) \theta) = f(u)& \mbox{in}\ \ (0,l)\times(0,+\infty),\\ \partial_t\theta-\partial_x(\kappa_\varepsilon(x) \partial_x\theta)+m(x) \partial_{xt}u = 0& \mbox{in}\ \ (0,l)\times(0,+\infty), \end{cases} $
where
$ l>0 $
,
$ a_\varepsilon,\kappa_\varepsilon $
 and
$ m $
 are regular enough functions, and the nonlinearity
$ f $
 is a continuously differentiable function satisfying suitable growth conditions. We show that rate of convergence, as
$ \varepsilon\to0^+ $
, of the global attractors of these problems is proportional the distance of the coefficients
$ \|a_\varepsilon-a_0\|_{L^p(0,l)}+\|\kappa_\varepsilon-\kappa_0\|_{L^p(0,l)} $
 for some
$ p\geq 2 $
.
Keywords: Thermoelastic system, global attractors, semigroups, rate of convergence, nonlinear dynamics.
Mathematics Subject Classification: Primary: 34D45, 35B40; Secondary: 35Q79, 37L05, 41A25, 47H20.
Citation: Fágner D. Araruna, Flank D. M. Bezerra, Milton L. Oliveira. Rate of attraction for a semilinear thermoelastic system with variable coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3211-3226. doi: 10.3934/dcdsb.2018316
References:
[1]

F. D. Araruna and F. D. M. Bezerra, Rate of attraction for a semilinear wave equation with variable coefficients and critical nonlinearities, Pacific J. Math., 266 (2013), 257-282.  doi: 10.2140/pjm.2013.266.257.

[2]

J. M. Arrieta, F. D. M. Bezerra and A. N. Carvalho, Rate of convergence of global attractors of some perturbed reaction-diffusion problems, Topological Methods in Nonlinear Analysis, 41 (2013), 229-253. 

[3]

J. M. Arrieta and A. N. Carvalho, Abstract parabolic equations with critical nonlinearities and applications to Navier-Stokes and heat equations, Transactions of the AMS, 352 (2000), 285-310.  doi: 10.1090/S0002-9947-99-02528-3.

[4]

J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dummbell domains Ⅲ. Continuity of attractors, J. Diff. Eqns., 247 (2009), 225-259.  doi: 10.1016/j.jde.2008.12.014.

[5]

J. M. Arrieta and E. Santamaría, Estimates on the distance of inertial manifolds, Discrete Contin. Dyn. Syst., 34 (2014), 3921-3944.  doi: 10.3934/dcds.2014.34.3921.

[6]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and Its Applications, 25, New York, 1992.

[7]

A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., 62 (1983), 441-491. 

[8]

F. D. M. Bezerra and M. J. D. Nascimento, Convergence estimates of the dynamics of a hyperbolic system with variable coefficients, Math. Methods Appl. Sci., 37 (2014), 663-675.  doi: 10.1002/mma.2823.

[9]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[10]

S. M. Bruschi, A. N. Carvalho, J. W. Cholewa and T. Dlotko, Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equations, J. D. Diff. Eqns., 18 (2006), 767-814.  doi: 10.1007/s10884-006-9023-4.

[11]

V. L. Carbone, A. N. Carvalho and K. Schiabel-Silva, Continuity of attractors for parabolic problems with localized large diffusion, Nonlinear Analysis, 68 (2008), 515-535.  doi: 10.1016/j.na.2006.11.017.

[12]

V. L. Carbone, M. J. D. Nascimento, K. Schiabel-Silva and R. P. Silva, Pullback attractors for a singularly nonautonomous plate equation, Electronic Journal of Differential Equations, 77 (2011), 1-13. 

[13]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[14]

A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numerical Functional Analysis and Optimization, 27 (2006), 785-829.  doi: 10.1080/01630560600882723.

[15]

H. Gao, Global attractor for the semilinear thermoelastic problem, Math. Methods Appl. Sci., 26 (2003), 1255-1271.  doi: 10.1002/mma.416.

[16]

H. Gao and J. E. Muñoz Rivera, Global existence and decay for the semilinear thermoelastic contact problem, J. Differential Equations, 186 (2002), 52-68.  doi: 10.1016/S0022-0396(02)00016-5.

[17]

J. K. Hale and A. Perissinotto, Global attractor and convergence for one-dimensional semilinear thermoelasticity, Dynamic Systems and Applications, 2 (1993), 1-9. 

[18]

D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Springer-Verlag, 840 New York, 1981.

[19]

D. B. Henry, A. Perissinitto and O. Lopes, On the essential spectrum of a semigroup of thermoelasticity, Applications, 21 (1993), 65-75.  doi: 10.1016/0362-546X(93)90178-U.

[20]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, 1978.

show all references

References:
[1]

F. D. Araruna and F. D. M. Bezerra, Rate of attraction for a semilinear wave equation with variable coefficients and critical nonlinearities, Pacific J. Math., 266 (2013), 257-282.  doi: 10.2140/pjm.2013.266.257.

[2]

J. M. Arrieta, F. D. M. Bezerra and A. N. Carvalho, Rate of convergence of global attractors of some perturbed reaction-diffusion problems, Topological Methods in Nonlinear Analysis, 41 (2013), 229-253. 

[3]

J. M. Arrieta and A. N. Carvalho, Abstract parabolic equations with critical nonlinearities and applications to Navier-Stokes and heat equations, Transactions of the AMS, 352 (2000), 285-310.  doi: 10.1090/S0002-9947-99-02528-3.

[4]

J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dummbell domains Ⅲ. Continuity of attractors, J. Diff. Eqns., 247 (2009), 225-259.  doi: 10.1016/j.jde.2008.12.014.

[5]

J. M. Arrieta and E. Santamaría, Estimates on the distance of inertial manifolds, Discrete Contin. Dyn. Syst., 34 (2014), 3921-3944.  doi: 10.3934/dcds.2014.34.3921.

[6]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and Its Applications, 25, New York, 1992.

[7]

A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., 62 (1983), 441-491. 

[8]

F. D. M. Bezerra and M. J. D. Nascimento, Convergence estimates of the dynamics of a hyperbolic system with variable coefficients, Math. Methods Appl. Sci., 37 (2014), 663-675.  doi: 10.1002/mma.2823.

[9]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[10]

S. M. Bruschi, A. N. Carvalho, J. W. Cholewa and T. Dlotko, Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equations, J. D. Diff. Eqns., 18 (2006), 767-814.  doi: 10.1007/s10884-006-9023-4.

[11]

V. L. Carbone, A. N. Carvalho and K. Schiabel-Silva, Continuity of attractors for parabolic problems with localized large diffusion, Nonlinear Analysis, 68 (2008), 515-535.  doi: 10.1016/j.na.2006.11.017.

[12]

V. L. Carbone, M. J. D. Nascimento, K. Schiabel-Silva and R. P. Silva, Pullback attractors for a singularly nonautonomous plate equation, Electronic Journal of Differential Equations, 77 (2011), 1-13. 

[13]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[14]

A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numerical Functional Analysis and Optimization, 27 (2006), 785-829.  doi: 10.1080/01630560600882723.

[15]

H. Gao, Global attractor for the semilinear thermoelastic problem, Math. Methods Appl. Sci., 26 (2003), 1255-1271.  doi: 10.1002/mma.416.

[16]

H. Gao and J. E. Muñoz Rivera, Global existence and decay for the semilinear thermoelastic contact problem, J. Differential Equations, 186 (2002), 52-68.  doi: 10.1016/S0022-0396(02)00016-5.

[17]

J. K. Hale and A. Perissinotto, Global attractor and convergence for one-dimensional semilinear thermoelasticity, Dynamic Systems and Applications, 2 (1993), 1-9. 

[18]

D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Springer-Verlag, 840 New York, 1981.

[19]

D. B. Henry, A. Perissinitto and O. Lopes, On the essential spectrum of a semigroup of thermoelasticity, Applications, 21 (1993), 65-75.  doi: 10.1016/0362-546X(93)90178-U.

[20]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, 1978.

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