Abstract: We prove the existence of strong time-periodic solutions and their asymptotic stability with the total energy of the perturbations decaying to zero at an exponential decay rate as t goes to infinity for a semilinear (nonlinearly coupled) magnetoelastic system in bounded, simply connected, three-dimensional domain. The mathematical model includes a mechanical dissipation and a periodic forcing function of period T. In the second part of the paper, we consider a magnetoelastic system in the form of a semilinear initial boundary value problem in a bounded, simply-connected two-dimensional domain. We use LaSalle invariance principle to obtain results on the asymptotic behavior of solutions. This second result was obtained for the system under the action of only one dissipation (the natural dissipation of the system).

Existence of time-periodic solutions for a magnetoelastic system in bounded domains (with M. Mohebbi)
Journal of Elasticity 113 (2013), pp. 113-133.

Abstract: We investigate the existence of periodic solutions for a semilinear (nonlinearly coupled) magnetoelastic system in bounded, simply connected, three-dimensional domains with boundaries of class $C^2$. The mathematical model includes a nonlinear mechanical dissipation like $\rho(u^{'})=|u^{'}|^{p}u^{'}$ and a periodic forcing function of period $T$. We prove the existence of $T$-periodic weak solutions when $p \in [3,4]$ (p=0 being a simpler case). In the corresponding two-dimensional case, the existence result holds under the assumption that $p \geq 2$.

Decay rates of magnetoelastic waves in an unbounded conductive medium (with R.C. Charão, G.P. Menzala)
Eletronic Journal of Differential Equations 2011, No. 127, 14pp.

Abstract: We study the uniform decay of the total energy of solutions for a system in magnetoelasticity with localized damping near infinity in an exterior 3-D domain. Using appropriate multipliers and recent work by Charao and Ikekata [3], we conclude that the energy decays at the same rate as (1+t)^{-1} when t -> infinity.

Global attractor for a class of nonlinear lattices (with J.M. Pereira)
Journal of Mathematical Analysis and Applications . 370 (2010), No. 2, 726-739. 

Abstract: We consider a class of nonlinear lattices with nonlinear damping

$\ddot u_n(t)+(-1)^p \Delta^p u_n(t)+\alpha u_n(t)+h(u_n(t))+ g(n,\dot u_n(t))=f_n$

where $N \in Z$, $t \in R^{+}$ is a real positive constant, p is any positive integer and $\Delta$ is the discrete one-dimensional Laplace operator. Under suitable conditions on $h$ and $g$ we prove the existence of a global attractor for the continuous semigroup associated with the above equation.

Our proofs are based on a difference inequality due to M. Nakao [M. Nakao, Global attractors for nonlinear wave equations with nonlinear dissipative terms, J. Differential Equations 227 (2006) 204–229].

Large time behavior of multidimensional nonlinear lattices with nonlinear damping (wih J.M. Pereira, G.P. Menzala)
Communications in Applied Analysis. 14 (2010),  No. 2, 155-176.

Abstract: In this paper we study the asymptotic behavior of solutions of multidimensional nonlinear lattices subject to cyclic boundary conditions under the effect of a nonlinear dissipation. We establish the existence of a global attractor.

Energy decay rates of magnetoelastic waves in a bounded conductive medium (with R.C. Charão, G.P. Menzala)
Discrete and Continuous Dynamical Systems-A 25 (2009), No. 3, 797-821.

Abstract: We consider a coupled system of evolution equations modeling the propagation of elastic waves interacting with a magnetic field in a bounded simply connected region of $R^3$ with boundary of class $C^2$ A nonlinear dissipative mechanism is allowed to be effective in an small subregion of $\Omega$. We prove that the total energy decays as t tends to infinity .

CV

Address:

Departamento de Matemática,
Universidade Federal de Santa Catarina,
CEP 88040-900 Florianópolis, SC
Brasil

E-mails: j.c.oliveira@ufsc.br, jauber.coliveira@gmail.com

phone: (+55)(48) 37213693