Papers by Marcelo Cavalcanti
Zeitschrift für Angewandte Mathematik und Physik, Oct 19, 2021
Journal of Dynamics and Differential Equations, Feb 25, 2022
Differential and Integral Equations, 2000
ABSTRACT In this paper we consider the existence, uniqueness and asymptotic behaviour of the solu... more ABSTRACT In this paper we consider the existence, uniqueness and asymptotic behaviour of the solutions of nonlinear degenerate equations on manifolds. The existence is proved by making use of the Galerkin method and the uniform decay is obtained considering the perturbed energy method.
Birkhäuser Basel eBooks, 2005
We study the global existence and uniform decay rates of solutions of the problem
Applied Mathematics and Optimization, Aug 2, 2018
In this article, we consider the energy decay of a viscoelastic wave in an heterogeneous medium. ... more 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.
Journal of Differential Equations, Feb 1, 2017
In this paper, we study the existence at the H 1-level as well as the stability for the damped de... more In this paper, we study the existence at the H 1-level as well as the stability for the damped defocusing Schrödinger equation in R d. 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 theory. In particular, when d = 1 or d = 2, we obtain the uniqueness. Decay estimates for the L 2-level and (H 1 ∩ L p+2)-level energies are established with the help of direct multipliers method, coupled with refined energy estimates and a lower semi-continuity argument.
Journal of Mathematical Analysis and Applications, May 1, 2003
We study the global existence of solutions of the nonlinear degenerate wave equation (ρ 0) (*) ρ(... more We study the global existence of solutions of the nonlinear degenerate wave equation (ρ 0) (*) ρ(x)y − ∆y = 0 in Ω × ]0, ∞[, y = 0 on Γ 1 × ]0, ∞[, ∂y ∂ν + y + f (y) + g(y) = 0 on Γ 0 × ]0, ∞[, y(x, 0) = y 0 , (√ ρy)(x, 0) = (√ ρy 1)(x) in Ω, where y denotes the derivative of y with respect to parameter t, f (s) = C 0 |s| δ s and g is a nondecreasing C 1 function such that k 1 |s| ξ +2 g(s)s k 2 |s| ξ +2 for some k 1 , k 2 > 0 with 0 < δ, ξ 1/(n − 2) if n 3 or δ, ξ > 0 if n = 1, 2. The existence of solutions is proved by means of the Faedo-Galerkin method. Furthermore, when ξ = 0 the uniform decay is obtained by making use of the perturbed energy method.
Methods and applications of analysis, 2008
This paper is concerned with the study of the wave equation on compact surfaces and locally distr... more This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by utt − ∆ M u + a(x) g(ut) = 0 on M × ]0, ∞[ , where M ⊂ R 3 is a smooth (of class C 3) oriented embedded compact surface without boundary, such that M = M 0 ∪ M 1 , where M 1 := {x ∈ M; m(x) • ν(x) > 0} , AND M 0 = M\M 1. Here, m(x) := x − x 0 , (x 0 ∈ R 3 fixed) and ν is the exterior unit normal vector field of M. For i = 1,. .. , k, assume that there exist open subsets M 0i ⊂ M 0 of M with smooth boundary ∂M 0i such that M 0i are umbilical, or more generally, that the principal curvatures k 1 and k 2 satisfy |k 1 (x) − k 2 (x)| < ε i (ε i considered small enough) for all x ∈ M 0i. Moreover suppose that the mean curvature H of each M 0i is non-positive (i.e. H ≤ 0 on M 0i for every i = 1,. .. , k). If a(x) ≥ a 0 > 0 on an open subset M * ⊂ M that contains M\ ∪ k i=1 M 0i and if g is a monotonic increasing function such that k|s| ≤ |g(s)| ≤ K|s| for all |s| ≥ 1, then uniform decay rates of the energy hold.
Differential and Integral Equations, 2002
Communications in Contemporary Mathematics, Oct 1, 2004
The nonlinear and damped extensible plate (or beam) equation is considered [Formula: see text] wh... more The nonlinear and damped extensible plate (or beam) equation is considered [Formula: see text] where Ω is any bounded or unbounded open set of Rn, α&gt;0 and f, g are power like functions. The existence of global solutions is proved by means of the Fixed Point Theorem and continuity arguments. To this end we avoid handling the nonlinearity M(∫Ω|∇u|2dx) in the a priori estimates of energy. Furthermore, uniform decay rates of the energy are also obtained by making use of the perturbed energy method for domains with finite measure.
Advances in Differential Equations, 2001
ABSTRACT This paper is devoted to the existence of global solutions of the \newline Kirchhoff-Car... more ABSTRACT This paper is devoted to the existence of global solutions of the \newline Kirchhoff-Carrier equation $$u_{tt}-M\bigl(t,\int_{\Omega}\left|\nabla u\right|^2dx\bigr)\Delta u=0$$ subject to nonlinear boundary dissipation. Assuming that $M(t,\lambda )\geq m_0&gt;0$, we prove the existence and uniqueness of regular solutions without any smallness on the initial data. Moreover, uniform decay rates are obtained by assuming a nonlinear feedback acting on the boundary.
arXiv (Cornell University), Nov 26, 2020
We study the stabilization and the wellposedness of solutions of the wave equation with subcritic... more 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 observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we truncate the nonlinearities, and thereby construct approximate solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures. We include an appendix on the latter topic here to make the article self contained and supplement details to proofs of some of the theorems which can be already be found in the lecture notes of [7]. Once we establish essential observability properties for the approximate solutions, it is not difficult to prove that the solution of the original problem also possesses a similar feature via a delicate passage to limit. In the last part of the paper, we establish various decay rate estimates for different growth conditions on the nonlinear dissipative effect. We in particular generalize the known results on the subject to a considerably larger class of dissipative effects.
Southeast Asian Bulletin of Mathematics, May 1, 2000
In this paper, we study a hyperbolic model based on the equation y tt − y + n j =1 b j (x, t) ∂y ... more In this paper, we study a hyperbolic model based on the equation y tt − y + n j =1 b j (x, t) ∂y t ∂x j = 0 with nonlinear boundary conditions given by ∂y ∂ν + f (y) + g(y t) = 0. We prove the existence and the uniqueness of global solutions. Also, we obtain the uniform decay of the energy without control of its derivative sign.
Electronic Journal of Qualitative Theory of Differential Equations, 1998
This paper is concerned with the boundary exact controllability of the equation u − ∆u − t 0 g(t ... more This paper is concerned with the boundary exact controllability of the equation u − ∆u − t 0 g(t − σ)∆u(σ)dσ = −∇p where Q is a finite cylinder Ω×]0, T [ , Ω is a bounded domain of R n , u = (u 1 (x, t), • • • , u 2 (x, t)), x = (x 1 , • • • , x n) are n− dimensional vectors and p denotes a pressure term. The result is obtained by applying HUM (Hilbert Uniqueness Method) due to J.L.Lions. The above equation is a simple model of dynamical elasticity equations for incompressible materials with memory.
Siam Journal on Control and Optimization, 2018
This paper is concerned with the study of the uniform decay rates of the energy associated with m... more 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 unbound...
Journal of Differential Equations, Dec 1, 2015
In this paper, we consider coupled wave-wave, Petrovsky-Petrovsky and wave-Petrovsky systems in N... more 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, where the decay rate is given in terms of the general growth of the convolution kernel at infinity and the arbitrary regularity of the initial data. We finish our paper by considering the uncoupled wave and Petrovsky equations with complementary frictional damping and infinite memory, and showing a strong stability estimate depending only on the general growth of the convolution kernel at infinity.
Differential and Integral Equations, 2004
The viscoelastic Euler-Bernoulli equation with nonlinear and nonlocal damping utt + Δ 2 u − t 0 g... more The viscoelastic Euler-Bernoulli equation with nonlinear and nonlocal damping utt + Δ 2 u − t 0 g(t − τ)Δ 2 u(τ) dτ + a(t)ut = 0 in Ω × R + , where a(t) = M Ω |∇u(x, t)| 2 dx , is considered in bounded or unbounded domains Ω of R n. The existence of global solutions and decay rates of the energy are proved.
Transactions of the American Mathematical Society, Apr 13, 2009
This paper is concerned with the study of the wave equation on compact surfaces and locally distr... more This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by u tt − ∆ M u + a(x) g(u t) = 0 on M × ]0, ∞[ , where M ⊂ R 3 is a smooth oriented embedded compact surface without boundary. Denoting by g the Riemannian metric induced on M by R 3 , we prove that for each > 0, there exist an open subset V ⊂ M and a smooth function f : M → R such that meas(V) ≥ meas(M) − , Hessf ≈ g on V and inf x∈V |∇f (x)| > 0. In addition, we prove that if a(x) ≥ a 0 > 0 on an open subset M * ⊂ M which contains M\V and if g is a monotonic increasing function such that k|s| ≤ |g(s)| ≤ K|s| for all |s| ≥ 1, then uniform and optimal decay rates of the energy hold.
Differential and Integral Equations, Jul 1, 2009
Advanced Nonlinear Studies, 2021
In this paper we study the existence as well as uniform decay rates of the energy associated with... more 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 {\Omega\subset\mathbb{R}^{n}} , n ≤ 3 {n\leq 3} , is a bounded domain with smooth boundary ∂ Ω = Γ {\partial\Omega=\Gamma} and α = 2 , 3 {\alpha=2,3} . Our goal is to consider a different approach than the one used in [B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z. 254 2006, 4, 729–749], so instead than using the properties of pseudo-differential operators introduced by cited authors, we consider a nonlinear damping, so that we remark that no growth assumptions on g ( z ) {g(z)} are made near the origin.
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Papers by Marcelo Cavalcanti