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In this survey we provide an introduction to the theory of impulsive dynamical systems in both the autonomous and nonautonomous cases. In the former, we will show two different approaches which have been proposed to analyze such kind of dynamical systems which can experience some abrupt changes in their evolution (impulses). But, unlike the autonomous framework, the nonautonomous one is being developed right now and some progress is being obtained over the recent years. We will provide some results on how the theory of autonomous impulsive dynamical systems can be extended to cover such nonautonomous situations, which are more often to occur in the real world.

We investigate a non-isothermal viscous relaxation of some nonlocal Cahn-Hilliard equations. This perturbation problem generates a family of solution operators exhibiting dissipation and conservation. The solution operators admit a family of compact global attractors that are bounded in a more regular phase-space

We consider semilinear parabolic equations involving an operator that is X-elliptic with respect to a family of vector fields X with suitable properties. The vector fields determine the natural functional setting associated to the problem and the admissible growth of the non-linearity. We prove the global existence of solutions and characterize their longtime behavior. In particular, we show the existence and finite fractal dimension of the global attractor of the generated semigroup and the convergence of solutions to an equilibrium solution when time tends to infinity.

We consider the damped and driven Navier–Stokes system with stress free boundary conditions and the damped Euler system in a bounded domain Ω ⊂ R 2 . We show that the damped Euler system has a (strong) global attractor in H 1 ( Ω ) . We also show that in the vanishing viscosity limit the global attractors of the Navier–Stokes system converge in the non-symmetric Hausdorff distance in H 1 ( Ω ) to the strong global attractor of the limiting damped Euler system (whose solutions are not necessarily unique). [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of complex fluids. We also show that the attractors tend to the set of all complete bounded trajectories of the original problem when the parameter of the approximations goes to zero. [ABSTRACT FROM AUTHOR]

This paper is concerned with the nonlinear viscoelastic equation | ∂ t u | ρ ∂ t t u − Δ ∂ t t u − Δ u + ∫ 0 ∞ μ ( s ) Δ u ( t − s ) d s + f ( u ) = h , suitable to modeling extensional vibrations of thin rods with nonlinear material density ϱ ( ∂ t u ) = | ∂ t u | ρ , and presence of memory effects. This class of equations was studied by many authors, but well-posedness in the whole admissible range ρ ∈ [ 0 , 4 ] and for f growing up to the critical exponent were established only recently. The existence of global attractors was proved in presence of an additional viscous or frictional damping. In the present work we show that the sole weak dissipation given by the memory term is enough to ensure existence and optimal regularity of the global attractor A ρ for ρ < 4 and critical nonlinearity f . [ABSTRACT FROM AUTHOR]

We study the long-time behavior of porous-elastic system, focusing on the interplay between nonlinear damping and source terms. The sources may represent restoring forces, but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. Under some restrictions on the parameters, we also prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin. We also prove the existence of a global attractor. [ABSTRACT FROM AUTHOR]

We consider the subcritical SQG equation in its natural scale-invariant Sobolev space and prove the existence of a global attractor of optimal regularity. The proof is based on a new energy estimate in Sobolev spaces to bootstrap the regularity to the optimal level, derived by means of nonlinear lower bounds on the fractional Laplacian. This estimate appears to be new in the literature and allows a sharp use of the subcritical nature of the $$L^\infty $$ bounds for this problem. As a by-product, we obtain attractors for weak solutions as well. Moreover, we study the critical limit of the attractors and prove their stability and upper semicontinuity with respect to the strength of the diffusion. [ABSTRACT FROM AUTHOR]

This paper studies the existence and regularity properties of solutions to the Zakharov and Klein--Gordon--Schrödinger systems at low regularity levels. The main result is that the nonlinear part of the solution flow falls in a smoother space than the initial data. This relies on a new bilinear Xs,b estimate, which is proved using delicate dyadic and angu lar decompositions of the frequency domain. Such smoothing estimates have a number of implications for the long-term dynamics of the system. In this work, we give a simplified proof of the existence of global attractors for the Klein-- Gordon--Schrödinger flow in the energy space for dimensions d = 2, 3. Second, we use smoothing in conjunction with a high-low decomposition to show global well-posedness of the Klein--Gordon-- Schroödinger evolution on R4 below the energy space for sufficiently small initial data. [ABSTRACT FROM AUTHOR]

Informational Structures (IS) and Informational Fields (IF) have been recently introduced to deal with a continuous dynamical systems-based approach to Integrated Information Theory (IIT). IS and IF contain all the geometrical and topological constraints in the phase space. This allows one to characterize all the past and future dynamical scenarios for a system in any particular state. In this paper, we develop further steps in this direction, describing a proper continuous framework for an abstract formulation, which could serve as a prototype of the IIT postulates.

This paper investigates the global dynamics of a mean field model of the electroencephalogram developed by Liley, Cadusch, and Dafilis [Network, 13 (2002), pp. 67-113]. The model is presented as a system of coupled ordinary and partial differential equations with periodic boundary conditions. Existence, uniqueness, and regularity of weak and strong solutions of the model are established in appropriate function spaces, and the associated initial-boundary value problems are proved to be well- posed. Sufficient conditions are developed for the phase spaces of the model to ensure nonnegativity of certain quantities in the model, as required by their biophysical interpretation. It is shown that the semigroups of weak and strong solution operators possess bounded absorbing sets for the entire range of biophysical values of the parameters of the model. Challenges involved in establishing a global attractor for the model are discussed and it is shown that there exist parameter values for which the constructed semidynamical systems do not possess a compact global attractor due to the lack of the compactness property. Finally, using the theoretical results of the paper, instructive insights are provided into the complexity of the behavior of the model and computational analysis of the model. [ABSTRACT FROM AUTHOR]

In this work we obtain some continuity properties on the parameter q at p = q for the Takeuchi-Yamada problem which is a degenerate p-laplacian version of the Chafee-Infante problem. We prove the continuity of the ows and the equilibrium sets, and the upper semicontinuity of the global attractors. [ABSTRACT FROM AUTHOR]

We consider the evolution of weak vanishing viscosity solutions to the critically dissipative surface quasi-geostrophic equation. Due to the possible non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical system and prove the existence of a global attractor of optimal Sobolev regularity. To achieve this, we derive a new Sobolev estimate involving Holder norms, which complement the existing estimates based on commutator analysis. [ABSTRACT FROM AUTHOR]

This paper investigates the periodic initial value problem for the two-dimensional Burgers-Ginzburg-Landau (2D Burgers-GL) equations, which can be derived from the so-called modulated modulation equations (MME) that govern the dynamics of the modulated amplitudes of some periodic critical modes. The well-posedness of the solutions and the global attractors for the 2D Burgers-GL equations are obtained via delicate a priori estimates, the Galerkin method, and operator semigroup method. [ABSTRACT FROM AUTHOR]

Recently, in Bonfoh and Enyi [Commun. Pure Appl. Anal. 15 2016, 1077-1105], we considered the conserved phasefield system {τφt - Δ(δφt - Δφ + g(φ) - u) = 0, εut + φt - Δu = 0, in a bounded domain of Rd , d = 1, 2, 3, where τ > 0 is a relaxation time, δ > 0 is the viscosity parameter, ε ∈ (0, 1] is the heat capacity, φ is the order parameter, u is the absolute temperature and g : R → R is a nonlinear function. The system is subject to the boundary conditions of either periodic or Neumann type. We proved a well-posedness result, the existence and continuity of the global and exponential attractors at ε = 0. Then, we proved the existence of inertial manifolds in one space dimension, and in the case of two space dimensions in rectangular domains. Stability properties of the intersection of inertial manifolds with a bounded absorbing set were also proven. In the present paper, we show the above-mentioned existence and continuity properties at (ε, δ) = (0, 0). To establish the existence of inertial manifolds of dimension independent of the two parameters δ and ε, we require ε to be dominated from above by δ. This work shows the convergence of the dynamics of the above mentioned problem to the one of the Cahn-Hilliard equation, improving and extending some previous results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fractional Ginzburg-Landau equations near the Bardeen-Cooper-Schrieffer-Bose-Einstein-condensate (BCS-BEC) crossover of atomic Fermi gases. This fractional Ginzburg-Landau equations can be viewed as a generalization of the integral differential equations proposed by Machida and Koyama (Phys. Rev. A 74:033603, 2006). By using the Galerkin method and a priori estimates, together with the properties of Sobolev spaces, we first establish the existence and uniqueness of weak solutions to these equations and then we prove the existence of global attractors. [ABSTRACT FROM AUTHOR]

A global attractor in L2 is shown for weakly dissipative p-Laplace equations on the entire Euclid space, where the weak dissipativeness means that the order of the source is lesser than p - 1. Half-time decomposition and induction techniques are utilized to present the tail estimate outside a ball. It is also proved that the equations in both strongly and weakly dissipative cases possess an (L2;Lr)-attractor for r belonging to a special interval, which contains the critical exponent p. The obtained attractor is proved to be ap- proximated by the corresponding attractor inside a ball in the sense of upper strictly and lower semicontinuity. [ABSTRACT FROM AUTHOR]

This paper is concerned with a thermoelastic suspension bridge equations with memory effects. For the suspension bridge equations without memory, there are many classical results. However, the suspension bridge equations with both viscoelastic and thermal memories were not studied before. The object of the present paper is to provide a result on the global attractor to a thermoelastic suspension bridge equation with past history. [ABSTRACT FROM AUTHOR]

The properties of three dimensional space solution of strongly damped nonlinear wave equation by 3D was expanded to N dimensional space ( N > 3 ) . A standard Galerkin method and the Sobolev embedding theorem were utilized to study the existence of weak solution under the space. The inner product was used to make the solution' s dissipation estimates, and the Gronwall lemma was used to prove the existence of the attractor. [ABSTRACT FROM AUTHOR]

In a previous article [7], we proposed a model of phase separation in a binary mixture confined to a bounded region which may be contained within porous walls. The boundary conditions were derived from a mass conservation law and variational methods. In the present paper, we study the problem further. Using a Faedo-Galerkin method, we obtain the existence and uniqueness of a global solution to our problem, under more general assumptions than those in [7]. We then study its asymptotic behavior and prove the existence of an exponential attractor (and thus of a global attractor) with finite dimension.

In this paper we introduce the concept of collective tube conditions which assures a suitable behaviour for a family of dynamical systems close to impulsive sets. Using the collective tube conditions, we develop the theory of upper and lower semicontinuity of global attractors for a family of impulsive dynamical systems. [ABSTRACT FROM AUTHOR]

The dynamics of a (nonlinear) Berger plate in the absence of rotational inertia are considered with inhomogeneous boundary conditions. In our analysis, we consider boundary damping in two scenarios: (i) free plate boundary conditions, or (ii) hinged-type boundary conditions. In either situation, the nonlinearity gives rise to complicating boundary terms. In the case of free boundary conditions we show that well-posedness of finite-energy solutions can be obtained via highly nonlinear boundary dissipation. Additionally, we show the existence of a compact global attractor for the dynamics in the presence of hinged-type boundary dissipation (assuming a geometric condition on the entire boundary (Lagnese, 1989)). To obtain the existence of the attractor we explicitly construct the absorbing set for the dynamics by employing energy methods that: (i) exploit the structure of the Berger nonlinearity, and (ii) utilize sharp trace results for the Euler–Bernoulli plate in Lasiecka and Triggiani (1993). We provide a parallel commentary (from a mathematical point of view) to the discussion of modeling with Berger versus von Karman nonlinearities: to wit, we describe the derivation of each nonlinear dynamics and a discussion of the validity of the Berger approximation. We believe this discussion to be of broad value across engineering and applied mathematics communities. [ABSTRACT FROM AUTHOR]

We consider a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. Several results were already proven by two of the present authors. However, in the two-dimensional case, the uniqueness of weak solutions was still open. Here we establish such a result even in the case of degenerate mobility and singular potential. Moreover, we show the weak-strong uniqueness in the case of viscosity depending on the order parameter, provided that either the mobility is constant and the potential is regular or the mobility is degenerate and the potential is singular. In the case of constant viscosity, on account of the uniqueness results, we can deduce the connectedness of the global attractor whose existence was obtained in a previous paper. The uniqueness technique can be adapted to show the validity of a smoothing property for the difference of two trajectories which is crucial to establish the existence of an exponential attractor. The latter is established even in the case of variable viscosity, constant mobility and regular potential. [ABSTRACT FROM AUTHOR]

We investigate the large time behavior of a conserved phase-field system that describes the phase separation in a material with viscosity effects. We prove a well-posedness result, the existence of the global attractor and its upper semicontinuity, when the heat capacity tends to zero. Then we prove the existence of inertial manifolds in one space dimension, and for the case of a rectangular domain in two space dimension. We also construct robust families of exponential attractors that converge in the sense of upper and lower semicontinuity to those of the viscous Cahn-Hilliard equation. Continuity properties of the intersection of the inertial manifolds with bounded absorbing sets are also proven. This work extends and improves some recent results proven by A. Bonfoh for both the conserved and non-conserved phase-field systems. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the asymptotic behavior of quasilinear parabolic equations posed in a family of unbounded domains that degenerates onto a lower dimensional set. Considering an auxiliary family of weighted Sobolev spaces we show the existence of global attractors and we analyze convergence properties of the solutions as well of the attractors. [ABSTRACT FROM AUTHOR]

We give a survey of recent results on flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are considered. The focus of the discussion here is on the interesting mathematical aspects of physical phenomena occurring in aeroelasticity, such as flutter and divergence. This leads to a partial differential equation treatment of issues such as well-posedness of finite energy solutions, and long-time (asymptotic) behavior. The latter includes theory of asymptotic stability, convergence to equilibria, and to global attracting sets. We complete the discussion with several well known observations and conjectures based on experimental/numerical studies. [ABSTRACT FROM AUTHOR]

I. Stratis In this work, we investigate the analyticity properties of solutions of Kuramoto-Sivashinsky-type equations in two spatial dimensions, with periodic initial data. In order to do this, we explore the applicability in three-dimensional models of a spectral method, which was developed by the authors for the one-dimensional Kuramoto-Sivashinsky equation. We introduce a criterion, which provides a sufficient condition for analyticity of a periodic function u∈ C ∞, involving the rate of growth of ∇ n u, in suitable norms, as n tends to infinity. This criterion allows us to establish spatial analyticity for the solutions of a variety of systems, including Topper-Kawahara, Frenkel-Indireshkumar, and Coward-Hall equations and their dispersively modified versions, once we assume that these systems possess global attractors. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider scalar reaction-diffusion equations with non-dissipative nonlinearities generating global semiflows which exhibit blow-up in infinite time. This type of equations was only recently approached and the corresponding dynamical systems are known as slowly non-dissipative systems. The existence of unbounded solutions, referred to as grow-up solutions, requires the introduction of some objects interpreted as equilibria at infinity. By extending known results, we are able to obtain a complete decomposition of the associated non-compact global attractor. The connecting orbit structure is determined based on the Sturm permutation method, which yields a simple criterion for the existence of heteroclinic connections. [ABSTRACT FROM AUTHOR]

Abstract. A variety of models describing the interaction between flows and oscillating structures are discussed. The main aim is to analyze conditions under which structural instability (flutter) induced by a fluid flow can be suppressed or eliminated. The analysis provided focuses on effects brought about by: (i) different plate and fluid boundary conditions, (ii) various regimes for flow velocities: subsonic, transonic, or supersonic, (iii) different modeling of the structure which may or may not account for in-plane accelerations (full von Karman system), (iv) viscous effects, (v) an assortment of models related to piston-theoretic model reductions, and (iv) considerations of axial flows (in contrast to so called normal flows). The discussion below is based on conclusions reached via a combination of rigorous PDE analysis, numerical computations, and experimental trials. [ABSTRACT FROM AUTHOR]

Recently, in Bonfoh [Ann. Mat. Pura Appl. 2011;190:105–144], we investigated the dynamics of a nonconserved phase-field system whose singular limit is the viscous Cahn–Hilliard equation. More precisely, we proved the existence of the global attractor, exponential attractors, and inertial manifolds and we showed their continuity with respect to a singular perturbation parameter. In the present paper, we extend most of these results to a conserved phase-field system whose singular limit is the nonviscous Cahn–Hilliard equation. These equations describe phase transition processes. Here, we give a direct proof of the existence of inertial manifolds that differs from our previous method that was based on introducing a change of variables and an auxiliary problem. [ABSTRACT FROM AUTHOR]

In recent years a class of vibrating plates with nonlinear strain of p -Laplacian type was studied by several authors. The present paper contains a first thermoelastic model of that class of problems including both Fourier and non-Fourier heat laws. Our main result establishes the existence of global and exponential attractors for the strongly damped problem through a stabilizability inequality. In addition, for the weakly damped problem, we establish the exponential stability of its Galerkin semiflows. [ABSTRACT FROM AUTHOR]

In this paper we give two existence results for a class of degenerate diffusion equations with p-Laplacian. One is on a unique global strong solution, and the other is on a global attractor. It is also shown that the global attractor coincides with the unstable set of the set of all stationary solutions. As a by-product, an a-priori estimate for solutions of the corresponding elliptic equations is obtained.

It is shown that non-autonomous quasi-homogeneous dynamical systems admit a compact global attractor. The general results obtained here are applied to differential equations both in finite dimensional spaces and in infinite dimensional spaces, such as ordinary differential equations in Banach space and some types of evolutional partial differential equations.

This article is devoted to the study of linear periodic dynamical systems, possessing the property of uniform exponential stability. It is proved that if the Cauchy operator of these systems possesses a certain compactness property, then the asymptotic stability implies the uniform exponential stability. We also show applications to different classes of linear evolution equations, such as ordinary linear differential equations in the space of Banach, retarded and neutral functional differential equations, some classes of evolution partial differential equations.

This article is devoted to the study linear non-autonomous dynamical systems possessing the property of uniform exponential stability. We prove that if the Cauchy operator of these systems possesses a certain compactness property, then the uniform asymptotic stability implies the uniform exponential stability. For recurrent (almost periodic) systems this result is precised. We also show application for different classes of linear evolution equations: ordinary linear differential equations in a Banach space, retarded and neutral functional differential equations, and some classes of evolution partial differential equations.

We consider the memory relaxation of an Euler-Bernoulli plate equation with nonlinear source term and internal frictional damping of arbitrary polynomial growth. The main focus is the existence of a smooth global attractor for the associated dynamical system. [ABSTRACT FROM AUTHOR]

We consider a Cahn-Hilliard equation which is the conserved gradient flow of a nonlocal total free energy functional. This functional is characterized by a Helmholtz free energy density, which can be of logarithmic type. Moreover, the spatial interactions between the different phases are modeled by a singular kernel. As a consequence, the chemical potential $$\mu $$ contains an integral operator acting on the concentration difference $$c$$ , instead of the usual Laplace operator. We analyze the equation on a bounded domain subject to no-flux boundary condition for $$\mu $$ and by assuming constant mobility. We first establish the existence and uniqueness of a weak solution and some regularity properties. These results allow us to define a dissipative dynamical system on a suitable phase-space, and we prove that such a system has a (connected) global attractor. Finally, we show that a Neumann-like boundary condition can be recovered for $$c$$ , provided that it is supposed to be regular enough. [ABSTRACT FROM AUTHOR]

We introduce and analyze the nonlocal variants of two Cahn- Hilliard type equations with reaction terms. The first one is the so-called Cahn-Hilliard-Oono equation which models, for instance, pattern formation in diblock-copolymers as well as in binary alloys with induced reaction and type-I superconductors. The second one is the Cahn-Hilliard type equation introduced by Bertozzi et al. to describe image inpainting. Here we take a free energy functional which accounts for nonlocal interactions. Our choice is motivated by the work of Giacomin and Lebowitz who showed that the rigorous physical derivation of the Cahn-Hilliard equation leads to consider nonlocal functionals. The equations also have a transport term with a given velocity field and are subject to a homogenous Neumann boundary condition for the chemical potential, i.e., the first variation of the free energy functional. We first establish the well-posedness of the corresponding initial and boundary value problems in a weak setting. Then we consider such problems as dynamical systems and we show that they have bounded absorbing sets and global attractors. [ABSTRACT FROM AUTHOR]

In this study, we consider the nonlinear evolution equation of parabolic type We analyze the long time dynamics (in the sense of global attractors) under very general conditions on the nonlinearity $$f$$ . Since we do not assume any polynomial growth condition on it, the main difficulty arises at first in the proof of well-posedness. In fact, the very first contribution to this problem is a pioneering paper (Efendiev and Ôtano, Differ Int Equ 20:1201-1209, ) where the well-posedness result has been shown by exploiting the technique from the theory of maximal monotone operators. However, from some physical aspects, to obtain the solution in variational sense might be demanding which requires limiting procedure on the approximate solutions. In this work, we are interested in variational (weak) solution. The critical issue in the proof of well-posedness is to deal with the limiting procedure on $$f$$ which is overcome utilizing the weak convergence tecniques in Orlicz spaces (Geredeli and Khanmamedov, Commun Pure Appl Anal 12:735-754, ; Krasnosel'skiĭ and Rutickiĭ, Convex functions and Orlicz spaces, ). Then, proving the existence of the global attractors in $$L^2(\Omega )$$ and in more regular space $$W_0^{1,p}(\Omega )$$ , we show that they coincide. In addition, if $$f$$ is monotone and $$g=0$$ , we give an explicit estimate of the decay rate to zero of the solution. [ABSTRACT FROM AUTHOR]