# American Institute of Mathematical Sciences

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

## Rate of attraction for a semilinear thermoelastic system with variable coefficients

 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  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
 $p\geq 2$
.
Citation: Fágner D. Araruna, Flank D. M. Bezerra, Milton L. Oliveira. Rate of attraction for a semilinear thermoelastic system with variable coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3211-3226. doi: 10.3934/dcdsb.2018316
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