G. Perla Menzala

Federal University of Rio de Janeiro | UFRJ · Instituto de Matemática (IM)

We study a dynamical thin shallow shell whose elastic deformations are described by a nonlinear system of Marguerre–Vlasov’s type under the presence of thermal effects. Our main result is the proof of a global existence and uniqueness of a weak solution in the case of clamped boundary conditions. Standard techniques for uniqueness do not work direc...

This work is devoted to study the nature of vibrations arising in a multidimensional nonlinear periodic lattice structure with memory. We prove the existence of a global attractor. In the homogeneous case under a restriction on the nonlinear term we obtain decay rates of the total energy. These rates could be exponential, polynomial or several othe...

We study dynamic elastic deformations of a quasilinear plate model of Timoshenko's type under thermal effects, which are modeled by Cattaneo's law. We prove uniform exponential stabilization of the total energy as time approaches infinity. We show global wellposedeness of the model and build a convenient Lyapunov function, which allow us to conclud...

We consider an evolution system describing an electro/magneto/thermoelastic phenomenon in a bounded domain of R-3. The resulting anisotropic model is a coupling between two hyperbolic equations, two elliptic equations and one parabolic equation. Assuming that a dissipative mechanism is present on the boundary and suitable boundary conditions are gi...

We consider a dynamical nonlinear model for shallow shells of the Marguerre-Vlasov’s type in the presence of thermal effects. Results on existence and uniqueness of global weak solutions are already available. We consider the above model depending on a parameter ε>0 and study its weak limit as ε→0 + . The limit model turns out to be a nonlinear Tim...

We consider a linear evolution model describing a piezoelectric phenomenon under thermal effects as suggested by R. D. Mindlin [Int. J. Solids Structures 10, 625–637 (1974; Zbl 0282.73068)] and W. Nowacki [“Some general theorems of thermopiezoelectricity”, J. Thermal Stresses 1, No. 2, 171–182 (1978; doi:10.1080/01495737808926940)]. We prove the eq...

We study the best possible energy decay rates for a class of linear secondorder dissipative evolution equations in a Hilbert space. The models we consider are generated by a positive selfadjoint operator A having a bounded inverse. Our discussion applies to important examples such as the classical wave equation, the dynamical wave equation with Wen...

This paper is motivated by a piezoelectric/piezomagnetic phenomenon in the presence of thermal effects. The evolution system we consider is linear and coupled between one hyperbolic, two elliptic and one parabolic equation. We show the equivalence between “the exponential decay of the total energy of our system” and an “observability inequality for...

We consider a coupled system of Schrödinger equations with time-periodic coefficients iu t =-Deltau+V(x,t)u+g(x,t)v, iv t =-Deltav+W(x,t)v+g(x,t)u on the Hilbert space ℋ=L 2 (ℝ n )×L 2 (ℝ n ), where g,V and W are periodic time-dependent potentials, with period T. We denote by (U(t,s)) (t,s)∈ℝ×ℝ its associated propagator. By using a multiplier metho...

A shape optimization problem in three spatial dimensions for an elasto-dynamic piezoelectric body coupled to an acoustic chamber is introduced. Well-posedness of the problem is established and first order necessary optimality conditions are derived in the framework of the boundary variation technique. In particular, the existence of the shape gradi...

We study the stabilization of solutions of a coupled system of Korteweg-de Vries (KdV) equations in a bounded interval under the effect of a localized damping term. We use multiplier techniques combined with the so-called “compactness-uniqueness argument”. The problem is then reduced to proving a Unique Continuation Property (UCP) for weak solution...

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 R. C. Charão and R. Ikehata [J. Math. Anal. Appl. 380, No. 1, 46–56 (2011; Zbl 1218.35036)], we conclude that the energy decays at the same rat...

A new boundary observability inequality for the Maxwell equations and the elastodynamic system is obtained. We use modified multipliers to obtain such an inequality as long as a geometric condition on the region holds and important parameters of the model are (numerically) related. This allow us to use the HUM to conclude a "simultaneous'' boundary...

This work is devoted to study the asymptotic behavior of the total energy associated with a coupled system of anisotropic hyperbolic models: the elastodynamic equations and Maxwell’s system in the exterior of a bounded body in ℝ 3 . Our main result says that in the presence of nonlinear damping, a unique solution of small initial data exists global...

The optimization of shape and topology of piezo-patches or layered piezo-electrical material attached to structural parts, such as elastic bodies, plates and shells, plays a major role in the design of smart structures, as piezo-mechanic-acoustic devices in loudspeakers or energy harvesters. While the design for time-harmonic motions is genuinely f...

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.

We present a result on "simultaneous" exact controllability for two models that describe two hyperbolic dynamics. One is the system of Maxwell equations and the other a vector-wave equation with a pressure term. We obtain the main result using modified multipliers in order to generate a necessary observability estimate which allow us to use the Hil...

We find uniform rates of decay of the total energy of the coupled system of anisotropic electromagnetic/elasticity model in exterior domains provided mild dissipative effects are present. The decay of the total energy is of polynomial type. The conclusions of this paper improve previous results on the subject.

We consider the Maxwell system with variable anisotropic coecients in a bounded domain of R3. The boundary conditions are of Silver-Muller's type. We proved that the total energy decays exponentially fast to zero as time approaches infinity. This result is well known in the case of isotropic coecients. We make use of modified multipliers with the h...

Anisotropic Maxwell equations with electric conductivity are considered. Electromagnetic waves propagate in the exterior of a bounded connected obstacle with Lipschitz boundary. Our main result says that we can obtain uniform rates of decay of the total energy as $t \rightarrow + \infty$. No special requirements on the geometry of the obstacle are...

We study the decay of the energy of solutions of the system of magneto-elasticity in a bounded, three-dimensional conductive medium. We prove that all solutions do decay as t → ∞ in the energy-space when the domain is simply connected. We also describe the large time behavior of solutions when the domain is not simply connected. Our results are sim...

A one dimensional version of the dynamic Marguerre-Vlasov system in the presence of thermal effects is considered. The system depends on a parameter ε>0 in a singular way as ε→0. Our interest is twofold: 1) To find the limit system as ε→0 and 2) To study the asymptotic behavior as t→+∞ of the total energy E ε (t) and compare it with the total energ...

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.

The aim of this work is to consider the Korteweg–de Vries equation in a finite interval with a very weak localized dissipation namely the H−1-norm. Our main result says that the total energy decays locally uniform at an exponential rate. Our analysis improves earlier works on the subject (Q. Appl. Math. 2002; LX(1):111–129; ESAIM Control Optim. Cal...

A coupled system of dynamic hyperbolic equations in electromagnetic- elasticity theory in the exterior of an open bounded obstacle O in 3-D is considered. In the presence of dissipative effects we obtain uniform decay rates of the solution as t → + ∞. We do not require geometric assumptions on the obstacle or extra assumptions on the initial data....

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 ex...

We consider a coupled system of Kuramoto--Sivashinsky (KS) equations in a bounded interval depending on a suitable parameter $\nu > 0$. As $\nu$ tends to zero, we obtain a coupled system of Korteweg--de Vries (KdV) equations known to describe strong interactions of two long internal gravity waves in a stratified fluid. Existence and uniqueness of g...

We consider two quasi-electrostatic models in a multilayered piezoelectric structure. Both models only differ on the boundary conditions. For each one of them we prove a boundary observability inequality which together with the Hilbert Uniqueness Method introduced by J.L. Lions allow us to drive both systems to rest at time T with the same control...

We consider the dynamical system of elasticity in the exterior of a bounded open domain in 3-D with smooth boundary. We prove that under the effect of "weak" dissipation, the total energy decays at a uniform rate as $t o +infty$, provided the initial data is "small" at infinity. No assumptions on the geometry of the obstacle are required. The resul...

The full nonlinear dynamic von Kárm´n system depending on a small parameter ε > 0 is considered. We study the asymptotic behaviour of the total energy associated with the model for large t and ε → 0. Introducing appropriate boundary feedback, we show that the total energy of a solution of the corresponding damped model decays exponentially as t → +...

We consider a transmission problem for a model describing the evolution of sound in a compressible fluid. Assuming that dissipative mechanisms of memory type are effective in part of the boundary, the coefficients are piecewise constants and suitable geometric conditions on the domain and the interfaces, we prove that the total energy associated wi...

We consider a linear coupled system of quasi-electrostatic equations which govern the evolution of a 3-D layered piezoelectric body. Assuming that a dissipative effect is effective at the boundary, we study the uniform stabilization problem. We prove that this is indeed the case, provided some geometric conditions on the region and the interfaces h...

We consider a class of nonlinear beam equations in the whole space ℝ n . Using previous important work due to S. P. Levandosky and W. A. Strauss [Methods Appl. Anal. 7, No. 3, 479–487 (2000; Zbl 1029.35182)] we prove that, locally, the H 1 -norm of a strong solution approaches zero as t→+∞ as long as the spatial dimension n>6. The problem remains o...

Local and global existence of localized solutions of a discrete nonlinear Schrödinger (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 itse...

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 othe...

In this paper we study a nonlinear lattice with memory and show that the problem is globally well posed. Furthermore we find uniform rates of decay of the total energy. Our main result shows that the memory effect is strong enough to produce a uniform rate of decay. That is, if the relaxation function decays exponentially then, the corresponding so...

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→+∞.

We prove that the energy of solutions of the modified von Kármán system of a thermoelastic plate decays with the rate E(t)≤CE(0)exp−ωt1+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 + E2(0)) and provides...

A transmission problem for a class of dynamic coupled system of hyper-bolic equations having piecewise constant coefficients in a bounded three-dimensional domain is considered. Assuming that in the entire boundary, dissipative mechanisms are present and that suitable geometric conditions on the domain and the interfaces are satisfied, we prove tha...

A locally uniform stabilization result of the solutions of a coupled system of Korteweg- de Vries equations in a bounded domain is established. The main novelty is that internally only a localized damping mechanism is considered.

We consider a dynamical one-dimensional nonlinear von Kármán model for beams depending on a parameter ${\varepsilon}> 0$ and study its asymptotic behavior for $t$ large, as ${\varepsilon} \to 0$. Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponentially uniformly with resp...

We study the stabilization of solutions of the Korteweg-de Vries (KdV) equation in a bounded interval under the effect of a localized damping mechanism. Using multiplier techniques we deduce the exponential decay in time of the solutions of the underlying linear equation. A locally uniform stabilization result of the solutions of the nonlinear KdV...

We consider a transmission problem for a system of electromagneto-elasticity having piecewise constant coefficients in a bounded domain. Under suitable geometric conditions imposed on the domain and the interfaces where the coefficients have a jump discontinuity, results on uniform boundary stabilization are established. Exact boundary controllabil...

We consider the dynamical full von Kármán system of equations for viscoelastic plates. The main result says that the uniform rate of decay of the energy function associated with the system depends on the decay rate of relaxation functions. Our result can be seen as a refinement of our previous work [Q. Appl. Math. 57, No. 1, 181-200 (1999; Zbl 1021...

We consider a transmission problem for Maxwell's equations with a dissipative boundary condition of memory type. Under suitable geometric conditions imposed on the domain and the interfaces where the coefficients are allow to have a jump discontinuity, results on uniform stabilization are established.

A one-dimensional version of the so-called Marguerre-Vlasov system of equations describing the vibrations of shallow shells is considered. The system depends on a parameter 0 in a singular way and undergoes the effect of damping mechanisms. We show that the system converges to a nonlinear beam equation while the energy decays exponentially uniforml...

We study a one-dimensional coupled system of equations of Benjamin-Bona-Mahony's type which are space-periodic. We prove that the total energy associated with the above system decays exponentially. Furthermore, we show an "almost" sharp result, that is, the constant appearing in the exponential decay rate has to be an eigenvalue of the dissipative...

We study the resonances (or sattering frequenaes) associated with the syutem of elastic wave equations in an unbounded region Ω which is the union of a (bounded) cavity C, an (unbounded) exterior E and a thin channel Z conneding C with E. We show that as the channel gets narrower there are resonances of Ω which converge to the eigenvalues of the (v...

We consider a dynamical one-dimensional nonlinear von Kármán model depending on one parameter ε > 0 and study its weak limit as ε → 0. We analyse various boundary conditions and prove that the nature of the limit system is very sensitive to them. We prove that, depending on the type of boundary condition we consider, the nonlinearity of Timoshenko'...

We prove global existence and uniqueness of solutions of some important nonlinear lattices which include the Fermi-Pasta-Ulam (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.

We consider the full nonlinear dynamic von Kármán system of equations which models large deflections of thin plates and show how the so-called Timoshenko and Berger models for thin plates may be obtained as singular limits of the von Kármán system when a suitable parameter tends to zero. We also show that in the case where the plate is of infinite...

We consider the dynamical von Kármán equations for viscoelastic plates under the presence of a long-range memory. We find uniform rates of decay (in time) of the energy, provided that suitable assumptions on the relaxation functions are given. Namely, if the relaxation decays exponentially, then the first-order energy also decays exponentially. Whe...

We prove that the one-dimensional von Kármán system of equations describing the planar motion of a uniform prismatic beam of length L approaches (weakly) to a nonlocal beam equation of Timoshenko's type as a suitable parameter tends to zero.

New results on the validity of the so-called unique continuation property (UCP) for two nonlinear dispersive equations are given. We show that the UCP holds for the Benjamin-Bona-Mahony equation and Boussinesq's equation. Our strategy in both cases rely on appropriate Carleman-type estimates for partial differential operators closely related to our...

. We consider a dynamical von K'arm'an system in the presence of thermal effects. Our model includes the possibility of a rotational inertia term in the system. We show that the total energy of the solution of such system decays exponentially as t ! +1. The decay rates we obtain are uniform on bounded sets of the energy space. The main ingredients...

We present various results on the existence and location of resonances for a perturbed system of elastic wave equations, for perturbations which are independent of time and also for those that are periodic functions of time. We also establish the continuous dependence of the resonances on parameters and on the perturbation.

We consider the dynamical von K\'arm\'an system describing the nonlinear vibrations of a thin plate. We take into account thermal effects as well as a rotational inertia term in the system. Our main result states that the total energy of the system, $E (t)$, satisfies the following estimate: there exist $C>0$ and $\omega > 0$ such that $$ E (t) \le...

We consider the von K'arm'an system for a bounded smooth thermoelastic plate clamped on its boundary. We prove explicit exponential decay rates showing that the energy of solutions corresponding to initial data of energy equal to R decays as t !1 like exp (Gamma!t=(1 + R 2 )) for some universal positive constant !. Resumo -- Consideramos o sistema...

Existence of local, in a generic case, and global, in the case of even potential and positive nonlinear force constant, solutions of a lattice of the Fermi-Pasta-Ulam type, with higher nonlinearity is proven. It is shown that the lattice and its continuum limit display drastically different behaviour. 1. INTRODUCTION During the last years nonlinear...

On considère le système de von Kármán pour une plaque thermoélastique bornée et régulière. On suppose que la plaque est encastrée. On démontre des estimations explicites sur le taux de décroissance en temps des solutions, montrant que l'énergie des solutions d'énergie initiale égale à R décroit comme exp (−ωt/(1 + R2)) lorsque t → ∞, où ω > 0 est u...

We consider a beam equation with a nonlocal nonlinearity of Kirchhoff type on an unbounded domain. We show that smooth global solutions decay (in time) at a uniform rate as $t\to +\infty$. Our model is closely related to a nonlinear Schrödinger equation with a time-dependent dissipation. We use this observation to obtain intermediate information on...

We prove that the solutions of a dynamial Timoshenko type equation enjoy the so-called Unique Continuation Property. The result is established using Carleman type estimates for a differential operator which appears naturally in the discussion of our problem. An inequality due to F. Treves plays a central role in our discussion.

We consider the system of elastic waves in three dimensions under the presence of an impurity of the medium which we represent by a real-valued function q(x) (or q(x,t)). The medium is assumed to be isotropic and occupies the whole space Ω = ℝ3. We study the location of the scattering frequencies associated with such phenomenon. We conclude that th...

We consider the system of elastic waves in three dimensions under the presence of an impurity of the medium which we represent by a real-valued function q(x) (or q(x,t)). The medium is assumed to be isotropic and occupies the whole space Ω = R3. We study the location of the scattering frequencies associated with such phenomenon. We conclude that th...

We consider the quasilinear hyperbolic equation $$ u_{tt}-M\Bigl( \int_\Omega|\text{ grad }u|^2\,dx\Bigr)\,\Delta u=0, \tag1 $$ where $x\in\Omega=\Bbb R^n$, $t$ denotes time and $M(s)$ is a smooth function satisfying $M(s)>0$ for all $s\ge 0$. We prove that there are no non-trivial ``breathers" for equation (1). Here, a "breather" means a time peri...

We consider mathematical models of evolution which are conservative and include in the simplest case, an equation describing the unidirectional propagation of weakly nonlinear, dispersive long waves suffering disturbances due to the possible unevennes of the botton surface. Our main result gives rates of decay of the amplitude in terms of the alter...

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].

We consider a family of dispersive equations whose simplest representative would be a Benjamin–Bona–Mahony equation with a Burger's type dissipation. The effect of possible unevenness of the bottom surface is considered and our main result gives decay rates of the solutions in Lβ(ℝ) spaces, 2 ≦ β ≦ + ∞.

We consider smooth solutions of a perturbed system of linear elasticity in 3-D and concentrate our attention on the following question: Is it possible that all such perturbed elastic waves propagate on spherical shells? We prove that the answer is negative for a suitable class of perturbations. The main technical difficulties in proving our results...

The time dependent system of linear thermoelasticity for isotropic bodies but totally inhomogeneous is considered. Recent results (in the homogeneous case) due to D. Henry et al. [2] and J. Rivera [7, 8] show that uniform rates of decay of the total energy are valid only in one dimension for such systems. We show that in case of isotropic inhomogen...

We briefly discuss some important results on Huygens’ principle in the sense of Hadamard’s minor premise. We also indicate a negative result we obtained concerning the above principle and the system of elastic waves in the presence of an impurity of the medium. We also mention some open problems in the subject as well as possible interesting genera...

We consider a real potential q(x, t) which has compact support in space x ∈ R2 and it is “damped.” We study the solutions of utt − Δu + q(x, t) u = 0 showing that under additional technical assumptions on q the scattering operator exists. Our proofs are elementary.

We prove that there are many C∞ solutions of the semilinear wave equations initial data and compact support, with the property that they do not propagate on spherical shells. Our method is elementary and works for a large class of nonlinearities, which includes the case f(s) = s3.

Let M > 0and consider the forward cone $$\Omega _M = \{ (x,t), |x| \leqq t - M,t \geqq M\} \subseteq \mathbb{R}^{n + 1} .$$ Let u=u(x, t) be a finite energy solution of the semilinear wave equation $$u_{tt} - \Delta u + q(x,t)|u|^{p - 1} u = 0$$ for x∋ ℝn (n≧3).Here q ≧ 0and 1≦p<1+4/n- 2.In this paper we discuss the following question: What is the...

We consider finite energy solutions of the perturbed wave equation □u+q(x,t)u=0 where x ε ℝ3, t ε ℝ. We analyse two type of problems: First, we give suitable conditions on q and we prove that there exist infinite many "resonances" λj associated with q. Secondly, we study the problem of determining q from the scattering operator associated with the...

We consider finite energy solutions of the perturbed wave equation with “impurities” which depend upon space and time: in the case . We prove that under suitable assumptions on the potential q(x, t), the scattering operator exist and we analyse some of its properties. Finally, we present a uniqueness result on an inverse scattering problem associat...

We study the asymptotic behavior in time of the solutions of a system of nonlinear Klein-Gordon equations. We have two basic results: First, in the L∞(ℝ3) norm, solutions decay like 0(t−3/2) as t→+∞ provided the initial data are sufficiently small. Finally we prove that finite energy solutions of such a system decay in loca...

Synopsis We prove that classical solutions of the perturbed wave equation in ℝ ⁿ × ℝ ( n = odd ≧ 3) do not satisfy Huygens' principle in the presence of symmetries. The difficulties arising from the singularities of the Riemann function (for large space dimensions) are overcome by considering a class of potentials and initial data which are radial...

The existence of an infinite sequence of scattering frequencies for the equation □u + qu = 0 is established, where q is a real valued potential which may assume negative values. This result generalizes some of the results obtained by Lax and Phillips in Comm. Pure Appl. Math. 22 (1969), 737–787.

The semilinear wave equation □u + m2u + ¦u¦p − 2 u(V∗ ¦u¦p) = 0 in Ω= R3, −∞ < t < ∞, is studied where □ denotes the d'Alembertian operator and ∗ means spatial convolution. Under mild assumptions on the real-valued function V and 2 ⩽ p ⩽ 3 the well-posedness of the Cauchy problem is proved. Furthermore, some properties of the solutions of the equat...

We study the well-posedness of the Cauchy problem and the asymptotic behavior of solutions of the nonlinear wave equation in Euclidean space.

Without Abstract

Under suitable conditions on M,ƒ, we prove the existence and uniqueness of a global classical solution of the non-linear evolution equation for and . The Fourier transform and the energy method are the basic tools.

Synopsis We study the nonlinear equation in ℝ ³ , where Δ denotes the Laplacian operator, and R and K are real-valued functions satisfying suitable conditions. We use a variational formulation to show the existence of a non-trivial weak solution of the above equation for some real number λ. Because of our assumptions on R and K we shall look for s...

The inverse scattering problem for the perturbed wave equation (1) □u + V(x)u = 0 in Ω=Rn (n = odd ⩾ 3) is considered. Here the potentials V(x) are real, smooth, with compact support and non-negative. We apply the Lax and Phillips theory, together with some properties of solutions of a Dirichlet problem associated with the operator −Δ + V(x) to sho...