Article
Mathematics, Applied
Yu-Ying Duan, Ti-Jun Xiao
Summary: This paper investigates the long-time behavior of a porous-elastic system with infinite memory and nonlinear frictional damping. The authors prove the dissipativity of the dynamical system generated by the solutions under certain conditions on the memory kernel g and the frictional damping h. Moreover, they establish the asymptotic smoothness, quasi-stability, existence of a global attractor, and finite dimensionality of the attractor based on a more general condition on g.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Junior, Mauro L. Santos
Summary: This paper focuses on the existence of attractors for a nonlinear porous elastic system subjected to delay-type damping in the volume fraction equation. The study is conducted from the perspective of quasi-stability for infinite dimensional dynamical systems, leading to the results of global and exponential attractors.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2021)
Article
Mathematics, Applied
Xiaoming Peng, Yadong Shang
Summary: This paper investigates the long time behavior of a quasilinear viscoelastic equation with nonlinear damping, and establishes the existence of global attractors under suitable assumptions.
Article
Mathematics, Applied
Kais Ammari, Farhat Shel, Louis Tebou
Summary: In this paper, we investigate the regularity of two damped abstract elastic systems, where the damping and coupling involve fractional powers of principal operators. By proving the analyticity of the underlying semigroup and the certain Gevrey classes for different parameter ranges, we analyze the characteristics of the systems and provide some application examples.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Quang-Minh Tran, Thi-Thi Vu, Mirelson M. Freitas
Summary: This paper deals with blow-up solutions of the initial boundary value problems for a porous elastic system with nonlinear damping and source terms. The lower bound and upper bound of the lifespan of the blow-up solution are estimated, and the blow-up rate is also estimated by considering both linear and nonlinear weak damping terms.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
M. L. Santos, D. S. Almeida Junior, S. M. S. Cordeiro
Summary: In this paper, we investigate a one-dimensional porous-elastic system with nonlinear localized damping. By establishing an energy decay model and utilizing observability inequality, unique continuation property, and the reduction principle, we derive specific results that generalize and improve previous literature outcomes.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2022)
Article
Mathematics, Applied
Marek Galewski
Summary: This paper examines nonlinear problems involving monotone potential mappings and their strongly continuous perturbations using monotonicity methods and variational arguments. The study focuses on the Palais-Smale condition satisfied by functional obtained through the direct method of calculus of variations, the relationship between minimizing sequences and Galerkin approximations, and conditions on the derivative of the action functional leading to convergent bounded Palais-Smale sequences. Additionally, comments are made on the convergence of Palais-Smale sequences in Rabier's mountain pass theorem.
ADVANCES IN NONLINEAR ANALYSIS
(2021)
Article
Mathematics, Applied
Jose H. Rodrigues, Madhumita Roy
Summary: This paper investigates the long-time dynamics of a nonlinearly forced 3-D wave equation subject to nonlinear boundary dissipation. By utilizing the theory of global attractors and developing new techniques in Neumann hyperbolic problems, the problem of critical nonlinearity supported on the boundary has been tackled successfully.
APPLIED MATHEMATICS AND OPTIMIZATION
(2022)
Article
Mathematics, Applied
Mirelson M. Freitas, Anderson J. A. Ramos, Mauro L. Santos
Summary: This paper focuses on studying the asymptotic behavior of a binary mixture problem of solids with fractional damping and sources terms. The existence of global attractors with finite fractal dimension and exponential attractors is proven. Additionally, the upper-semicontinuity of global attractors as the fractional exponent tends to zero is also established.
APPLIED MATHEMATICS AND OPTIMIZATION
(2021)
Article
Mathematics, Applied
Chang Zhang, Fengjuan Meng, Cuncai Liu
Summary: This paper focuses on the well-posedness and long-time behavior of solutions for nonlocal diffusion porous medium equations with nonlinear term. It establishes the well-posedness of solutions in L-1(omega) and proves the existence of a global attractor through the existence of a compact absorbing set. Index theory is applied to consider the dimension of the attractor, demonstrating the presence of an infinite dimensional attractor under suitable conditions.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Physics, Multidisciplinary
Miao Yu, Xin Fang, Dianlong Yu, Jihong Wen, Li Cheng
Summary: This paper demonstrates that nonlinear elastic metamaterials can generate robust hyper-damping effects and provide a method for efficient low-frequency and broadband vibration suppression. The findings expand the application scope of nonlinear metamaterials.
Article
Mathematics, Applied
B. Feng, M. M. Freitas, D. S. Almeida, A. J. A. Ramos, R. Q. Caljaro
Summary: This paper studies the global attractors for a new partially damped porous-elasticity system by considering a truncated version without blow-up on the second wave speed. The global well-posedness of the system is established using the Faedo-Galerkin method. By considering only one damping term on the volume fraction, we prove the existence of an absorbing set for the solution semigroup, regardless of the coefficients of the system. Finally, using Lyapunov and recent quasi-stability methods, we prove the existence of smooth global attractors with finite fractal dimension.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2023)
Article
Mathematics, Applied
Andres Contreras, Juan Peypouquet
Summary: This work focuses on evolution equations, forward-backward discretizations, and the connection between differential equations and variational analysis. The contributions include estimating the distance between iterates of sequences generated by these schemes, approximating solutions of evolution equations on a bounded time frame, and developing a mathematical methodology for deducing the behavior of sequences generated by forward-backward algorithms.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2021)
Article
Mathematics, Applied
Yuxuan Chen, Yanan Li, Zhijian Yang
Summary: This study investigates the stability of strong exponential attractors for the Kirchhoff wave model with structural nonlinear damping and dissipative index θ ∈ [1/2, 1). It demonstrates the existence of a family of strong bi-space exponential attractors, which are also optimal regularity and Holder continuous at a specific point θ0 given optimal subcritical growth of the nonlinearity f(u). The method employed in this paper overcomes previous difficulties in obtaining Lipschitz continuity and quasi-stability of the evolution semigroup, leading to the desired result. (c) 2023 Elsevier Ltd. All rights reserved.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Abdelbaki Choucha, Salah Mahmoud Boulaaras, Djamel Ouchenane, Bahri Belkacem Cherif, Mohamed Abdalla
Summary: This paper considers a swelling porous elastic system with viscoelastic damping and distributed delay terms in the second equation. The coupling provides new contributions to the theory of asymptotic behaviors of swelling porous elastic soils, and a general decay result is established using the multiplier method.
JOURNAL OF FUNCTION SPACES
(2021)
Article
Mathematics, Applied
B. Feng, M. M. Freitas, D. S. Almeida Junior, A. J. A. Ramos
Summary: This paper discusses a porous elastic problem within a past history framework and studies its long-time behavior using an autonomous dynamical system. Instead of directly demonstrating the system has a bounded absorbing set, it shows the system is a gradient system and asymptotically smooth, proving the existence of a global attractor characterized as the unstable manifold of the set of stationary solutions. Additionally, quasi-stability of the system is obtained through a stabilizability inequality, leading to the finite fractal dimension of the global attractor.
APPLICABLE ANALYSIS
(2022)
Article
Mathematics, Applied
Tomas Caraballo, Alexandre N. Carvalho, Jose A. Langa, Alexandre N. Oliveira-Sousa
Summary: In this paper, we study the stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove the robustness of nonuniform hyperbolicity for linear evolution processes and provide conditions for the uniqueness and continuous dependence of projections associated with nonuniform exponential dichotomies. An example of an evolution process in a Banach space that exhibits nonuniform exponential dichotomy is presented, and the permanence of nonuniform hyperbolicity under perturbations is studied. Finally, we prove the persistence of nonuniform hyperbolic solutions for nonlinear evolution processes under perturbations.
ASYMPTOTIC ANALYSIS
(2022)
Article
Materials Science, Multidisciplinary
D. S. Almeida Junior, A. J. A. Ramos, M. M. Freitas, M. J. Dos Santos, T. El Arwadi
Summary: This paper examines a porous-elastic system where dissipation mechanisms impact both the elastic and porous structures. The one-dimensional porous-elastic system is defined on bounded domains in space, and polynomial stability is proven under specific relationships between damping parameters. The optimality of the rate of polynomial decay is also established.
MATHEMATICS AND MECHANICS OF SOLIDS
(2022)
Article
Mathematics, Applied
M. M. Freitas, R. Q. Caljaro, M. L. Santos, A. J. A. Ramos
Summary: In this paper, we investigate the singular limit dynamics of two parallel wave equations when the spring coefficient approaches infinity. We establish the existence, finiteness of fractal dimension, and smoothness of global attractors. Our results show that the singular limit of the parallel wave equations is a single wave equation as alpha approaches infinity, and we also prove the upper-semicontinuity of global attractors in this limit.
APPLIED MATHEMATICS AND OPTIMIZATION
(2022)
Article
Mathematics, Applied
M. C. Bortolan, A. N. Carvalho, J. A. Langa, G. Raugel
Summary: This work investigates Morse-Smale semigroups under nonautonomous perturbations and introduces the concept of Morse-Smale evolution processes of hyperbolic type. The stability of the phase diagram of the attractors is proven, with intersecting stable and unstable manifolds. The complete proofs of local and global lambda-lemmas in the infinite-dimensional case, originally due to D. Henry, are included here for completeness.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Hongyong Cui, Arthur C. Cunha, Jose A. Langa
Summary: This paper studies the conditions for ensuring that a random uniform attractor has a finite fractal dimension, with two main criteria being the smoothing property and the squeezing property of the system. The upper bound of the fractal dimension consists of the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property.
JOURNAL OF NONLINEAR SCIENCE
(2022)
Article
Mathematics
M. M. Freitas, A. J. A. Ramos, D. S. Almeida Junior, L. G. R. Miranda, A. S. Noe
Summary: This article explores the asymptotic dynamics of a nonlinear swelling porous-elastic system with memory term and fractional damping. The existence, smoothness, and finite dimensionality of global attractors are established using the quasi-stability approach. The existence of exponential attractors is also proven. These novel results make significant contributions to the theory of nonlinear dynamics in swelling porous elastic soils.
BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA
(2023)
Article
Mathematics, Applied
M. M. Freitas, M. L. Santos, M. J. Dos Santos, A. J. A. Ramos
Summary: In this paper, we investigate the long-term behavior of solutions to the one-dimensional porous-elasticity problem with porous dissipation and nonlinear feedback force. We prove that as the parameter J tends to zero, the porous-elasticity problem converges to a quasi-static problem for microvoids motion. By utilizing the recent quasi-stability theory, we obtain a finite dimensional global attractor with additional regularity in J. Finally, we compare the porous-elasticity problem with the quasistatic problem in terms of the upper-semicontinuity of their attractors as J -> 0.
ASYMPTOTIC ANALYSIS
(2023)
Article
Materials Science, Multidisciplinary
Mirelson M. Freitas, Dilberto S. Almeida Junior, Mauro L. Santos, Anderson J. A. Ramos, Ronal Q. Caljaro
Summary: This paper studies the long-time dynamics of a Timoshenko system and proves the existence of smooth finite dimensional global attractor and exponential attractor.
MATHEMATICS AND MECHANICS OF SOLIDS
(2023)
Article
Mathematics, Applied
Alberto L. C. Costa, Mirelson M. Freitas, Renhai Wang
Summary: This paper focuses on the asymptotic behavior of solutions to nonautonomous Lame systems that model the physical phenomenon of isotropic elasticity. It is discovered that the system generates a nonautonomous dynamical system and possesses a minimal universe pullback attractor. The upper-semicontinuity of these pullback attractors is also established as the perturbation parameter tends to zero. Quasi-stability ideas are utilized to overcome the difficulty caused by the critical growthness of the nonlinearity.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2022)
Article
Mathematics, Applied
Mirelson M. Freitas, Tijani A. Apalara, Anderson J. A. Ramos, Rafael S. Costa
Summary: This paper investigates the long-time dynamics of a semilinear system modeling a binary mixture of solids with frictional dissipation mechanism only affecting the elastic equation and subjected to a nonlinear source term. The coupling presents new contributions to the theory of nonlinear dynamics of partially damped semilinear systems of a binary mixture of solids. Using quasi-stability methods, we prove the existence of a smooth finite-dimensional global attractor regardless of the system's wave speeds. Furthermore, we establish the existence of exponential attractors.
APPLICABLE ANALYSIS
(2022)
Article
Mathematics, Applied
Alexandre N. Carvalho, Luciano R. N. Rocha, Jose A. Langa, Rafael Obaya
Summary: In this work, we investigate the structure of the skew-product attractor for a planar diffusively coupled ordinary differential equation. We provide a non-autonomous structure that completely describes the dynamics of this model and give a Morse decomposition for the skew-product attractor. Our findings indicate that the complexity of the isolated invariant sets is related to the complexity of the attractor, and when beta is asymptotically almost periodic, the isolated invariant sets will be almost periodic hyperbolic global solutions.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2023)
Article
Physics, Mathematical
M. M. Freitas, R. Q. Caljaro, A. J. A. Ramos, H. C. M. Rodrigues
Summary: In this paper, a system modeling a mixture of three interacting continua with localized nonlinear damping and external forces is considered. The main goal is to construct a smooth global attractor with a finite fractal dimension using the recent quasi-stability theory. The convergence of these attractors with respect to a parameter epsilon that multiplies the external forces is also studied. This study generalizes and improves upon the previous paper by Freitas et al. [Discrete Contin. Dyn. Syst. B 27, 3563 (2021)].
JOURNAL OF MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
M. M. Freitas, A. O. Ozer, G. Liu, A. J. A. Ramos, E. R. N. Fonseca
Summary: This paper presents a study on a fully-dynamic piezoelectric beam model with a nonlinear force acting on the beam's longitudinal displacements. The motion equations follow a system of non-compactly coupled wave equations. By discarding magnetic effects and decoupling the equations to a single wave equation, the authors investigate the existence of smooth finite-dimensional global attractors. Additionally, the existence of a finite set of determining functionals for the long-time behavior of the system is proved. Furthermore, a singular limit problem is considered to transition from the fully-dynamic beam model to the commonly-used electrostatic/quasi-static beam model, allowing for a comparison of attractors in the two models.
EVOLUTION EQUATIONS AND CONTROL THEORY
(2023)
Article
Mathematics, Applied
M. M. Freitas, A. J. A. Ramos, M. Aouadi, D. S. Almeida Junior
Summary: In this paper, the long-time dynamics of the Bresse system under mixed homogeneous Dirichlet-Neumann boundary conditions is studied, where the heat conduction is governed by Cattaneo's law. The damping effect is only applied to the shear angle displacement, while the vertical and longitudinal displacements are not restricted. By making general assumptions on the source term and utilizing the semigroup theory, the global well-posedness and the existence of global attractors with finite fractal dimension in natural space energy are established. Furthermore, the upper semicontinuity with respect to the relaxation time tau approaching zero is proven.
DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL
(2023)
Article
Mathematics
Daniele Cassani, Zhisu Liu, Giulio Romani
Summary: This article investigates the strongly coupled nonlinear Schrodinger equation and Poisson equation in two dimensions. The existence of solutions is proved using a variational approximating procedure, and qualitative properties of the solutions are established through the moving planes technique.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Giovanni Alessandrini, Romina Gaburro, Eva Sincich
Summary: This paper considers the inverse problem of determining the conductivity of a possibly anisotropic body Ω, subset of R-n, by means of the local Neumann-to-Dirichlet map on a curved portion Σ of its boundary. Motivated by the uniqueness result for piecewise constant anisotropic conductivities, the paper provides a Hölder stability estimate on Σ when the conductivity is a priori known to be a constant matrix near Σ.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Nuno Costa Dias, Cristina Jorge, Joao Nuno Prata
Summary: This article studies the time dependent Euler-Bernoulli beam equation with discontinuous and singular coefficients, and obtains an explicit formulation of the differential problem using an extension of the Hormander product of distributions. The dynamics of the Euler-Bernoulli beam model with discontinuous flexural stiffness and structural cracks are further explored, and the relationship between the characteristic frequencies of the beam and the singularities in the flexural stiffness is investigated.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Baoquan Zhou, Hao Wang, Tianxu Wang, Daqing Jiang
Summary: This paper is Part I of a two-part series that presents a mathematical framework for approximating the invariant probability measures and density functions of stochastic generalized Kolmogorov systems with small diffusion. It introduces two new approximation methods and demonstrates their utility in various applications.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Yun Li, Danhua Jiang, Zhi-Cheng Wang
Summary: In this study, a nonlocal reaction-diffusion equation is used to model the growth of phytoplankton species in a vertical water column with changing-sign advection. The species relies solely on light for metabolism. The paper primarily focuses on the concentration phenomenon of phytoplankton under conditions of large advection amplitude and small diffusion rate. The findings show that the phytoplankton tends to concentrate at certain critical points or the surface of the water column under these conditions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Monica Conti, Stefania Gatti, Alain Miranville
Summary: The aim of this paper is to study a perturbation of the Cahn-Hilliard equation with nonlinear terms of logarithmic type. By proving the existence, regularity and uniqueness of solutions, as well as the (strong) separation properties of the solutions from the pure states, we finally demonstrate the convergence to the Cahn-Hilliard equation on finite time intervals.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Qi Qiao
Summary: This paper investigates a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By using the geometric singular perturbation theory, the existence of a positive traveling wave connecting two constant steady states is confirmed. The monotonicity of the wave is analyzed for different parameter ranges, and spectral instability is observed in some exponentially weighted spaces.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Xiaolong He
Summary: This article employs the CWB method to construct quasi-periodic solutions for nonlinear delayed perturbation equations, and combines the techniques of Green's function estimate and the reducibility method in KAM theory to solve the linear equation, thus extending the applicability of the CWB method. As an application, it studies the positive quasi-periodic solutions for a class of Lotka-Volterra equations with quasi-periodic coefficients and time delay.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Nicolas Camps, Louise Gassot, Slim Ibrahim
Summary: In this paper, we consider the probabilistic local well-posedness problem for the Schrodinger half-wave equation with a cubic nonlinearity in quasilinear regimes. Due to the lack of probabilistic smoothing in the Picard's iterations caused by high-low-low nonlinear interactions, we need to use a refined ansatz. The proof is an adaptation of Bringmann's method on the derivative nonlinear wave equation [6] to Schrodinger-type equations. In addition, ill-posedness results for this equation are discussed.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Elie Abdo, Mihaela Ignatova
Summary: In this study, we investigate the Nernst-Planck-Navier-Stokes system with periodic boundary conditions and prove the exponential nonlinear stability of constant steady states without constraints on the spatial dimension. We also demonstrate the exponential stability from arbitrary large data in the case of two spatial dimensions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Peter De Maesschalck, Joan Torregrosa
Summary: This paper provides the best lower bound for the number of critical periods of planar polynomial centers known up to now. The new lower bound is obtained in the Hamiltonian class and considering a single period annulus. The key idea is the perturbation of a vector field with many cusp equilibria, which is constructed using elements of catastrophe theory.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Leyi Jiang, Taishan Yi, Xiao-Qiang Zhao
Summary: This paper studies the propagation dynamics of a class of integro-difference equations with a shifting habitat. By transforming the equation using moving coordinates and establishing the spreading properties of solutions and the existence of nontrivial forced waves, the paper contributes to the understanding of the propagation properties of the original equation.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Mckenzie Black, Changhui Tan
Summary: This article investigates a family of nonlinear velocity alignments in the compressible Euler system and shows the asymptotic emergent phenomena of alignment and flocking. Different types of nonlinearity and nonlocal communication protocols are studied, resulting in a variety of different asymptotic behaviors.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Lorenzo Cavallina
Summary: In this paper, the concept of variational free boundary problem is introduced, and a unified functional-analytical framework is provided for constructing families of solutions. The notion of nondegeneracy of a critical point is extended to this setting.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Ying-Chieh Lin, Kuan-Hsiang Wang, Tsung-Fang Wu
Summary: In this study, we investigate a linearly coupled Schrodinger system and establish the existence of positive ground states under suitable assumptions and by using variational methods. We also relax some of the conditions and provide some results on the existence of positive ground states to a linearly coupled Schrodinger system in a bounded domain.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
