Marcelo M. Cavalcanti

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• Professor
• Professor (Full) at State University of Maringá

This paper is concerned with a semilinear Rao–Nakra sandwich beam under the action of three nonlinear localized frictional damping terms in which the core viscoelastic layer is constrained by the pure elasticity or piezoelectric outer layers. The main goal is to prove its asymptotic behavior by applying minimal amount of support to the damping. We...

This paper is concerned with the decay estimate of solutions to the semilinear wave equation subject to two localized dampings in a bounded domain. The first one is of the nonlinear Kelvin–Voigt type which is distributed around a neighborhood of the boundary and the second is a frictional damping depending in the first one. We show uniform decay ra...

We study the stabilization and the well-posedness of solutions of the quintic wave equation with locally distributed damping. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well k...

We study the wellposedness and stabilization for a Cauchy–Ventcel problem in an inhomogeneous medium Ω⊂R2$\Omega \subset \mathbb {R}^2$ with dynamic boundary conditions subject to a exponential growth source term and a nonlinear damping distributed around a neighborhood ω of the boundary according to the geometric control condition. We, in particul...

We are concerned with the existence as well as the exponential stability in H1-level for the damped defocusing Schrödinger equation posed in a two-dimensional exterior domain Ω with smooth boundary ∂Ω. The proofs of the existence are based on the properties of pseudo-differential operators introduced in Dehman et al. [Math. Z. 254, 729–749 (2006)]...

This paper is concerned with the decay estimate of solutions to the semilinear wave equation subject to two localized dampings in a bounded domain. The first one is of the nonlinear Kelvin-Voigt type and is distributed around a neighborhood of the boundary according to the Geometric Control Condition. While the second one is a frictional damping an...

In this paper, we consider the problem of recovering solutions for matrix factorizations of the Helmholtz equation in a multidimensional bounded domain from their values on a part of the boundary of this domain, i.e., the Cauchy problem. An approximate solution to this problem is constructed based on the Carleman matrix method.

In this paper, we discuss the asymptotic stability as well as the wellposedness of the viscoelastic damped wave equation posed on a bounded domain \(\Omega \) of \({\mathbb {R}}^2,\)$$\begin{aligned} \partial _{t}^2u - \Delta u+ \displaystyle \int _0^\infty g(s)\hbox {div}[a(x)\nabla u(\cdot ,t-s)]\,\mathrm{{d}}s + b(x) \partial _{t}u + f(u)=0, \hb...

In this paper, we consider a weakly coupled system consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain. By assuming a more general type of relaxation functions, we establish explicit and general decay rate results, using th...

We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable o...

The long time behavior of a class of degenerate parabolic equations in a bounded domain will be considered in the sense that the nonnegative diffusion coefficient a(x) is allowed to vanish in a set of positive measure in the interior of the domain. We prove decay rates for a class of semilinear reaction diffusion equations and a nonlinear equation...

In this paper, we prove a stability result for a nonlinear wave equation, defined in a bounded domain of \({\mathbb {R}}^N\), \(N\ge 2\), with time-dependent coefficients. The smooth boundary of \(\Omega \) is \(\Gamma =\Gamma _0\cup \Gamma _1\) such that \(\Sigma ={\overline{\Gamma }}_0\cap {\overline{\Gamma }}_1\ne \emptyset \). On \(\Gamma _0\)...

In this manuscript, we analyze the exponential stability of a strongly coupled semilinear system of Klein-Gordon type, posed in an inhomogeneous medium Ω , subject to local dampings of different natures distributed around a neighborhood of the boundary according to the geometric control condition (GCC). The first one is of the type viscoelastic and...

In this paper we study the existence and uniqueness of local solutions for the coupled nonlinear system u′′−Δu+M∫Ω|∇u|2dxΔu+v=fv′−Δv+u′=g for bounded or unbounded domains making use of the diagonalization theorem for self-adjoint operators.

In this paper we study the stability of the energy associated to an initial boundary value problem involving a semilinear wave equation. The domain is an unbounded subset of R2 with unbounded boundary. On the function responsible by the semilinearity we consider a very general decay rate control. The main tools are the Trudinger–Moser inequality co...

We study in this paper the well-posedness and stability for two linear Schrödinger equations in d-dimensional open bounded domain under Dirichlet boundary conditions with an infinite memory. First, we establish the well-posedness in the sense of semigroup theory. Then, a decay estimate depending on the smoothness of initial data and the arbitrarily...

We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable o...

We are concerned with the well-posedness of solutions as well as the asymptotic behaviour of the energy related to the viscoelastic wave equation with localized memory with past history and supercritical source and damping terms, posed on a bounded domain Ω⊂R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfon...

We are concerned with the numerical exact controllability of the semilinear wave equation on the interval (0, 1). We introduce a Picard iterative scheme yielding a sequence of approximated solutions which converges towards a solution of the null controllability problem, provided that the initial data are small enough. The boundary control, which is...

This work is concerned with a semilinear non-homogeneous Timoshenko system under the effect of two nonlinear localized frictional damping mechanisms. The main goal is to prove its uniform stability by imposing minimal amount of support for the damping and, as expected, without assuming any relation on the non-constant coefficients. This fact genera...

In this paper, we consider a coupled semilinear wave and plate equations subject to an internal nonlinear damping locally distributed on an inhomogeneous medium Ω with smooth boundary ∂Ω. We are able to prove that the coupled system is well-posed in the sense of semigroups theory and more importantly, the associated energy decays uniformly to zero...

In this article we exploite the uniform decay for damped linear wave equation with Zaremba boundary condition, obtained in a previous work, to treat the same problem in nonlinear context. We need a uniqueness assumption, usual for this type of nonlinear problem. The result is deduced from an observation estimate for nonlinear problem proved by a co...

In this paper we study the existence as well as uniform decay rates of the energy associated with the nonlinear damped Schrödinger equation, i ⁢ u t + Δ ⁢ u + | u | α ⁢ u - g ⁢ ( u t ) = 0 in ⁢ Ω × ( 0 , ∞ ) , iu_{t}+\Delta u+|u|^{\alpha}u-g(u_{t})=0\quad\text{in }\Omega\times(0,\infty), subject to Dirichlet boundary conditions, where Ω ⊂ ℝ n {\Ome...

We are concerned with the numerical exact controllability of the semi-linear wave equation on the interval (0, 1). We introduce a Picard iterative scheme yielding a sequence of approximated solutions which converges towards a solution of the null controllability problem, provided that the initial data are small enough. The boundary control, which i...

In this paper, motivated by recent papers on the stabilization of evolution problems with nonlocal degenerate damping terms, we address an extensible beam model with degenerate nonlocal damping of Balakrishnan-Taylor type. We discuss initially on the well-posedness with respect to weak and regular solutions. Then we show for the first time how hard...

We give a complete answer to the questions concerning existence and regularity of periodic solutions to a class of linear partial differential operators. The results depend on Diophantine conditions and also on a control on the sign of the imaginary part of the symbol, which is related to the Nirenberg–Treves condition (P). This control is based on...

In this work, we study at the L^2– level global well-posedness as well as long-time stability of an initial-boundary value problem, posed on a bounded interval, for a generalized higher order nonlinear Schrödinger equation, modeling the propagation of pulses in optical fiber, with a localized damping term. In addition, we implement a precise and ef...

In the present paper, we are concerned with the semilinear viscoelastic wave equation in an inhomogeneous medium Ω subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we conside...

In this paper, we study the semilinear beam equation with a locally distributed nonlinear damping on a smooth bounded domain. We first construct approximate solutions, and we show that the aforementioned approximate solutions decay uniformly in the weak phase space by using an observability inequality associated to the linear problem and a unique c...

We study in this paper the well-posedness and stability for two linear Schr\"odinger equations in $d$-dimensional open bounded domain under Dirichlet boundary conditions with an infinite memory. First, we establish the well-posedness in the sens of semigroup theory. Then, a decay estimate depending on the smoothness of initial data and the arbitrar...

This book includes different topics associated with integral and integro-differential equations and their relevance and significance in various scientific areas of study and research. Integral and integro-differential equations are capable of modelling many situations from science and engineering. Readers should find several useful and advanced met...

In this paper we prove stability results for a semilinear hyperbolic coupled system subject to a viscoelastic localized damping acting in the first equation and a frictional localized one acting in the second equation of the system. We divide the proof into two parts. In the first part the equations are posed in a homogeneous medium Ω with the damp...

Inspired by the highly cited work due to Lasiecka and Tataru (1993) [20], a semilinear model of the wave equation in an inhomogeneous medium with simultaneous interior and boundary feedbacks is considered. While for a homogeneous medium Ω and for certain geometries nonlinear feedback is strong enough to derive uniform decay rates of the energy asso...

In this paper, we study the defocusing nonlinear Schr\"{o}dinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using the theory of monotone operators. We show that approximate solutions decay exponentially fast in the $L^2$-sen...

This paper is concerned with the study of a transmission problem of viscoelastic waves with hereditary memory, establishing the existence, uniqueness and exponential stability for the solutions of this problem. The proof of the stabilization result combines energy estimates and results due to Gérard (Commun Partial Differ Equ 16:1761–1794 (1991)) o...

In this article, we consider the energy decay of a viscoelastic wave in an heterogeneous medium. To be more specific, the medium is composed of two different homogeneous medium with a memory term located in one of the medium. We prove exponential decay of the energy of the solution under geometrical and analytical hypothesis on the memory term.

Contribution to Djairo Guedes De Figueiredo's birthday!

Sob o nome de Introdução as Equações Diferenciais Parciais guns métodos de resolução de problemas clássicos governados pelas equações diferenciais parciais. Usamos o método de Fedo-Galerkin associado à teoria de operadores monótonos.

Este ensaio de livro apresenta a teoria associada a semigrupos lineares e não lineares e aplicações.

We study a nonlocal evolution equation modeling the deformation of a bridge, either a footbridge or a suspension bridge. Contrarily to the previous literature we prove the asymptotic stability of the considered model with a minimum amount of damping which represents less cost of material. The result is also numerically proved.

Cavalcanti, Marcelo Moreira; Domingos Cavalcanti, Valéria Neves Introdução à teoria das distribuições e aos espaços de Sobolev. (Portuguese) [Introduction to distribution theory and Sobolev spaces] Editora da Universidade Estadual de Maringá (Eduem), Maringá, 2009. 452 pp. ISBN: 978-85-7628-195-5

Cavalcanti, Marcelo M.; Domingos Cavalcanti, Valéria N.; Komornik, Vilmos Introdução à análise funcional. (Portuguese) [Introduction to functional analysis] Editora da Universidade Estadual de Maringá (Eduem), Maringá, 2011. 481 pp. ISBN: 978-85-7628-407-9

We consider the Klein-Gordon system posed in an inhomogeneous medium Ω with smooth boundary ∂Ω subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geo...

In this work we present a finite difference scheme used to solve a higher order nonlinear Schrödinger equation. These equations are related to models of propagation of solitons travelling in fiber optics. The scheme is designed to preserve the numerical \(L^2\) norm, and control the energy for a suitable choose on the equation’s parameters. We prov...

This talk is concerned with the defocusing nonlinear Schrödinger equation with a locally distributed damping on a smooth bounded domain. We first construct approximate solutions for this model by using the theory of monotone operators. We show that these approximate solutions decay exponentially fast in the L_2-sense by using the multiplier techniq...

In this work we study the exact boundary controllability of a generalized wave equation in a nonsmooth domain with a nontrapping obstacle. In the more general case, this work contemplates the boundary control of a transmission problem admitting several zones of transmission. The result is obtained using the technique developed by David Russell, tak...

This talk is concerned with the defocusing nonlinear Schr¨odinger equation with a locally distributed damping on a smooth bounded domain. We first construct approximate solutions for this model by using the theory of monotone operators. We show that these approximate solutions decay exponentially fast in the L^2-sense by using the multiplier techni...

In this paper, we consider the Cauchy–Ventcel problem in an inhomogeneous medium with dynamic boundary conditions subject to a nonlinear damping distributed around a neighborhood [Formula: see text] of the boundary according to the Geometric Control Condition. Uniform decay rates of the associated energy are established and, in addition, the exact...

In this paper, we study the defocusing nonlinear Schrödinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using the theory of monotone operators. We show that approximate solutions decay exponentially fast in the $L^2$-sense b...

In the present paper, we are concerned with the semilinear viscoelastic wave equation subject to a locally distributed dissipative effect of Kelvin-Voigt type, posed on a bounded domain with smooth boundary. We begin with an auxiliary problem and we show that its solution decays exponentially in the weak phase space. The method of proof combines an...

We consider a coupled semilinear wave system posed in an inhomogeneous medium, with smooth boundary, subject to a nonlinear damping distributed around a neighborhood of the boundary according to the Geometric Control Condition. We show that the energy of the coupled system goes uniformly to zero, for all initial data of finite energy taken in bound...

We consider a strongly coupled Klein-Gordon system posed in an inhomogeneous medium Ω with smooth boundary ∂Ω subject to a local damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energ...

This paper is concerned with locally damped semilinear wave equations defined on compact Riemannian manifolds with boundary. We present a construction of measure-controlled damping regions which are sharp in the sense that their summed interior and boundary measures are arbitrarily small. The construction of this class of open sets is purely geomet...

We study a nonlocal evolution equation modeling the deformation of a bridge, either a footbridge or a suspension bridge. Contrarily to the previous literature we prove the asymptotic stability of the considered model with a minimum amount of damping which represents less cost of material. The result is also numerically proved.

We consider the semilinear wave equation posed in an inhomogeneous media Ω with smooth boundary ∂Ω subject to a non linear damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the wave equation goes uniformly to zero for all initial data of finite energy phase-space.

We study a nonlocal evolution equation modeling the deformation of a bridge, either a footbridge or a suspension bridge. Contrarily to the previous literature, we prove the exponential asymptotic stability of the considered model with a small amount of damping (namely, on a small collar around the whole boundary) which represents less cost of mater...

The Klein Gordon equation subject to a nonlinear and locally distributed damping, posed in a complete and non compact n dimensional Riemannian manifold \((\mathcal {M}^n,\mathbf {g})\) without boundary is considered. Let us assume that the dissipative effects are effective in \((\mathcal {M}\backslash \Omega ) \cup (\Omega \backslash V)\), where \(...

We consider the Klein-Gordon system posed in an inhomogeneous medium with smooth boundary subject to a local viscoelastic damping distributed around a neighborhoodof the boundary according to the Geometric Control Condition. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in...

The following coupled damped Klein-Gordon-Schrödinger equations are considered i?t + ?? + iab(x)(-?) 2¹ b(x)? = f??? in ? × (0, 8), (a > 0) ftt - ?f + a(x)ft = |?|²?? in ? × (0, 8), where ? is a bounded domain of Rⁿ, n = 2, with smooth boundary G and ? is a neighbourhood of ?? satisfying the geometric control condition. Here ?? represents the chara...

We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary subject to a nonlinear damping distributed around a neighborhood ω of the boundary according to the geometric control condition. We show that the energy of the wave equation goes uniformly to zero for all initial data of finite energy phase-space.

We are concerned with the asymptotic behavior of two different cubic, defocusing and damped nonlinear Schrödinger equations on compact Riemannian manifolds without boundary. Two mechanisms of locally distributed damping are considered: a weak damping and a stronger one. In the first problem, we consider a two-dimensional case and prove that the cor...

We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary subject to a local viscoelastic damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finit...

This paper is concerned with the internal exact controllability of the following model of dynamical elasticity equations for incompressible materials with a pressure term, $$\phi''-\Delta \phi=-\nabla p,$$ and it is also devoted to the study of the uniform decay rates of the energy associated with the same model subject to a locally distributed non...

Unilateral problems related to the wave model subject to degenerate and localized nonlinear damping on a compact Riemannian manifold are considered. Our results are new and concern two main issues: (a) to prove the global well‐posedness of the variational problem; (b) to establish that the corresponding energy functional is not (uniformly) stable t...

In this paper we discuss the asymptotic stability as well as the well-posedness of the damped wave equation posed on a bounded domain Ω of Rn,n≥2,. ρ(x)utt-Δu+∫0∞g(s)div[a(x) u( ,t-s)]ds+b(x)ut=0, subject to a locally distributed viscoelastic effect driven by a nonnegative function a(x) and supplemented with a frictional damping b(x)≥0 acting on a...

This paper is concerned with the study of the uniform decay rates of the energy associated with mixed problems involving the wave equation with nonlinear localized damping. The domain is an unbounded open set of R² with finite measure and has an unbounded smooth boundary Γ = ΓN ∪ ΓD such that ΓN ∩ ΓD = ∅. On ΓD and ΓN we place the homogeneous Diric...

We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary ∂Ω subject to a non linear damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the wave equation goes uniformly to zero for all initial data of finite energy phase-space.

The main goal of the present paper is twofold: (i) to establish the well-posedness of a class of nonlinear degenerate parabolic equations and (ii) to investigate the related null controllability and decay rate properties. In a previous step, we consider an appropriate regularized system, where a small parameter α is involved. More precisely, the us...

In this paper, we obtain very general decay rate estimates associated to a wave–wave transmission problem subject to a nonlinear damping locally distributed employing arguments firstly introduced in Lasiecka and Tataru (1993) and we shall present explicit decay rate estimates as considered in Alabau-Boussouira (2005) and Cavalcanti et al. (2007). I...

A damped nonlinear wave equation with a degenerate and nonlocal damping term is considered. Well-posedness results are discussed, as well as the exponential stability of the solutions. The degeneracy of the damping term is the novelty of this stability approach.

We consider the wave equation with two types of locally distributed damping mechanisms: a frictional damping and a Kelvin-Voigt type damping. The location of each damping is such that none of them alone is able to exponentially stabilize the system; the main obstacle being that there is a quite big undamped region. Using a combination of the multip...

In this article, we are concerned with the asymptotic behavior of a class of degenerate parabolic problems involving porous medium type equations, in a bounded domain, when the diffusion coefficient becomes large. We prove the upper semicontinuity of the associated global attractor as the diffusion increases to infinity. © 2017, American Institute...

In this paper, we study the existence at the H¹-level as well as the stability for the damped defocusing Schrödinger equation in Rd. The considered damping coefficient is time-dependent and may vanish at infinity. To prove the existence, we employ the method devised by Özsarı, Kalantarov and Lasiecka [27], which is based on monotone operators theor...

We consider the long-time behavior of a nonlinear PDE with a memory term which can be recast in the abstract form ddtρ(ut)+Autt+γAθut+Au-∫0tg(s)Au(t-s)=0, where A is a self-adjoint, positive definite operator acting on a Hilbert space H, ρ(s) is a continuous, monotone increasing function, and the relaxation kernel g(s) is a continuous, decreasing f...

In this paper, the Shrödinger equation with a defocusing nonlinear term and dynamic boundary conditions defined on a smooth boundary of a bounded domain Ω ⊂ ℝN, N = 2, 3 is considered. Local well-posedness of strong H² solutions is also established. In the case, N = 2 local solutions are shown to be global, and existence of weak H¹ solutions in dim...

In this paper, we consider coupled wave-wave, Petrovsky-Petrovsky and wave-Petrovsky systems in N-dimensional open bounded domain with complementary frictional damping and infinite memory acting on the first equation. We prove that these systems are well-posed in the sense of semigroups theory and provide a weak stability estimate of solutions, whe...

New York Journal Of Mathematics

A nonlinear model described by von Karman equations with long memory is considered. Hadamard wellposedness of weak solutions, regularity of solutions and intrinsic decay rate estimates for the energy are established by assuming that the memory kernel satisfies the inequality introduced in Alabau-Boussouira and Cannarsa (2009): , where is a given co...

We consider the Timoshenko model for vibrating beams under effect of two nonlinear and localized frictional damping mechanisms acting on the transverse displacement and on the rotational angle. We prove that the damping placed on an arbitrarily small support, unquantitized at the origin and without assuming equal speeds of propagation of waves, lea...

The Schrödinger equation subject to a nonlinear and locally distributed damping, posed in a connected, complete, and noncompact n dimensional Riemannian manifold (ℳ, g) is considered. Assuming that (ℳ, g) is nontrapping and, in addition, that the damping term is effective in ℳ \ Ω, where Ω ⊂ ⊂ ℳ is an open bounded and connected subset with smooth b...

Wave equation defined on a compact Riemannian manifold (M, g) subject to a combination of locally distributed viscoelastic and frictional dissipations is discussed. The viscoelastic dissipation is active on the support of a(x) while the frictional damping affects the portion of the manifold quantified by the support of b(x) where both a(x) and b(x)...

This talk addresses uniform decay rate estimates of the energy related to the Klein-Gordon equation subject to a nonlinear and locally distributed damping, posed in a complete and non compact n dimensional Riemannian manifold (M^n; g) without boundary.

The Schrödinger equation subject to a nonlinear and locally distributed damping, posed in a connected, complete, and noncompact n dimensional Riemannian manifold ��� (M,g)� is considered. Assuming that ��� (M,g)� is non trapping and, in addition, that the damping term is effective in M���\Omega�, where � Omega ⊂⊂ �M is an open bounded and connected...

Let \(\mathcal{M}\subset\mathbb{R}^{3}\) be an oriented compact surface on which we consider the system: $$\left \{ \begin{array}{l@{\quad}l} u_{tt} - \Delta_{\mathcal{M}} u + a(x)g_{0}(u_{t})=0 & \text{in } \mathcal{M}\times\mathopen{]} 0,\infty[ ,\\ \partial_{\nu_{co}}u +u + b(x)g(u_t)=0 & \text{on } \partial \mathcal {M}\times\mathopen{]}0,+\inf...

This paper is concerned with the study of the uniform decay rates of the energy associated with the wave equation with nonlinear damping locally distributed u tt -Δu+a(x)g(u t )=0 in Ω×(0,∞) subject to Dirichlet boundary conditions where Ω⊂ℝ n n∈2 is an unbounded open set with finite measure and unbounded smooth boundary ∂Ω=Γ. The function a(x), re...