G. Perla Menzala

On a wave equation with a cubic convolution

Abstract We study the well-posedness of the Cauchy problem and the asymptotic behavior of solutions of the nonlinear wave equation u tt − Δu + m 2 + u(V ∗ u 2 ) = 0 in Euclidean space.

Boundary Observation and Exact Control of a Quasi-electrostatic Piezoelectric System in Multilayered Media

We study the evolution of a layered quasi-electrostatic piezoelectric system. Under suitable assumptions on the geometry of a region and the interfaces as well as a monotonicity condition on the coefficients, we prove a boundary observation inequality which together with the Hilbert uniqueness method introduced by Lions give us a solution of the exact controllability problem for the model under study.

Attractors for second order periodic lattices with nonlinear damping

We consider second order nonlinear lattices under the effect of nonlinear damping. The family we study is subject to cyclic boundary conditions and includes as distinguished examples the Fermi–Pasta–Ulam and sine-Gordon lattices. We prove global well posedness and existence of a global attractor.

On global existence of localized solutions of some nonlinear lattices

We prove global existence and uniqueness of solutions of some important nonlinea lattices which include the Fermi-Pasta-Ularn (FPU) lattice. Our result shows (on a particular example) that the FPU lattice with high nonlinearity and its continuum limit display drastically different behaviour with respect to blow up phenomenon

Uniform decay rates of the solutions of a nonlinear lattice

Abstract We consider a family of finite nonlinear Klein–Gordon lattices subject to cyclic boundary conditions under the effect of a dissipative mechanism. We show that the model is globally well posed in a natural Banach space and our main result says that the total energy associated with the model decays exponentially fast when t→+ ∞ .

The energy decay rate for the modified Von Kármán system of thermoelastic plates: An improvement☆

Abstract We prove that the energy of solutions of the modified von Karman system of a thermoelastic plate decays with the rate E(t)≤CE(0) exp −ωt 1+E(0) , as t → + ∞ where C and ω are positive constants which are independent of the solution. This improves an earlier result in which we claimed the decay rate to be of the order of exp ( −wt (1 + E 2 (0) ) and provides a simpler and complete proof.

On localized solutions of discrete nonlinear Schrödinger equation: An exact result

Abstract Local and global existence of localized solutions of a discrete nonlinear Schrodinger (DNLS) equation, with arbitrary on-site nonlinearity, is proved. In particular, it is shown that an initially localized excitation persists localized during infinite time. Moreover, if initial localization is stronger than | n | − d with any power d, it maintains itself as such during infinite time. The results are generalized to various types of inter-side and saturable nonlinearities, to lattices with long range interactions, as well as DNLS with dissipation.

Energy decay rates and the dynamical von Karman equations

Abstract We find uniform rates of decay of the solutions of the dynamical von Karman equations in the presence of dissipative effects. Our proof is elementary and uses ideas of a recent technique due to E. Zuazua while studying nonlinear dissipative wave equations [1].

Localized solutions of a nonlinear diatomic lattice

We consider a coupled system of differential-difference nonlinear equations. We study the dynamics of such a diatomic lattice showing global existence and uniqueness in an appropriate function space. Our approach based on energy estimates allows us to prove the result only in the case where nonlinear force constants are positive and equal. All other situations remain at this point as open problems.

Asymptotic behaviour in time of KdV type equations with time dependent coefficients

Abstract We study the asymptotic behaviour in time of the solutions of a class of evolution equations whose simplest representative would be the Korteweg de Vries equation with variable coefficients. Specific rates of decay are given in either the “conservative” or the dissipative case.