Carlos Raposo

Federal University of Pará | UFPA · Faculty of Mathematics

This manuscript deals with a thermoelastic laminated Timoshenko beam with a nonlocal integral condition on the transversal displacement and thermal dissipation in the equation that describes the dynamical of rotate angle. Using the Hille–Yosida Theorem, we prove the existence, uniqueness, and regularity of the solution. For the asymptotic behavior,...

This work deals with the solution and asymptotic analysis for a porous-elastic system with internal damping of the fractional derivative type. We consider an augmented model. The energy function is presented and establishes the dissipativity property of the system. We use the semigroup theory. The existence and uniqueness of the solution are obtain...

The present paper is devoted to studying the well-posedness and exponential stability of the one-dimensional system in the linear isothermal theory of swelling porous elastic soils with fluid saturation and Gurtin–Pipkin thermal law. For the well-posedness, we apply the well-known Hille–Yosida theorem of semigroup theory. To prove exponential stabi...

This paper deals with a one-dimensional system in the linear isothermal theory of swelling porous elastic soils subject to fractional derivative-type boundary damping. We apply the semigroup theory. We prove well-posedness by the Lumer–Phillips theorem. We show the lack of exponential stability and strong stability is proved by using general criter...

This paper concerns a viscoelastic Kirchhoff-type equation with the dispersive term, internal damping, and logarithmic nonlinearity. We prove the local existence of a weak solution via a modified lemma of contraction of the Banach fixed-point theorem. Although the uniqueness of a weak solution is still an open problem, we proved uniqueness locally...

This manuscript introduces a suspension bridge system where laminated beams model the deck. The action of frictional damping is considered. Well-posedness is proved using the Lumer-Phillips theorem, and the exponential stability is obtained by applying the Gearhart-Huang-Prüss theorem.

This article deals with the solution and asymptotic analysis for a porous-elastic system with fractional-order time delay. Semigroup theory is used. The existence and uniqueness of the solution are obtained by applying the Lumer-Phillips Theorem. Additionally, two results for the asymptotic behavior are presented concerning the (i) strong stability...

This work deals with a von Kármán system with internal damping. For the solution’s existence, we use nonlinear semigroup theory tools. We construct an evolution system by nonlinear Lipschitz perturbation of a semigroup of contractions. We apply the energy method for the asymptotic behavior, which uses suitable multipliers to construct a Lyapunov fu...

In this work, we use the theory of semigroups to study the well-posedness and the asymptotic behavior of the Timoshenko beam system with delayed viscoelastic damping acting only on the shear force. This system is different from all others related to time delay terms. We use the Lumer–Phillips theorem to prove the well-posedness of the problem and t...

This work deals with a von Kármán system with internal damping. For the solution’s existence, we use nonlinear semigroup theory tools. We construct an evolution system by nonlinear Lipschitz perturbation of a semigroup of contractions. We apply the energy method for the asymptotic behavior, which uses suitable multipliers to construct a Lyapunov fu...

In this work, we use the theory of semigroups to study the well-posedness and the asymptotic behavior of the Timoshenko beam system with delayed viscoelastic damping acting only on the shear force. This system is different from all others related to time delay terms. We use the Lumer-Phillips Theorem to prove the well-posedness of the problem and t...

In this work, we consider a laminated beam subjected to Kelvin–Voigt damping. Under the semigroup theory approach, applying the Lumer–Phillips Theorem, we establish the well-posedness of the associated initial value problem. This paper aims to prove exponential and polynomial stability results when the system is fully and partially damped. First, u...

This article focuses on a Timoshenko beam model introduced by Elishakoff. This model is free of the second frequency spectrum and solves the paradox of equal wave speeds, related to Timoshenko's model. Damping created by a fractional Laplacian is considered, which includes internal damping, Kelvin-Voigt damping, and intermediate damping. Exponentia...

This paper deals with long-time dynamics of nonlinear laminated beams modeled under the assumptions of Timoshenko beam theory. The model considered here is composed of two-layered beams and was proposed by Hansen and Spies. It describes the slip produced by a thin adhesive layer uniting the structure. As the adhesive stiffness γ\documentclass[12pt]...

This manuscript deals with global solution, polynomial stability and blow-up behavior at a finite time for the nonlinear system \begin{align*} \left\{ \begin{array}{rcl} & u'' - \Delta_{p} u + \theta + \alpha u' = \left\vert u\right\vert ^{p-2}u\ln \left\vert u\right\vert \\ &\theta' - \Delta \theta = u' \end{array}% \right. \end{align*} where $\De...

In this work, we analyze the stability of the semigroup associated with a Timoshenko beam model with distributed delay in the rotation angle equation. We show that the type of stability resulting from the semigroup is directly related to some model coefficients, which constitute the velocities of the system's component equations. In the case of sta...

In this paper, we investigate the existence, uniqueness, exponential decay, and blow‐up behavior of the viscoelastic beam equation involving the p$$ p $$‐Laplacian operator, strong damping, and a logarithmic source term, given by utt+Δ2u−Δpu+∫0tg(t−s)Δu(s)ds−Δut=ur−2ulnu,inQ=Ω×ℝ+$$ {u}_{tt}+{\Delta}^2u-{\Delta}_pu+{\int}_0^tg\left(t-s\right)\Delta...

In this work, we consider the existence of global solution and the exponential decay of a nonlinear porous elastic system with time delay. The nonlinear term as well as the delay acting in the equation of the volume fraction. In order to obtain the existence and uniqueness of a global solution, we will use the semigroup theory of linear operators a...

This manuscript deals with a Timoshenko system with damping and source. The existence and stability of the solution are analyzed taking into account the competition of the internal damping versus the logarithmic source. We use the potential well theory. For initial data in the stability set created by the Nehari surface, the existence of global sol...

In this work, we consider the existence of global solution and the exponential decay of a nonlinear porous elastic system with time delay. The nonlinear term as well as the delay acting in the equation of the volume fraction. In order to obtain the existence and uniqueness of a global solution, we will use the semigroup theory of linear operators a...

This paper aims to investigate the initial boundary value problem of the nonlinear viscoelastic Petrovsky type equation with nonlinear damping and logarithmic source term. We derive the blow-up results by the combination of the perturbation energy method, concavity method, and differential-integral inequality technique.

This manuscript deals with the global existence and asymptotic behavior of solutions for a Kirchhoff beam equation with internal damping. The existence of solutions is obtained by using the Faedo-Galerkin method. Exponential stability is proved by applying Nakao’s theorem.

In this paper we consider a model of laminated beams combining viscoelastic damping and strong time-delayed damping. The global well-posedness is proved by using the theory of semigroups of linear operators. We prove the lack of exponential stability when the speed wave propagations are not equal. In fact, we show in this situation, that the system...

The nonuniform thermoelastic laminated beam of the Lord–Shulman type is considered. The model is a two‐layered beam with structural damping due to the interfacial slip. The well‐posedness is proved by the semigroup theory of linear operators approach together with the Lumer–Phillips theorem. The stability results presented in this paper depend on t...

This paper deals with existence, uniqueness and energy decay of solutions to a degenerate hyperbolic equations given by \begin{align*} K(x,t)u'' - M\left(\int_\Omega |\nabla u|^2\,dx \right) \Delta u - \Delta u' = 0, \end{align*} with operator coefficient $K(x,t)$ satisfying suitable properties and $M(\,\cdot \,) \in C^1([0, \infty))$ is a function...

This manuscript deals with global solution, polynomial stability and blow-up behavior at a finite time for the nonlinear system u − ∆ p u + θ + αu = |u| p−2 u ln |u| θ − ∆θ = u where ∆ p is the nonlinear p-Laplacian operator, 2 ≤ p < ∞. Taking into account that the initial data is in a suitable stability set created from the Nehari manifold, the gl...

We study the stabilization of non-homogeneous viscoelastic waves for the vibrations of a flexible structure with thermodiffusion effect and a distributed forcing term as input disturbance. The coupled heat conduction is governed by Cattaneo-Vernotte’s law. Using the semigroup theory, we prove the existence and the uniqueness of the solution. Under...

In this paper, we study the global well-posedness and exponential stability for a Rao-Nakra sandwich beam equation with time-varying weight and time-varying delay. The system consists of one Euler-Bernoulli beam equation for the transversal displacement, and two wave equations for the longitudinal displacements of the top and bottom layers. By usin...

This paper is concerned with a stability result for a Kirchhoff beam equation with variable exponents and time delay. The exponential and polynomial stability results are proved based on Komornik’s inequality.

In this manuscript, we consider a system composed of two identical Timoshenko beams joined by an adhesive layer of negligible thickness, producing an interfacial slip. We introduce a dissipative mechanism not previously considered given by \(\alpha (-\Delta )^{\theta }u_{t}\), \(0\le \theta \le 1\), \(\alpha >0\), which includes the frictional damp...

In this article we study the well-posedness and exponential stability to the one-dimensional system in the linear isothermal theory of swelling porous elastic soils subject with time-varying weights and time-varying delay. We prove existence of global solution for the problems combining semigroup theory with the Kato’s variable norm technique. To p...

In this work, we obtain global solutions for nonlinear inequalities of p-Laplacian type in noncylindrical domains, for the unilateral problem with strong dissipation u − ∆pu − ∆u − f ≥ 0 in Q 0 , where ∆p is the nonlinear p-Laplacian operator with 2 ≤ p < ∞, and Q 0 is the noncylindrical domain. Our proof is based on a penalty argument by J. L. Lio...

This paper deals with the stability for a weakly coupled wave equations with a boundary dissipation of fractional derivative type. We have proved well posedness and polynomial stability using the semigroup theory and a sharp result provided by Borichev and Tomilov.

In this manuscript, we investigate the unilateral problem for a viscoelastic beam equation of p-Laplacian type. The competition of the strong damping versus the logarithmic source term is considered. We use the potential well theory. Taking into account the initial data is in the stability set created by the Nehari surface, we prove the existence a...

This paper deals with stability of solution for a one-dimensional model of Rao-Nakra sandwich beam with Kelvin-Voigt damping and time delay given by ?1h1utt ? E1h1uxx ? k(?u + v + awx) ? auxxt ? ?uxxt( ? , t ? ?) = 0, ?3h3vtt ? E3h3vxx + k(?u + v + ?wx) ? bwxxt = 0, ? hwtt + EIwxxxx ? k?(?u + v + ?wx)x ? cwxxt = 0. A sandwich beam is an engineering...

In this paper, we consider a system of magnetic effected piezoelectric beams with interior time-varying delay and time-dependent weights, in which the beam is clamped at the two side points subject to a single distributed state feedback controller with a time-varying delay. By combining the semigroup theory with Kato's variable norm technique, we o...

This manuscript focuses on in the transmission problem for one dimensional waves with time-varying weights on the frictional damping and time-varying delay. We prove global existence of solutions using Kato’s variable norm technique and we show the exponential stability by the energy method with the construction of a suitable Lyapunov functional.

In this paper, we are concerned with the existence and uniqueness of global strong solution of non-planar oscillations for a nonlinear coupled Kirchhoff beam equations with moving boundary.

This manuscript focuses on the study of the existence of a solution for an abstract Cauchy problem involving a wave equation with a monotone operator damping and nonlinear source term. We apply the potential well and prove the global weak solutions and the exponential stability for initial data in the set of stability created from the Nehari Manifo...

In this work we prove the well-posedness and establish the uniform stability of a thermoelastic Timoshenko system free of second spectrum. The heat conduction is governed by classical Fourier’s law. Our stability result holds for any parameters of the system.

A Kirchhoff equation type with memory term competing with a logarithmic source is considered. By using potential well theory, we obtained the global existence of solution for the initial data in a stability set created from Nehari Manifold and prove blow up results for initial data in the instability set.

This manuscript is concerned with long-time dynamics for a laminated beam which consists of two identical layers of uniform thickness, taking into account that an adhesive of small thickness is bonding the two surfaces thereby producing an interfacial slip. Using the variable norm technique of Kato, we prove the global well-posedness of solutions....

In this manuscript, by using the semigroup theory, the well-posedness and exponential stability for a Rao-Nakra sandwich beam equation with internal damping and time delay is proved. The system consists of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler-Bernoulli beam equation for the transversal di...

This paper is concerned with the existence and exponential stability of the global solution to a Klein–Gordon equation of Kirchhoff-Carrier type with strong damping −Δut and logarithmic source term uln|u|R2. We apply the potential well corresponding to the logarithmic nonlinearity. We prove the global weak solutions and the exponential stability fo...

In this manuscript, we prove the well-posedness and exponential stability for a thermoelastic structure given by a Rao-Nakra sandwich beam. The system consists of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler–Bernoulli beam equation for the transversal displacement, where, the thermal disturbances...

An extensible beam equation of Kirchhoff type with internal damping and source term is investigated. We apply the potential well and establish the global well-posedness of the initial and boundary value problem by using the Faedo-Galerkin approximations, taking into account that the initial data is located in a suitable set of stability created fro...

The main result of this work is to obtain the exponential decay of the solutions of a piezoelectric beam model with magnetic effect and delay term. The dampings are inserted into the equation of longitudinal displacement. The terms of damping, whose weight associated with them varies over time, are of the friction type, and one of them has delay. T...

This manuscript focus on in the transmission problem for one dimensional waves with nonlinear weights on the frictional damping and time-varying delay. We prove global existence of solutions using Kato's variable norm technique and we show the exponential stability by the energy method with the construction of a suitable Lyapunov functional.

In this paper, we study the well-posedness and asymptotic stability to a thermoelastic laminated beam with nonlinear weights and time-varying delay. To the best of our knowledge, there are no results on the system and related Timoshenko systems with nonlinear weights. On suitable premises about the time delay and the hypothesis of equal-speed wave...

This paper is concerned with the well-posedness of global solution and exponential stability to the Timoshenko system subject with time-varying weights and time-varying delay. We consider two problems: full and partially damped systems. We prove existence of global solution for both problems combining semigroup theory with the Kato's variable norm...

This paper is concerned with system of magnetic effected piezoelectric beams with interior time-varying delay and time-dependent weights, in which the beam is clamped at the two side points subject to a single distributed state feedback controller with a time-varying delay. Under appropriate assumptions on the time-varying delay term and time-depen...

The purpose of this paper is to study the Timoshenko system with the nonlocal time-delayed condition. The well-posedness is proved by Hille-Yosida theorem. Exploring the dissipative properties of the linear operator associated with the full damped model, we obtain the exponential stability by using Gearhart-Huang-Prüss theorem.

The purpose of this paper is to study the Timoshenko system with nonlocal time-delayed condition. The well-posedness is proved by Hille-Yosida theorem. Exploring the dissipative properties of the linear operator associated to full damped model, we obtain the exponential stability by using Gearhart-Huang-Prüss theorem.

In this paper, we consider a one-dimensional system governed by two partial differential equations. Such a system models phenomena in engineering, such as vibrations in beams or deformation of elastic bodies with porosity. By using the HUM method, we prove that the system is boundary exactly controllable in the usual energy space. We will also dete...

This paper focuses on the long-time dynamics of a thermoelastic laminated beam modeled from the well-established Timoshenko theory. From mathematical point of view, the study system consists of three hyperbolic motion equations coupled with the parabolic equation governed by Fouriers law of heat conduction and, in consequence , does not belong to o...

In this manuscript, we prove the well-posedness and exponential stability for a thermoelastic structure given by a laminated Timoshenko beam model consisting of two identical layers, taking into account that an adhesive of the small thickness is bonding these layers and produce a interfacial slip. We consider the action of the temperature differenc...

We consider the wave equation with a weak internal damping with non-constant delay and nonlinear weights given by $ \begin{eqnarray*} \label{NLS} u_{tt}(x,t) - u_{xx}(x,t)+\mu_1(t)u_t(x,t) +\mu_2(t)u_t(x,t-\tau(t)) = 0 \end{eqnarray*} $ in a bounded domain. Under proper conditions on nonlinear weights $ \mu_1(t), \mu_2(t) $ and non-constant delay $...

In this work we study the global solution, uniqueness and asymptotic behaviour of the nonlinear equation u tt − ∆ p u − ∆ p u t = |u| r−1 u where ∆ p u is the nonlinear p-Laplacian operator, 2 ≤ p < ∞. The global solutions are constructed by means of the Faedo-Galerkin approximations and the asymptotic behavior is obtained by Nakao method.

We study a laminated beam which consists of two identical layers of uniform thickness, taking into account that an adhesive of small thickness is bonding the two surfaces thereby producing an interfacial slip. We show that when the frictional damping acts on the effective rotation angle there is no need for any other kind of internal or boundary co...

In this manuscript we prove the property of growth determined by spectrum of the linear operator associated with the Timoshenko system with two histories.

In this manuscript we consider the transmission problem, in one space dimension, for linear dissipative waves with locally indirect stabilization. We study the wave propagation in a medium with a component with attrition and another being simply elastic. We show that for this type of material, the dissipation produced by the frictional part is stro...

This paper is concerned with long-time dynamics of laminated beams modeled from the well established Timoshenko system. Of particular interest is a model of two-layered beam proposed by Hansen and Spies which describes the slip effect produced by a thin adhesive layer uniting the structure. In a more general setting, involving a nonlinear foundatio...

In this work the asymptotic behavior of solution for one-dimensional equations of an homogeneous and isotropic porous elastic solid is analyzed. We use a classic result of P. Martinez [1] to obtain the general decay result. We give some example to illustrate the energy decay rates and consider also the case of the polynomial growth.

In this paper, we consider a one-dimensional transmission problem in a bounded domain with a delay in porous-elasticity. Using a semigroup theorem, under suitable assumption on the weight of the delay term, we establish the well-posedness of the system. Furthermore, using the method developed by Z. Liu and S. Zheng we show that the semigroup associ...

In this work we prove global solution for the nonlinear system utt−Δpu+θ=|u|r−1uθt−Δθ=utwhere Δp is the nonlinear p-Laplacian operator, 2≤p<∞. We apply the potential well theory. The global solution is constructed by means of the Faedo–Galerkin approximations, taking into account that the initial data is in appropriated set of stability created fro...

Well-posedness and exponential stability of nonlocal time-delayed of a wave equation with a integral conditions of the 1st kind forms the center of this work. Through semigroup theory we prove the well-posedness by the Hille-Yosida theorem and the exponential stability exploring the dissipative properties of the linear operator associated to damped...

We consider the hybrid laminated Timoshenko beam model. This structure is given by two identical layers uniform on top of each other, taking into account that an adhesive of small thickness is bonding the two surfaces and produces an interfacial slip. We suppose that the beam is fastened securely on the left while on the right it's free and has an...

We use the Sobolev spaces as in \cite{adams} and then, the existence and uniqueness of a generalized solution is proved applying the classical Faedo-Galerkin method for a mixed system of wave equations with an integral nonlocal condition in a cylinder.

In this work we consider a structure given by a laminated beam consisting of two identical layers uniform of length 1, taking into account that an adhesive of the small thickness is bonding the two layers and produce the interfacial slip. ' E 1 The memory effect with a dissipative relaxation function together with other stronger dissipative effects...

In this work we prove that the effect of the memory together with the frictional damping produces stabilization for the system of two identical laminated beams of uniform density taking into account that an adhesive of small thickness is bonding the two beams and produce the interfacial slip. It is assumed that the thickness of the adhesive is smal...

In this work we prove the exponential stability for a laminated beam consisting of two identical layers of uniform density, which is a system closely related to the Timoshenko beam theory, taking into account that an adhesive of small thickness is bonding the two layers and produce the interfacial slip. It is assumed that the thickness of the adhes...

We study a thermoviscoelastic Timoshenko system with heat conduction modeled by the Cattaneo law. Additionally, a viscoelastic damping in the equation for the displacement competing with a viscoelastic delay are considered. In this paper, we prove the exponential stability of the system applying semigroup theory.

In this work, was considered the Timoshenko system with weakly dissipation, and was proved the property of growth determined by spectrum of operator associated, was presented the type of semigroup and also indicated the best constant for the exponential stability

This work concerns the unilateral problem for the Klein-Gordon operator $$ \mathbb{L}=\frac{\partial^2 u}{\partial t^2}-M(|\nabla u|^2)\Delta u+M_1(|u|^2)u-f. $$ Using an appropriate penalization, we obtain a variational inequality for a perturbed equation, and then show the existence and uniqueness of solutions.

In this work we consider the Von Kármán system with frictional damping acting on the displacement and using the Method of Nakao we prove the exponential decay of the solution. The numerical scheme is presented for calculate the solution and to verify the long-time decay energy.

In this work we study the asymptotic behavior as t → ∞ of the solutions for the initial boundary value problem associated to the semi-linear wave equation with weak damping.

In this work we study the asymptotic behavior as t → ∞ of the solution for the Timoshenko system with delay term in the feedback. We use the semigroup theory for to prove the well-posedness of the system and for to establish the exponential stability. As far we know, there exist few results for problems with delay, where the asymptotic behavior is...

In this work we study existence of solutions for an abstract coupled system of nonlinear equations of extensible beams models and present the exponential decay for full energy of the system. Keywords: Nonlinear beam equation, abstract coupled system, existence and uniqueness of solution, asymptotic behavior.

We consider a control problem where the state variable is defined as the solution of a variational inequality. This system describes the vertical displacement of points of a thin plate with the presence of crack inside [7]. As the control we define the force that originates the deection of the plate. In order to get the system of optimality for the...

In this work we study a transmission problem for the model of beams developed by S. P. Timoshenko and J. M. Gere, Mechanics of materials, [D. Van Nostrand Company, Inc., New York (1972)]. We consider the case of mixed material, that is, a part of the beam has friction and the other is purely elastic. We show that for this type of material, the diss...

In this work we consider the Von Kármán system with internal damping acting on the displacement of the plate and using the Theorem due to Nakao, we prove the exponential decay of the solution.

In this work we establish existence, uniqueness and exponential decay of energy for the solutions of a system of wave equations coupled with locally distributed damping in a bounded smooth domain of any space dimension.

In this paper we study exact boundary controllability for a system of two linear wave equations coupled by lower order terms. We obtain square integrable control of Neuman type for initial state with finite energy, in nonsmooth domains of the plane.

In this paper we study the von Kármán plate model with long-range memory and we show the general decay of the solution as time goes to infinity. This result generalizes and improves on earlier ones in the literature because it allows certain relaxation functions which are not necessarily of exponential or polynomial decay.

We consider a transmission problem for the longitudinal displacement of a Euler-Bernoulli beam, where one small part of the beam is made of a viscoelastic material with Kelvin-Voigt constitutive relation. We use semigroup theory to prove existence and uniqueness of solutions. We apply a general results due to L. Gearhart [Trans. Am. Math. Soc. 236,...

In this article we study a thermoelastic system considering the linearized model proposed by Gurtin and Pipkin [8] instead of the Fourier's law for the heat flux. We use theory of semigroups [9, 11] combining Pruss' Theorem [10] and the idea developed in [5] to show that the system is not exponentially stable.