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This paper is concerned with the well-posedness and asymptotic behavior of solutions for the following extensible beam equation with non-local energy damping u tt + Δ 2 u - κ ϕ (‖ ∇ u ‖ 2 ) Δ u - φ (‖ Δ u ‖ 2 + ‖ u t ‖ 2) Δ u t + f (u) = h. More specifically, this is a complementary work to the paper by Sun and Yang (Discrete Contin Dyn Syst Ser B 27(6):3101–3129, 2022), where the authors consider this model assuming the hypothesis that φ ∈ C 1 (R +) with non-degenerate condition φ (s) > 0 , s ∈ R + . They prove the existence of strong global and exponential attractors and their robustness on the perturbed extensibility parameter κ . In this paper assuming φ (s) ⪆ γ s q which contemplates the degenerate condition φ (0) = 0 , we prove the existence of weak and regular solutions to the problem proposed and employing the method given in Temam (Springer-Verlag, New York, 1998) we show that the dynamic system (X , S t) given by the weak solutions of the problem has a compact global attractor in the weak topology of the phase space X. This class of nonlinear beams arising in connection with models for flight structures with non-local energy damping is proposed by Balakrishnan and Taylor (Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989). [ABSTRACT FROM AUTHOR]

In this paper, we propose a reaction–advection–diffusion dengue fever model with seasonal developmental durations and intrinsic incubation periods. Firstly, we establish the well-posedness of the model. Secondly, we define the basic reproduction number ℜ 0 for this model and show that ℜ 0 is a threshold parameter: if ℜ 0 < 1 , then the disease-free periodic solution is globally attractive; if ℜ 0 > 1 , the system is uniformly persistent. Thirdly, we study the global attractivity of the positive steady state when the spatial environment is homogeneous and the advection of mosquitoes is ignored. As an example, we use the model to investigate the dengue fever transmission case in Guangdong Province, China, and explore the impact of model parameters on ℜ 0 . Our findings indicate that ignoring seasonality may underestimate ℜ 0 . Additionally, the spatial heterogeneity of transmission may increase the risk of disease transmission, while the increase of seasonal developmental durations, intrinsic incubation periods and advection rates can all reduce the risk of disease transmission. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the dynamics of the modified Leslie–Gower model with a weak Allee effect and fear effect on predators. Assuming that the inherent growth rate of prey is much faster than that of predators, this becomes a singular perturbation problem. Based on the geometric singular perturbation theory, we prove the existence of canard cycles and relaxation oscillation and provide an asymptotic expression of the relaxation oscillation period. Furthermore, it demonstrates that, under certain conditions, the degenerate transcritical bifurcation point is a global attractor using geometric singular perturbation theory, the center manifold theorem and the blow-up method. Numerical simulations verify the theoretical results. [ABSTRACT FROM AUTHOR]

Of concern is a problem of strain gradient porous elastic theory with nonlinear damping terms, whose constitutive equations contain higher-order derivatives of the displacement in the basic postulates. The paper is based on the theory of 'consistency' due to Aouadi et al. [J. Therm. Stress. 43(2)(2020), 191–209] and Ieşan [American Institute of Physics, Conference Proceedings, 1329 (2011), 130–149], and contains four results. We firstly show that the system is global well posed by using maximal monotone operator. The second main result is the existence of global attractors which is proved by the method developed by Chueshov and Lasiecka [Long-time behavior of second order evolution equations with nonlinear damping. Mem. Amer. Math. Soc. vol. 195, no. 912, Providence, 2008; Von Karman evolution equations: well-posedness and long-time dynamics. Springer Monographs in Mathematics, Springer, New York, 2010]. By showing the system is gradient and asymptotically smooth, we establish the existence of global attractors with finite fractal dimension via a stabilizability inequality. Then we study the continuity of global attractors regarding the parameter in a residual dense set. The above results allow the damping terms with polynomial growth. Finally we discuss the exponential decay and global boundedness to the linear case of damping terms of the system. The assumption of equal-speed wave propagations is not needed for all of results obtained. [ABSTRACT FROM AUTHOR]

In this paper we prove several turnpike results for trajectories of discrete disperse dynamical systems introduced in 1980 by A. M. Rubinov, which have a prototype in mathematical economics. [ABSTRACT FROM AUTHOR]

This paper is concerned with the dimension of the global attractors for a time-dependent strongly damped subcritical Kirchhoff wave equation with a memory term. A careful analysis is required in the proof of a stabilizability inequality. The main result establishes the finite dimensionality of the global attractor. [ABSTRACT FROM AUTHOR]

In this paper, we consider the 3D tropical climate model with damping terms in the equation of u, v and θ , respectively. Firstly, we get some uniform estimates of strong solution. Secondly, we derive the result of the continuity of the semigroup { S (t) } t ≥ 0 in case of 4 ≤ α , β < 5 and 13 5 < γ < 5 via some usual inequalities. Finally, the system (1.1) is shown to possess an (V , V) -global attractor and an (V , H 2) -global attractor. [ABSTRACT FROM AUTHOR]

A semilinear beam/plate equation with non-constant material density in the context of energy damping models is considered. Well-posedness and asymptotic behavior of solutions are established. The main result deals with the existence of a compact global attractor for the proposed problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the long‐time behavior of a porous‐elastic system with infinite memory and nonlinear frictional damping. We prove that the dynamical system generated by the solutions of the equations is dissipative, only under the basic conditions (for the well‐posedness) on the memory kernel g$$ g $$ and the frictional damping h$$ h $$. Further, we come up with a condition on g$$ g $$, being more general than the usual one g′(t)≤−cg(t)$$ {g}&#x0005E;{\prime }(t)\le - cg(t) $$ (with a positive constant c$$ c $$), under which we prove the asymptotic smoothness and quasi‐stability (the latter needs some stronger condition on h$$ h $$) of the dynamical system. Accordingly, we obtain the existence of a global attractor and show the finite dimensionality of the attractor. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of the existence of trajectory and global attractors in the Kelvin–Voigt fluid model, taking into account memory along the trajectories of fluid motion. For the model under study, the concept of a weak solution on a finite segment and semi-axis is introduced and the existence of their solutions is proved. The necessary exponential estimates for the solutions are established. Then, based on these estimates, the existence of trajectory and global attractors in the problem under study is proved. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and uniqueness of global attractors for a class of degenerate parabolic equations with memory on R n . Since the corresponding equation includes the degenerate term div { a (x) ∇ u } , it requires us to give appropriate assumptions about the weight function a (x) for studying our problem. Based on this, we first obtain the existence of a bounded absorbing set, then verify the asymptotic compactness of a solution semigroup via the asymptotic contractive semigroup method. Finally, the existence and uniqueness of global attractors are proved. In particular, the nonlinearity f satisfies the polynomial growth of arbitrary order p − 1 (p ≥ 2 ) and the idea of uniform tail-estimates of solutions is employed to show the strong convergence of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time behavior of a nonlinear porous elasticity system. The system is subject to a viscoporous damping and a nonlinear source term which is locally Lipschitz and depends only on the volume fraction. The dynamical system associated with the solutions of the model is gradient, and under the hypothesis of equal speeds of propagation for the waves, we prove that it is also quasi-stable, which allows us to show the existence of a global attractor for the system, which is the main result of the paper. [ABSTRACT FROM AUTHOR]

The problem of Reissner–Mindlin–Timoshenko plate systems with nonlinear damping terms is considered. The main result is the existence of global attractors. By showing the system is gradient and asymptotic smoothness via a stabilizability inequality, we establish the existence of global attractors with finite fractal dimension. The continuity of global attractors regarding the parameter in a residual dense set is also proved. The above results allow the damping terms with polynomial growth. [ABSTRACT FROM AUTHOR]

We prove the existence of global solutions and of a global attractor for a regularized viscous 3D tropical climate model. The regularization is carried out by means of the Helmholtz filter, in the spirit of what is done for some families of LES-turbulence models, thus obtaining (as α-model) the 3D Bardina tropical climate model for which we consider the associated dynamics. [ABSTRACT FROM AUTHOR]

We study two damped and forced discrete nonlinear Schrödinger equations on the one-dimensional infinite lattice. Without damping and forcing they are represented by the integrable Ablowitz-Ladik equation (AL) featuring non-local cubic nonlinear terms, and its standard (nonintegrable) counterpart with local cubic nonlinear terms (DNLS). The global existence of a unique solution to the initial value problem for both, the damped and forced AL and DNLS, is proven. It is further shown that for sufficiently close initial data, their corresponding solutions stay close for sufficiently long times. Concerning the asymptotic behaviour of the solutions to the damped and forced AL equation, a sufficient condition for the existence of a restricted global attractor is established. It is shown that the damped and driven DNLS possesses a global attractor. Finally, we prove the congruence of the restricted global AL attractor and the global DNLS attractor for dynamics ensuing from initial data contained in an appropriate bounded subset in a Banach space. [ABSTRACT FROM AUTHOR]

We study the long time behavior of the solutions for a weakly damped forced nonlinear fractional Klein‐Gordon‐Schrödinger system iut+iνu+i|u|2u−(−Δ)αu+vu=f,vtt+γvt−Δv+v+v3−|u|2=g,$$ {\displaystyle \begin{array}{cc}\hfill i{u}_t&amp; &amp;#x0002B; i\nu u&amp;#x0002B;i{\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;2u-{\left(-\Delta \right)}&amp;#x0005E;{\alpha }u&amp;#x0002B; vu&amp;#x0003D;f,\hfill \\ {}\hfill {v}_{tt}&amp; &amp;#x0002B;\gamma {v}_t-\Delta v&amp;#x0002B;v&#x0002B;{v}&amp;#x0005E;3-{\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;2&amp;#x0003D;g,\hfill \end{array}} $$for a given α∈12,1$$ \alpha \in \left(\frac{1}{2},1\right) $$ considered in the whole space ℝ$$ \mathrm{\mathbb{R}} $$. We prove that this system provides an infinite dimensional dynamical system in Hα(ℝ)×H1(ℝ)×L2(ℝ)$$ {H}&amp;#x0005E;{\alpha}\left(\mathrm{\mathbb{R}}\right)\times {H}&amp;#x0005E;1\left(\mathrm{\mathbb{R}}\right)\times {L}&amp;#x0005E;2\left(\mathrm{\mathbb{R}}\right) $$ that possesses a global attractor Aα$$ {\mathcal{A}}_{\alpha } $$ in the same space and more particularly that this attractor is in fact a compact set of H2α(ℝ)×H1+α(ℝ)×Hα(ℝ)$$ {H}&amp;#x0005E;{2\alpha}\left(\mathrm{\mathbb{R}}\right)\times {H}&amp;#x0005E;{1&amp;#x0002B;\alpha}\left(\mathrm{\mathbb{R}}\right)\times {H}&amp;#x0005E;{\alpha}\left(\mathrm{\mathbb{R}}\right) $$. [ABSTRACT FROM AUTHOR]

A laminated beam system with thermodiffusion effects is considered. The main result is the long‐time dynamics of the system. By showing the system is gradient and asymptotic smoothness, the existence of a global attractor is obtained, which is characterized as unstable manifold of the set of stationary solutions. The quasi‐stability of the system and the finite fractal dimension of the global attractor are established by a stabilizability inequality. The continuity of global attractors regarding the parameter in a residual dense set is finally proved. [ABSTRACT FROM AUTHOR]

In this article we develop an analogue of Aubry-Mather theory for time periodic dissipative equation x ˙ = ∂ p H (x , p , t) , p ˙ = - ∂ x H (x , p , t) - f (t) p with (x , p , t) ∈ T ∗ M × T (compact manifold M without boundary). We discuss the asymptotic behaviors of viscosity solutions for the associated Hamilton-Jacobi equation ∂ t u + f (t) u + H (x , ∂ x u , t) = 0 , (x , t) ∈ M × T w.r.t. certain parameters, and analyze the meanings in controlling the global dynamics. We also discuss the prospect of applying our conclusions to many physical models. [ABSTRACT FROM AUTHOR]

In this paper, we consider the long time behavior for a coupled system on Riemannian manifold consisting of the plate equation and the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity. We obtain the existence of the global attractor for the coupled system by semi-group method and multiplier method. [ABSTRACT FROM AUTHOR]

We characterize when the unique equilibrium point of the positive quadrant of the twodimensional Lotka–Volterra system is a global attractor in that quadrant. Additionally, we classify the phase portraits of this class of Lokta–Volterra system. [ABSTRACT FROM AUTHOR]

The multi-dimensional hyperviscous magnetohydrodynamic equations are considered in this paper. The well-posedness of the multi-dimensional hyperviscous magnetohydrodynamic equations is proved. Global attractor of the multi-dimensional hyperviscous magnetohydrodynamic equations is proved in H 1 2 + n 4 × H 1 2 + n 4 and H 1 + n 2 × H 1 + n 2 . [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time dynamics of Bresse system under mixed homogeneous Dirichlet–Neumann boundary conditions. The heat conduction is given by Cattaneo's law. Only the shear angle displacement is damped via the dissipation from the Cattaneo's law, and the vertical displacement and the longitudinal displacement are free. Under quite general assumptions on the source term and based on the semigroup theory, we establish the global well-posedness and the existence of global attractors with finite fractal dimension in natural space energy. Finally, we prove the upper semicontinuous with respect to the relaxation time τ as it converges to zero. [ABSTRACT FROM AUTHOR]

This paper focuses on the networks of n reaction-diffusion systems of the Hindmarsh - Rose type. We prove the existence of the global attractor of these networks in the space (L² (Ω) x L² (Ω))n. It is also bounded in L∞ (Ω) for any network topology satisfied some necessary conditions of coupling function and also the dynamic of the system. [ABSTRACT FROM AUTHOR]

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 tends to infinity (effectively causing a no-slip condition between the two layers), we get the convergence (in some sense) to the Timoshenko beam system. The existence of smooth finite-dimensional global attractors and exponential attractors is proved using the recent quasi-stability theory. We also established that the long-time behavior of solutions of the nonlinear system is completely determined by the dynamics of large finite number of functionals. Finally, we compare the laminated beam model with the Timoshenko model in the sense of the upper semicontinuity of its attractors as γ → ∞ . [ABSTRACT FROM AUTHOR]

In this research, we viewed the weights of a neural network as points in a metric space. We proposed that the process of training a neural network can be seen as an iterated function system on this space. Additionally, we found that the most effective training method involves starting with initial weights that are very close to the minimum error. We also provided an ergodic characterization of this efficient training. Our findings suggest that there is potential for further optimization advancements through numerical experimentation and the study of dynamical systems theory. [ABSTRACT FROM AUTHOR]

In this article, a second-order backward difference scheme combined with a conforming finite element method (FEM) is applied to a two-dimensional Sobolev equation with Burgers' type non-linearity with a nonhomogeneous forcing function in L ∞ (L 2). Some new a priori estimates are derived for the semidiscrete solution, which helps to prove the convergence result. Then, based on the Stolz-Cesaro theorem, a priori bounds for the second-order backward difference scheme are derived, which are valid uniformly in time as t N → ∞ and uniformly in the dispersion coefficient μ as μ → 0. For the discrete problem at each time step, it is shown using one variant of Brouwer's fixed point theorem that, there exists a discrete solution and uniqueness follows in a standard way. Moreover, existence of a discrete global attractor is established. Optimal error estimates in ℓ ∞ (L 2) and ℓ ∞ (H 0 1) -norms are obtained, whose bounds may depend exponentially on time. Subsequently, error bounds are established, which are valid uniformly in time under uniqueness conditions. Additionally, it is shown that as μ → 0 , the completely discrete solution of the 2D Sobolev equation converges to the corresponding discrete solution of the Burgers' equation. Further, results on extrapolated Backward difference scheme are briefly discussed. Finally, some computational experiments are conducted, whose results confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

Of concern is a problem of strain gradient porous elastic theory with nonlinear damping terms, whose constitutive equations contain higher-order derivatives of the displacement in the basic postulates. The paper is based on the theory of ‘consistency’ due to Aouadi et al. [J. Therm. Stress. 43(2)(2020), 191–209] and Ieşan [American Institute of Physics, Conference Proceedings, 1329 (2011), 130–149], and contains four results. We firstly show that the system is global well posed by using maximal monotone operator. The second main result is the existence of global attractors which is proved by the method developed by Chueshov and Lasiecka [Long-time behavior of second order evolution equations with nonlinear damping. Mem. Amer. Math. Soc. vol. 195, no. 912, Providence, 2008; Von Karman evolution equations: well-posedness and long-time dynamics. Springer Monographs in Mathematics, Springer, New York, 2010]. By showing the system is gradient and asymptotically smooth, we establish the existence of global attractors with finite fractal dimension via a stabilizability inequality. Then we study the continuity of global attractors regarding the parameter in a residual dense set. The above results allow the damping terms with polynomial growth. Finally we discuss the exponential decay and global boundedness to the linear case of damping terms of the system. The assumption of equal-speed wave propagations is not needed for all of results obtained. [ABSTRACT FROM AUTHOR]

Incorporating humoral immunity, cell-to-cell transmission and degenerated diffusion into a virus infection model, we investigate a viral dynamics model in heterogenous environments. The model is assumed that the uninfected and infected cells do not diffuse and the virus and B cells have diffusion. Firstly, the well-posedness of the model is discussed. And then, we calculated the reproduction number R 0 account for virus infection, and some useful properties of R 0 are obtained by means of the Kuratowski measure of noncompactness and the principle eigenvalue. Further, when R 0 < 1 , the infection-free steady state is proved to be globally asymptotically stable. Moreover, to discuss the antibody response reproduction number R ~ 0 of the model and the global dynamics of virus infection, including the global stability infection steady state and the uniform persistence of infection, and to obtain the k-contraction of the model with the Kuratowski measure of noncompactness, a special case of the model is considered. At the same time, when R 0 > 1 and R ~ 0 < 1 ( R ~ 0 > 1 ), we obtained a sufficient condition on the global asymptotic stability of the antibody-free infection steady state (the uniform persistence and global asymptotic stability of infection with antibody response). Finally, the numerical examples are presented to illustrate the theoretical results and verify the conjectures. [ABSTRACT FROM AUTHOR]

This paper is devoted to the global attractors of the tropical climate model. We first establish the global well-posedness of the system. Then by studying the existence of bounded absorbing sets, the global attractor is constructed. The estimates of the Hausdorff dimension and of the fractal dimension of the global attractor are obtained in the end. [ABSTRACT FROM AUTHOR]

In this paper, we consider the hyperbolic Cahn–Hilliard equation with a proliferation term, which has applications in biology. First, we study the well-posedness and the regularity of the solutions, which then allow us to study the dissipativity and the high-order dissipativity and finally the existence of the exponential attractor with Dirichlet boundary conditions. Finally, we give numerical simulations that confirm the results. [ABSTRACT FROM AUTHOR]

This paper is concerned with longtime dynamics of semilinear Lamé systems ∂ t 2 u - μ Δ u - (λ + μ) ∇ div u + α ∂ t u + f (u) = b , defined in bounded domains of R 3 with Dirichlet boundary condition. Firstly, we establish the existence of finite dimensional global attractors subjected to a critical forcing f(u). Writing λ + μ as a positive parameter ε , we discuss some physical aspects of the limit case ε → 0 . Then, we show the upper-semicontinuity of attractors with respect to the parameter when ε → 0 . To our best knowledge, the analysis of attractors for dynamics of Lamé systems has not been studied before. [ABSTRACT FROM AUTHOR]

In this paper we prove the Belitskii–Lyubich conjecture for triangular and gradient mappings. These results resemble those obtained for the discrete Markus–Yamabe conjecture. However, the proofs are quite different, which sheds new light on the subject. We complete the picture by showing that the general versions of the two conjectures can be turned into theorems at little cost, simply by relaxing their spectral condition. [ABSTRACT FROM AUTHOR]

We will study in this article the nonlinear Cahn–Hilliard equation with proliferation and regularization terms with regular and logarithmic potentials. First, we consider the regular potential case, we show that the solutions blow up in finite time or exist globally in time. Furthermore, we prove that the model possess a global attractor. In addition, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. In the second part, we consider the logarithmic potential case and show the existence of a global solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time dynamics of a Timoshenko system modeling vibrations of beams with non-linear localized damping mechanisms acting on both displacement and angular rotation and subjected to non-linear source terms. Using recent quasi-stability methods, we prove the existence of smooth finite dimensional global attractor, which is characterized as unstable manifold of the set of stationary solutions. Moreover, the existence of exponential attractors is shown. These aspects were not previously considered for the Timoshenko system with localized damping. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of the existence of an exponential attractor for a family of problems, in which diffusion dλ$$ {d}_{\lambda } $$ blows up in localized regions inside the domain utλ−div(dλ(x)(|∇uλ|p(x)−2+η)∇uλ)+|uλ|p(x)−2uλ=B(uλ),inΩuλ=0,on∂Ωuλ(0)=u0λ∈L2(Ω),$$ \left\{\begin{array}{cc}{u}_t^{\lambda }-\operatorname{div}\left({d}_{\lambda }(x)\left({\left|\nabla {u}^{\lambda}\right|}^{p(x)-2}+\eta \right)\nabla {u}^{\lambda}\right)+{\left|{u}^{\lambda}\right|}^{p(x)-2}{u}^{\lambda }=B\left({u}^{\lambda}\right),\kern0.30em & \kern0.1em \mathrm{in}\kern0.4em \Omega \\ {}{u}^{\lambda }=0,\kern0.30em & \kern0.1em \mathrm{on}\kern0.4em \mathrm{\partial \Omega}\\ {}{u}^{\lambda }(0)={u}_0^{\lambda}\in {L}^2\left(\Omega \right),\kern0.30em & \end{array}\right. $$and their limit problem via the l$$ l $$‐trajectory method. [ABSTRACT FROM AUTHOR]

We show that for any fixed accuracy and time length T, a finite number of T-time length pieces of the complete trajectories on the global attractor are capable of uniformly approximating all trajectories within the accuracy in the natural strong metric after sufficiently large time when the observed dissipative system is asymptotically compact. Moreover, we obtain the strong equicontinuity of all the complete trajectories on the global attractor. These results follow by proving the existence of a strongly compact strong trajectory attractor. The notion of a trajectory attractor was previously constructed for a family of auxiliary systems including the originally considered one without uniqueness. Recently, Cheskidov and the author developed a new framework called evolutionary system, with which a (weak) trajectory attractor can be actually defined for the original system. In this paper, the theory of trajectory attractors is further developed in the natural strong metric for our purpose. We then apply it to both the 2D and the 3D Navier–Stokes equations and a general nonautonomous reaction–diffusion system. [ABSTRACT FROM AUTHOR]

In this paper, we propose a degenerate dengue fever model with distinct dispersal rates in heterogeneous environment. Firstly, we obtain the well-posedness of the model and the existence of the global attractor. Secondly, we define the basic reproduction number ℜ 0 and show that ℜ 0 is a threshold parameter: if ℜ 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable; if ℜ 0 > 1 , the system is uniformly persistent, and the endemic equilibrium is globally attractive when the spatial environment is homogeneous. Thirdly, by studying the asymptotic profiles of the positive steady state and ℜ 0 , we show the impact of the dispersal rate on disease transmission. We conclude with some numerical simulations to illustrate the effect of some model parameters on ℜ 0. Particularly, we find that spatial heterogeneity may increase the risk of disease transmission. [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time behaviour of solutions of a degenerate Klein–Gordon–Schrödinger-type system which is defined in a bounded domain. First, we proved the existence, uniqueness and continuity of the solutions on the initial data, then the asymptotic compactness of the solutions and finally the existence of a global compact attractor. [ABSTRACT FROM AUTHOR]

This paper studies firstly the well-posedness and the asymptotic behavior of a Cahn–Hilliard–Oono type model, with cubic nonlinear terms, which is proposed for image segmentation. In particular, the existences of the global attractor and the exponential attractor have been proved, and it shows that the fractal dimension of the global attractor will tend to infinity as α → 0. The difficulty here is that we no longer have the conservation of mass. Furthermore, this model with logarithmic nonlinear terms has been studied as well. One difficulty here is to make sure that the logarithmic terms can pass to the limit under the standard Galerkin scheme. Another difficulty is to prove additional regularities on the solutions which is essential to prove a strict separation from the pure states 0 and 1 in one and two space dimensions. It eventually shows that the dimension of the global attractor is finite by proving the existence of the exponential attractor. [ABSTRACT FROM AUTHOR]

In this paper we investigate the long-term behavior of the solutions of the one-dimensional porous-elasticity problem with porous dissipation and nonlinear feedback force. We prove that the porous-elasticity problem converges to a quasi-static problem for the microvoids motion as a suitable parameter J tends to zero. Finite dimensional global attractor with additional regularity in J is obtained using the recent quasi-stability theory. Finally, we compare the porous-elasticity problem with quasi-static problem, in the sense of the upper-semicontinuity of their attractors as J → 0. [ABSTRACT FROM AUTHOR]

The aim of this study is to determine a 3D incompressible Brinkman–Forchheimer-extended Darcy fluid flow. Based on global well-posedness and regularity of solutions with a periodic boundary condition, the determining modes for weak and regular solutions is achieved via the generalized Grashof number for a 3D non-autonomous Brinkman–Forchheimer-Darcy fluid flow in porous medium. Furthermore, the asymptotic determination of the complete trajectories inside an attractor via Fourier functionals is shown for a 3D autonomous Brinkman–Forchheimer-extended Darcy model. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to investigate the modified Leray-alpha equation on the two-dimensional sphere S 2 , the square torus T 2 and the three-torus T 3 . In the strategy, we prove the existence and the uniqueness of the weak solutions and also the existence of the global attractor for the equation. Then we establish the upper and lower bounds of the Hausdorff and fractal dimensions of the global attractor on both S 2 and T 2 . Our method is based on the estimates for the vorticity scalar equations and the stationary solutions around the invariant manifold that are constructed by using the Kolmogorov flows. Finally, we will use the results on T 2 to study the lower bound for attractor's dimensions in the case of T 3 . [ABSTRACT FROM AUTHOR]

In this paper we study the longtime dynamics of a class of thermoelastic Timoshenko beams with history in a nonlinear elastic foundation. Our main result establishes the existence of a global attractor with finite fractal dimension without requiring the so-called equal wave speeds assumption. In addition, the attractor belongs to the phase space of strong solutions. The results are based on properties of gradient systems and a concept of quasi-stability. We believe this is the first study on the existence of global attractors for semilinear Timoshenko systems with hybrid dissipation (heat and memory). [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of a bounded weak solution for a triply nonlinear thermistor problem in Sobolev spaces. Furthermore, we prove the existence of an absorbing set and, consequently, the universal attractor. [ABSTRACT FROM AUTHOR]

We consider the periodic nonlinear Schrödinger equation with nonlinearity given by | u | p − 1 u for odd p > 1 in dimension 1. We first establish that the difference between the nonlinear evolution and a phase rotation of the linear evolution is in a smoother space. We then study forced and damped defocusing nonlinear Schrödinger equations of the above type and establish an analogous smoothing statement that extends globally in time. As a corollary we establish both existence and smoothness for global attractors in the energy space. [ABSTRACT FROM AUTHOR]

A Cauchy problem for a modification of the Swift-Hohenberg equation in RN with a mildly integrable potential is considered. Applying the dissipative mechanism of fourth order parabolic equations in unbounded domains, it is shown that the equation generates a semigroup of global solutions possessing a global attractor in the scale of Bessel potential spaces and in H²(RN) in particular. [ABSTRACT FROM AUTHOR]

We consider a diffuse interface model for an incompressible binary viscoelastic fluid flow. The model consists of the Navier–Stokes–Voigt equations where the instantaneous kinematic viscosity has been replaced by a memory term incorporating hereditary effects coupled with the Cahn–Hilliard equation with Flory–Huggins potential. The resulting system is subject to no‐slip condition for the (volume averaged) fluid velocity u$$ \mathbf{u} $$ and to no‐flux boundary conditions for the order parameter ϕ$$ \phi $$ and the chemical potential μ$$ \mu $$. We first establish the well‐posedness of the initial and boundary value problem. Then, taking advantage of an Ekman‐type damping, we obtain a dissipative estimate. Thus, we can define a dissipative dynamical system on a suitable phase space. Also, we show that any weak solution is such that ϕ$$ \phi $$ regularizes in finite time, and in dimension two, it stays uniformly away from pure phases; that is, the instantaneous strict separation property holds. Finally, we prove the existence of the global attractor, and exploiting the strict separation property, we demonstrate that an exponential attractor exists in dimension two. [ABSTRACT FROM AUTHOR]

In this paper we prove several turnpike results for trajectories of discrete disperse dynamical systems introduced in 1980 by A. M. Rubinov, which have a prototype in mathematical economics. [ABSTRACT FROM AUTHOR]

The well-posedness and global attraction are proved for nonautonomous 2D nonlinear damped Schrödinger equations with almost periodic external source in a bounded region. The proofs rely on a development of the method of energy equation. The research is inspired by the problems of laser and maser coherent radiation. [ABSTRACT FROM AUTHOR]