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In this paper, the asymptotic behavior of a semilinear heat equation with long time memory and non-local diffusion is analyzed in the usual set-up for dynamical systems generated by differential equations with delay terms. This approach is different from the previous published literature on the long time behavior of heat equations with memory which is carried out by the Dafermos transformation. As a consequence, the obtained results provide complete information about the attracting sets for the original problem, instead of the transformed one. In particular, the proved results also generalize and complete previous literature in the local case.

In this paper, we study the asymptotic behavior of the solutions of a nonautonomous differential inclusion modeling a reaction-diffusion equation with a discontinuous nonlinearity. We obtain first several properties concerning the uniqueness and regularity of non-negative solutions. Then we study the structure of the pullback attractor in the positive cone, showing that it consists of the zero solution, the unique positive nonautonomous equilibrium and the heteroclinic connections between them, which can be expressed in terms of the solutions of an associated linear problem. Finally, we analyze the relationship of the pullback attractor with the uniform, the cocycle and the skew product semiflow attractors.

In this work we consider a family of nonautonomous partial differential inclusions governed by \begin{document}$ p $\end{document} -laplacian operators with variable exponents and large diffusion and driven by a forcing nonlinear term of Heaviside type. We prove first that this problem generates a sequence of multivalued nonautonomous dynamical systems possessing a pullback attractor. The main result of this paper states that the solutions of the family of partial differential inclusions converge to the solutions of a limit ordinary differential inclusion for large diffusion and when the exponents go to \begin{document}$ 2 $\end{document} . After that we prove the upper semicontinuity of the pullback attractors.

We study properties of \begin{document}$ \omega $\end{document} -limit sets of multivalued semiflows like chain recurrence or the existence of cyclic chains. First, we prove that under certain conditions the \begin{document}$ \omega $\end{document} -limit set of a trajectory is chain recurrent, applying this result to an evolution differential inclusion with upper semicontinous right-hand side. Second, we give conditions ensuring that the \begin{document}$ \omega $\end{document} -limit set of a trajectory contains a cyclic chain. Using this result we are able to check that the \begin{document}$ \omega $\end{document} -limit set of every trajectory of a reaction-diffusion equation without uniqueness of solutions is an equilibrium.

In this article, we study the structure of the global attractor for a non-local one-dimensional quasilinear problem. The strong relation of our problem with a non-local version of the Chafee-Infante problem allows us to describe the structure of its attractor. For that, we made use of the Conley index and the connection matrix theories in order to find geometric information such as the existence of heteroclinic connections between the equilibria. In this way, the structure of the attractor is completely described.

In this paper we study the dynamical system generated by the solutions of optimal control problems. We obtain suitable conditions under which such systems generate multivalued semiprocesses. We prove the existence of uniform attractors for the multivalued semiprocess generated by the solutions of controlled reaction-diffusion equations and study its properties.

In this paper we obtain the existence of global attractors for the dynamicalsystems generated by weak solution of the three-dimensional Navier-Stokesequations with damping. We consider two cases, depending on the values of the parameter β controlling the damping term. First, we prove that for β≥4 weaksolutions are unique and establish the existence of the global attractor forthe corresponding semigroup. Second, for 3≤β

In this paper, we apply resampling techniques to a modified compartmental SEIR model which takes into account the existence of undetected infected people in an epidemic. In particular, we implement numerical simulations for the evolution of the first wave of the COVID-19 pandemic in Spain in 2020. We show, by using suitable measures of goodness, that the point estimates obtained by the bootstrap samples improve the ones of the original data. For example, the relative error of detected currently infected people is equal to 0.061 for the initial estimates, while it is reduced to 0.0538 for the mean over all bootstrap estimated series.

In this paper, it is first addressed the well-posedness of weak solutions to a nonlocal partial differential equation with long time memory, which is carried out by exploiting the nowadays well-known technique used by Dafermos in the early 70's. Thanks to this Dafermos transformation, the original problem with memory is transformed into a non-delay one for which the standard theory of autonomous dynamical system can be applied. Thus, some results about the existence of global attractors for the transformed problem are {proved}. Particularly, when the initial values have higher regularity, the solutions of both problems (the original and the transformed ones) are equivalent. Nevertheless, the equivalence of global attractors for both problems is still unsolved due to the lack of enough regularity of solutions in the transformed problem. It is therefore proved the existence of global attractors of the transformed problem. Eventually, it is highlighted how to proceed to obtain meaningful results about the original problem, without performing any transformation, but working directly with the original delay problem.

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.

The long-time behavior of solutions (more precisely, the existence of random pullback attractors) for an integro-differential parabolic equation of diffusion type with memory terms, more particularly with terms containing both finite and infinite delays, as well as some kind of randomness, is analyzed in this paper. We impose general assumptions not ensuring uniqueness of solutions, which implies that the theory of multivalued dynamical system has to be used. Furthermore, the emphasis is put on the existence of random pullback attractors by exploiting the techniques of the theory of multivalued nonautonomous/random dynamical systems.

In this paper we study the robustness of dynamically gradient multivalued semiflows. As an application, we describe the dynamical properties of a family of Chafee-Infante problems approximating a differential inclusion studied in J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, International Journal of Bifurcation and Chaos, 16 (2006), 2965-2984, proving that the weak solutions of these problems generate a dynamically gradient multivalued semiflow with respect to suitable Morse sets. Ministerio de Educación, Cultura y Deporte Ministerio de Economía y Competitividad Junta de Andalucía Fundação de Amparo à Pesquisa do Estado de São Paulo Conselho Nacional de Desenvolvimento Científico e Tecnológico

In this paper, we consider a system of ordinary differential equations with non-local discrete diffusion and finite delay and with either a finite or an infinite number of equations. We prove several properties of solutions such as comparison, stability and symmetry. We create a numerical simulation showing that this model can be appropriate to model dynamical life tables in actuarial or demographic sciences. In this way, some indicators of goodness and smoothness are improved when comparing with classical techniques.

In this paper we study cocycle attractors, pullback attractors and uniform attractors for multi-valued non-autonomous dynamical systems. We first consider the relationship between the three attractors and find that, under suitable conditions, they imply each other. Then, for generalized dynamical systems, we find that these attractors can be characterized by complete trajectories, which implies that the uniform attractor is lifted invariant, though it has no standard invariance by definition. Finally, we study both upper and lower semi-continuity of these attractors. A weak equi-attraction method is introduced to study the lower semi-continuity, and we show with an example the advantages of this method. A reaction-diffusion system and a scalar ordinary differential inclusion are studied as applications.

In this work we consider a differential inclusion governed by a p-Laplacian operator with a diffusion coefficient depending on a parameter in which the space variable belongs to an unbounded domain. We prove the existence of a global attractor and show that the family of attractors behaves upper semicontinuously with respect to the diffusion parameter. Both autonomous and nonautonomous cases are studied.

In this paper we prove that the global attractor generated by strong solutions of a reaction-diffusion equation without uniqueness of the Cauchy problem is bounded in suitable Lr -spaces. In order to obtain this result we prove first that the concepts of weak and mild solutions are equivalent under an appropriate assumption. Also, when the nonlinear term of the equation satisfies a supercritical growth condition the existence of a weak attractor is established.

In this paper we study the theory of Morse decompositions with an infinite number of components in the multivalued framework, proving that for a disjoint infinite family of weakly invariant sets (being all isolated but one) a Lyapunov function ordering them exists if and only if the multivalued semiflow is dynamically gradient. Moreover, these properties are equivalent to the existence of a Morse decomposition. This theorem is applied to a reaction-diffusion inclusion with an infinite number of equilibria.

In this paper we investigate the long term behavior of a stochastic lattice dynamical system with a diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and a different multiplicative white noise at each node. We prove that this stochastic lattice equation generates a random dynamical system that possesses a global random attractor. In particular, we first establish an existence theorem for weak solutions to general random evolution equations, which is later applied to the specific stochastic lattice system to show that it has weak solutions and the solutions generate a random dynamical system. We then prove the existence of a random attractor of the underlying random dynamical system by constructing a random compact absorbing set and using an embedding theorem. The major novelty of this work is that we consider a different multiplicative white noise term at each different node, which significantly improves the previous results in the literature where the same multiplicative noise was considered at all the nodes. As a consequence, the techniques used in the existing literature are not applicable here and a new methodology has to be developed to study such systems.

The structure of attractors for differential equations is one of the main topics in the qualitative theory of dynamical systems. However, the theory is still in its infancy in the case of multivalued dynamical systems. In this paper we study in detail the structure and internal dynamics of a scalar differential equation, both in the autonomous and non-autonomous cases. To this aim, we will also show a general result on the characterization of a pullback attractor for a multivalued process by the union of all the complete bounded trajectories of the system.

Adaptive management of traffic congestion in the Internet is a complex problem that can gain useful insights from a dynamical approach. In this paper we propose and analyze a one-dimensional, discrete-time nonlinear model for Internet congestion control at the routers. Specifically, the states correspond to the average queue sizes of the incoming data packets and the dynamical core consists of a monotone or unimodal mapping with a unique fixed point. This model generalizes a previous one in that additional control parameters are introduced via the data packet drop probability with the objective of enhancing stability. To make the analysis more challenging, the original model was shown to exhibit the usual features of low-dimensional chaos with respect to several system and control parameters, e.g., positive Lyapunov exponents and Feigenbaum-like bifurcation diagrams. We concentrate first on the theoretical aspects that may promote the unique stationary state of the system to a global attractor, which in our case amounts to global stability. In a second step, those theoretical results are translated into stability domains for robust setting of the new control parameters in practical applications. Numerical simulations confirm that the new parameters make it possible to extend the stability domains, in comparison with previous results. Therefore, the present work may lead to an adaptive congestion control algorithm with a more stable performance than other algorithms currently in use.

The purpose of this paper is to give a characterization of the structure of non-autonomous attractors of the problem $u_t= u_{xx} + \lambda u - \beta(t)u^3$ when the parameter $\lambda > 0$ varies. Also, we answer a question proposed in [11], concerning the complete description of the structure of the pullback attractor of the problem when $1, Comment: 32 pages, 04 figures

In this paper, we study the dynamical properties inside the global attractor for multivalued semiflows. Given a disjoint finite family of isolated weakly invariant sets, we prove, extending a previous result from the single-valued case, that the existence of a Lyapunov function, the property of being a dynamically gradient semiflow and the existence of a Morse decomposition are equivalent properties. We apply this abstract theorem to a reaction–diffusion inclusion.

In this paper we describe some dynamical properties of a Morse decomposition with a countable number of sets. In particular, we are able to prove that the gradient dynamics on Morse sets together with a separation assumption is equivalent to the existence of an ordered Lyapunov function associated to the Morse sets and also to the existence of a Morse decomposition -that is, the global attractor can be described as an increasing family of local attractors and their associated repellers.

In this paper we study the asymptotic behaviour of solutions of a lattice dynamical system of a logistic type. Namely, we study a system of infinite ordinary differential equations which can be obtained after the spatial discretization of a logistic equation with diffusion. We prove that a global attractor exists in suitable weighted spaces of sequences.

In this paper we study the structure of the global attractor for a reaction-diffusion equation in which uniqueness of the Cauchy problem is not guarantied. We prove that the global attractor can be characterized using either the unstable manifold of the set of stationary points or the stable one but considering in this last case only solutions in the set of bounded complete trajectories.

In this paper we study the structural properties of global attractors of multi-valued semiflows generated by regular solutions of reaction-diffusion system without uniqueness of the Cauchy problem. Under additional gradient-like condition on the nonlinear term we prove that the global attractor coincides with the unstable manifold of the set of stationary points, and with the stable one when we consider only bounded complete trajectories. As an example we consider a generalized Fitz-Hugh-Nagumo system. We also suggest additional conditions, which provide that the global attractor is a bounded set in $(L^\infty(\Omega))^N$ and compact in $(H_0^1 (\Omega))^N$.

In this paper we first prove a rather general theorem about existence of solutions for an abstract differential equation in a Banach space by assuming that the nonlinear term is in some sense weakly continuous. &nbsp We then apply this result to a lattice dynamical system with delay, proving also the existence of a global compact attractor for such system.

We study a one-dimensional ordinary differential equation modelling optical conveyor belts, showing in particular cases of physical interest that periodic solutions exist. Moreover, under rather general assumptions it is proved that the set of periodic solutions is bounded.

In this paper we study a system of ordinary differential equations with non-local discrete diffusion, which can be considered as a spatial discretization of a non-local reaction-diffusion equation. We prove several properties of solutions concerning comparison, stability, symmetry or the conservation of mass. We propose that this model can be appropriate to modeling dynamical life tables in actuarial or demographic sciences. We show that it allows to improve some indicators of goodness and smoothness when comparing with classical techniques.

We prove the existence and connectedness of a global attractor for the multivalued semiflow generated by the weak solutions of the three-dimensional Benard system that satisfy a system of energy inequalities.

In this paper we study the asymptotic behavior of solutions of a first-order stochastic lattice dynamical system with a multiplicative noise.We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true.Using the theory of multi-valued random dynamical systems we prove the existence of a random compact global attractor.

We prove the Kneser property (i.e. the connectedness and compactness of the attainability set at any time) for lattice dynamical systems in which we do not know whether the property of uniqueness of the Cauchy problem holds or not. Using this property, we can check that the global attractor of the multivalued semiflow generated by such system is connected.

In this paper, we prove the Kneser property for a reaction–diffusion equation on an unbounded domain satisfying the Poincare inequality with an external force taking values in the space H − 1 . Using this property of solutions we check also the connectedness of the associated global pullback attractor. We study also similar properties for systems of reaction–diffusion equations in which the domain is the whole R N . Finally, the results are applied to a generalized logistic equation.

In this paper we prove the existence of solutions for the 3D Benard system in the class of functions which are strongly continuous with respect to the second component of the vector (that is, the one corresponding to the parabolic equation). We construct then a multivalued semiflow generated by such solutions and obtain the existence of a global φ −attractor for the weak-strong topology. Moreover, a family of multivalued semiflows is defined on suitable convex bounded subsets of the phase space, proving for them the existence of a global attractor (which is the same for every semiflow of the family) for the weak-strong topology.

We prove the existence of solutions for a Navier–Stokes model in two dimensions with an external force containing infinite delay effects in the weighted space C γ ( H ) . Then, under additional suitable assumptions, we prove the existence and uniqueness of a stationary solution and the exponential decay of the solutions of the evolutionary problem to this stationary solution. Finally, we study the existence of pullback attractors for the dynamical system associated to the problem under more general assumptions.

In this article, we study the asymptotic behaviour of solutions of a first-order stochastic lattice dynamical system with an additive noise. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions so that uniqueness of the Cauchy problem fails to be true. Using the theory of multi-valued random dynamical systems, we prove the existence of a random compact global attractor.

In this paper we construct a dynamical process (in general, multivalued) generated by the set of solutions of an optimal control problem for the three-dimensional Navier–Stokes system. We prove the existence of a pullback attractor for such multivalued process. Also, we establish the existence of a uniform global attractor containing the pullback attractor. Moreover, under the unproved assumption that strong globally defined solutions of the three-dimensional Navier–Stokes system exist, which guaranties the existence of a global attractor for the corresponding multivalued semiflow, we show that the pullback attractor of the process coincides with the global attractor of the semiflow.

In this paper we prove the existence of uniform global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of non-autonomous complex fluids.

In this paper we consider a lattice dynamical system generated by a parabolic equation modeling suspension flows. We prove the existence of a global compact connected attractor for this system and the upper semicontinuity of this attractor with respect to finite-dimensional approximations. Also, we obtain a sequence of approximating discrete dynamical systems by the implementation of the implicit Euler method, proving the existence and the upper semicontinuous convergence of their global attractors.

For a large class of noncoercive operator inclusions, including those generated by maps of Sk type, we obtain a general theorem on the existence of solutions. We apply this result to a particular example. This theorem is proved using the method of Faedo–Galerkin approximations.

In this paper we make a thorough comparison between the theory of global attractors for multivalued semiflows and the theory of trajectory attractors, two methods which are useful for studying the asymptotic behavior of solution for equations without uniqueness of the Cauchy problem. We show that under some conditions the formula A = U(0) takes place for the global attractor A of a multivalued semiflow and the trajectory attractor U of the associated translation semigroup. We apply these results to reaction–diffusion equations and hyperbolic equations, obtaining also new theorems concerning the existence of related trajectory attractors.

The long-time behavior of an integro-differential parabolic equation of diffusion type with memory terms, expressed by convolution integrals involving infinite delays and by a forcing term with bounded delay, is investigated in this paper. The assumptions imposed on the coefficients are weak in the sense that uniqueness of solutions of the corresponding initial value problems cannot be guaranteed. Then, it is proved that the model generates a multivalued non-autonomous dynamical system which possesses a pullback attractor. First, the analysis is carried out with an abstract parabolic equation. Then, the theory is applied to the particular integro-differential equation which is the objective of this paper. The general results obtained in the paper are also valid for other types of parabolic equations with memory.

The Kneser theorem for ordinary differential equations without uniqueness says that the attainability set is compact and connected at each instant of time. We establish corresponding results for the attainability set of weak solutions for the 3D Navier-Stokes equations satisfying an energy inequality. First, we present a simplified proof of our earlier result with respect to the weak topology in the space $H$. Then we prove that this result also holds with respect to the strong topology on $H$ provided that the weak solutions satisfying the weak version of the energy inequality are continuous. Finally, using these results, we show the connectedness of the global attractor of a family of setvalued semiflows generated by the weak solutions of the NSE satisfying suitable properties.

In this work we present the existence and uniqueness of pullback and random attractors for stochastic evolution equations with infinite delays when the uniqueness of solutions for these equations is not required. Our results are obtained by means of the theory of set-valued random dynamical systems and their conjugation properties.

The existence of a pullback attractor for a reaction-diffusion equations in an unbounded domain containing a non-autonomous forcing term taking values in the space $H^{-1}$, and with a continuous nonlinearity which does not ensure uniqueness of solutions, is proved in this paper. The theory of set-valued non-autonomous dynamical systems is applied to the problem.

We study the Kneser property (i.e. the compactness and connectedness for the attainability set of solutions) for a reaction–diffusion system including as a particular case the complex Ginzburg–Landau equation and the Lotka–Volterra system with diffusion. Using this property we obtain also that the global attractor of this system in both the autonomous and non-autonomous cases is connected.

We prove the existence of a random global attractor for the multivalued random dynamical system associated to a nonlinear multivalued parabolic equation with a stochastic term of amplitude of the order of e . The equation was initially suggested by North and Cahalan (following a previous deterministic model proposed by M.I. Budyko), for the modeling of some non-deterministic variability (as, for instance, the cyclones which can be treated as a fast varying component and are represented as a white-noise process) in the context of energy balance climate models. We also prove the convergence (in some sense) of the global attractors, when e → 0 , i.e., the convergence to the global attractor for the associated deterministic case ( e = 0 ).

In this paper, we prove the existence of solutions for first order lattice dynamical systems with continuous nonlinear term obtained via discretization of a reaction–diffusion system. Since the uniqueness of the Cauchy problem is not guaranteed, we define a multivalued semiflow and prove the existence of a global compact attractor.

In this paper we prove the existence of a global attractor with respect to the weak topology of a suitable Banach space for a parabolic scalar differential equation describing a non-Newtonian flow. More precisely, we study a model proposed by Hebraud and Lequeux for concentrated suspensions.