Multivalued Processes Research Articles

Dynamics of Nonautomous Impulsive Multivalued Processes

In this paper we study the asymptotic behaviour of multivalued processes which are under the influence of impulsive action. We provide conditions to guarantee the existence of a pullback attractor and we illustrate the results with several examples.

Solvability and pullback attractor for a class of differential hemivariational inequalities with its applications

In this paper, we consider an abstract system which consists of a nonlinear differential inclusion and a parabolic hemivariational inequality (DPHVI) in Banach spaces. The objective of this paper is four fold. The first target is to deal with the existence of solutions and the properties which involve the boundedness and continuous dependence results of the solution set to parabolic hemivariational inequality. The second aim is to investigate the existence of mild solutions to DPHVI by means of a fixed point technique. The third one is to study the existence of a pullback attractor for the multivalued processes governed by DPHVI. Finally, the fourth goal is to demonstrate a concrete application of our main results arising from the dynamic thermoviscoelasticity problems.

Boundedness and Finite-Time Stability for Multivalued Doubly-Nonlinear Evolution Systems Generated by a Microwave Heating Problem

Doubly-nonlinear evolutionary systems are considered. Sufficient conditions of the boundedness of solutions of such systems are derived. Similar results for a one-dimensional microwave heating problem are proved. The notions of global process and of a local multivalued process are introduced. Sufficient conditions for the finite-time stability of a global process and of a local multivalued process are obtained. For local multivalued processes sufficient conditions for the finite-time instability are derived. For the one-dimensional microwave heating problem conditions of the finite-time stability are shown.

Pullback Attractors for a Class of Semilinear Second‐Order Nonautonomous Evolution Equations with Hereditary Characteristics

In this paper, we investigate the long‐time behavior for the nonautonomous semilinear second‐order evolution equation (∂2u/∂t2) − Δu − Δ(∂u/∂t) − Δ(∂2u/∂t2) = f(t, u(x, t − ρ(t))) + g(t, x), in(τ, ∞) × Ω with some hereditary characteristics, where Ω is an open‐bounded domain of ℝN(N ≥ 3) with smooth boundary ∂Ω. Firstly, we establish the existence of solutions for the second‐order nonautonomous evolution equation by the standard Faedo–Galerkin method, but without the uniqueness of solutions. Then by proving the pullback asymptotic compactness for the multivalued process {U(t, τ)} on , we obtain the existence of pullback attractors in the Banach spaces for the multivalued process generated by a class of second‐order nonautonomous evolution equations with hereditary characteristics and ill‐posedness.

Trajectory and global attractors for generalized processes

In this work the theory of generalized processes is used to describe the dynamics of a nonautonomous multivalued problem and, through this approach, some conditions for the existence of trajectory attractors are proved. By projecting the trajectory attractor on the phase space, the uniform attractor for the multivalued process associated to the problem is obtained and some conditions to guarantee the invariance of the uniform attractor are given. Furthermore, the existence of the uniform attractor for a class of \begin{document}$ p $\end{document} -Laplacian non-autonomous problems with dynamical boundary conditions is established.

Boundedness and Finite-Time Stability for Multivalued Doubly-Nonlinear Evolution Systems Generated by a Microwave Heating Problem

Doubly-nonlinear evolutionary systems are considered. Sufficient conditions of the boundedness of solutions of such systems are derived. Analogical results for a one-dimensional microwave heating problem are proved. The notions of global process and of a local multivalued process are introduced. Sufficient conditions for the finite-time stability of a global process and of a local multivalued process are shown. For local multivalued processes sufficient conditions for the finite-time instability are derived. For the one-dimensional microwave heating problem conditions of the finite-time stability are shown.

Pullback attractors in the weighted space for multi-valued process generated by the non-autonomous nonclassical diffusion equations with unbounded delays without uniqueness of solutions

ABSTRACTIn this article, we consider the asymptotic behavior of the following non-autonomous nonclassical diffusion equations with infinite delays Under suitable conditions on f and G, we prove the existence of pullback attractors in the weighted space for multi-valued process generated by non-autonomous nonclassical diffusion equations with unbounded delays without uniqueness of solutions while the external force .

A selected pullback attractor

We introduce a new class of multivalued processes generated by generalized processes and a new concept of pullback attractor, named a selected pullback attractor, and we establish suitable conditions for the existence of such a selected pullback attractor. Finally, we give and application to nonautonomous problems with dynamic boundary conditions.

Pullback attractors for a strongly damped delay wave equation in ℝn

In this paper, we prove the existence of a pullback attractor for a strongly damped delay wave equation in [Formula: see text]. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth conditions, so that some new methods to obtain the existence of pullback attractors for multi-valued processes on an unbounded domain are introduced.

Finite lattice approximation of infinite lattice systems with delays and non-Lipschitz nonlinearities

In this paper, we consider the dynamics of multi-valued processes generated by nonautonomous lattice systems with delays. In particular, the effects of small delays on the asymptotic behavior of multi-valued nonautonomous lattice systems and finite lattice approximation of infinite delay lattice systems are presented. We do not assume any Lipschitiz condition on the nonlinear term, just a continuity assumption together with growth condition, so that uniqueness of solutions of the problem fails to be true.

Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay

In this work we prove the existence of solution for a p-Laplacian non-autonomous problem with dynamic boundary and infinite delay. We ensure the existence of pullback attractor for the multivalued process associated to the non-autonomous problem we are concerned.

Pullback Attractors for Multivalued Processes and Application to Nonautonomous Problems with Dynamic Boundary Conditions

In this work we prove, by using the Faedo-Galerkin method, the existence of at least one solution for a class of p-Laplacian nonautonomous problems with dynamic boundary conditions. We also extend to the multivalued context results on the existence of \(\mathcal {D}\)-pullback attractor - the pullback attractor for a determined basin of attraction \(\mathcal {D}\) - and we prove the existence of such attractors for the class of nonautonomous problems we are concerned about.

Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays

In this paper, we prove the existence of a pullback attractor in higher regularity space for the multi-valued process associated with the $2D$-non-autonomous incompressible non-Newtonian fluid with delays and without the uniqueness of solutions.

Structure of the pullback attractor for a non-autonomous scalar differential inclusion

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.

Uniform attractors for multi-valued process generated by non-autonomous nonclassical diffusion equations with delay in unbounded domain without uniqueness of solutions

In this article, the existence of a uniform attractor is proved for the multi-valued process generated by non- autonomous nonclassical diffusion equations (NDEs) with delays in unbounded domain without uniqueness of solutions, when the external force g(x, t) belongs to the space L 2∗ (R ,L 2 (R N )) and is a translation bounded function, but is not a translation

Pullback attractors for a damped wave equation with delays

Using a method to prove the pullback asymptotically compactness for the multi-valued processes, we present the existence of unique pullback attractors in CV,H and CD(A),V for the multi-valued process associated with a damped wave equation with delays and without the uniqueness of solutions.

Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations

We consider the following lattice nonclassical diffusion equation with delaysv̇i(t)+λ0vi(t)+(-1)pΔpvi(t)+ε(-1)pΔpv̇i(t)=fi(vi(t-ρ(t)))+gi(t),i∈Z,where ε∈(0,1],λ0 is a positive constant with λ0<1,p is any positive integer and ▵ is the discrete one-dimensional Laplace operator. Under suitable conditions on f and g we prove the existence of pullback attractors for the multi-valued process associated with the ε-small perturbed systems for which the uniqueness of solutions need not hold. Moreover, we compare the dynamics of the original systems and the ε-small perturbed systems, and show that their attractors are “close” in the sense of Hausdorff semidistance.

The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain

The existence of a uniform attractor in a space of higher regularity is proved for the multi-valued process associated with the nonautonomous reaction-diffusion equation on an unbounded domain with delays for which the uniqueness of solutions need not hold. A new method for checking the asymptotical upper-semicompactness of the solutions is used.

Global φ-attractor for a modified 3D Bénard system on channel-like domains

AbstractIn this paper we prove the existence of a global '-attractor in the weak topology of the natural phase space for the family of multi-valued processes generated by solutions of a nonautonomous modified 3D Bénard system in unbounded domains for which Poincaré inequality takes place

Nash Points for Nonzero-Sum Stochastic Differential Games with Separate Hamiltonians

We study a nonzero-sum stochastic differential game under the assumptions that the control sets are multidimensional convex compact, the game has separate dynamic and running costs and the multifunctions representing the optimal feedbacks have convex values. To prove the existence of Nash equilibria we reduce to study a system of uniformly parabolic equations strongly coupled by multivalued applications. We obtain the existence of Nash points in two different cases: (i) \(\mathbb{R}\)-valued process and general dynamic, (ii) multivalued process and affine dynamic.