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Attractors for general operators

Alain Miranville (2003)

Applications of Mathematics

In this article we introduce the notion of a minimal attractor for families of operators that do not necessarily form semigroups. We then obtain some results on the existence of the minimal attractor. We also consider the nonautonomous case. As an application, we obtain the existence of the minimal attractor for models of Cahn-Hilliard equations in deformable elastic continua.

Attractors for stochastic reaction-diffusion equation with additive homogeneous noise

Jakub Slavík (2021)

Czechoslovak Mathematical Journal

We study the asymptotic behaviour of solutions of a reaction-diffusion equation in the whole space $ℝ^{d}$ driven by a spatially homogeneous Wiener process with finite spectral measure. The existence of a random attractor is established for initial data in suitable weighted $L^{2}$ -space in any dimension, which complements the result from P. W. Bates, K. Lu, and B. Wang (2013). Asymptotic compactness is obtained using elements of the method of short trajectories.

Attractors of asymptotically periodic multivalued dynamical systems governed by time-dependent subdifferentials.

Yamazaki, Noriaki (2004)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Attractors of nonautonomous and stochastic differential inclusions

Caraballo, Tomás, Langa, José Antonio, Melnik, Valery S., Valero, José (2002)

Equadiff 10

Attractors of semigroups associated with nonlinear systems for diffusive phase separation.

Kenmochi, Nobuyuki (1996)

Abstract and Applied Analysis

Attractors of Strongly Dissipative Systems

A. G. Ramm (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

A class of infinite-dimensional dissipative dynamical systems is defined for which there exists a unique equilibrium point, and the rate of convergence to this point of the trajectories of a dynamical system from the above class is exponential. All the trajectories of the system converge to this point as t → +∞, no matter what the initial conditions are. This class consists of strongly dissipative systems. An example of such systems is provided by passive systems in network theory (see, e.g., MR0601947...

Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization

M. Efendiev, S. Zelik (2002)

Annales de l'I.H.P. Analyse non linéaire

Bifurcations for Turing instability without SO(2) symmetry

Toshiyuki Ogawa, Takashi Okuda (2007)

Kybernetika

In this paper, we consider the Swift–Hohenberg equation with perturbed boundary conditions. We do not a priori know the eigenfunctions for the linearized problem since the $SO (2)$ symmetry of the problem is broken by perturbation. We show that how the neutral stability curves change and, as a result, how the bifurcation diagrams change by the perturbation of the boundary conditions.

Blow-up of solutions to a periodic nonlinear dispersive rod equation.

Jin, Liangbing, Liu, Yongming (2010)

Documenta Mathematica

Cauchy problems in weighted Lebesgue spaces

Jan W. Cholewa, Tomasz Dłotko (2004)

Czechoslovak Mathematical Journal

Global solvability and asymptotics of semilinear parabolic Cauchy problems in $ℝ^{n}$ are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over $ℝ^{n}$ , $n \in ℕ$ . In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.

Center manifold and exponentially-bounded solutions of a forced Newtonian system with dissipation.

Garcia, Luis, Leiva, Hugo (2000)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Chaos and shadowing around a homoclinic tube.

Li, Yanguang (2003)

Abstract and Applied Analysis

Cone invariance and squeezing properties for inertial manifolds for nonautonomous evolution equations

Norbert Koksch, Stefan Siegmund (2003)

Banach Center Publications

In this paper we summarize an abstract approach to inertial manifolds for nonautonomous dynamical systems. Our result on the existence of inertial manifolds requires only two geometrical assumptions, called cone invariance and squeezing property, and some additional technical assumptions like boundedness or smoothing properties. We apply this result to processes (two-parameter semiflows) generated by nonautonomous semilinear parabolic evolution equations.

Continuation of the connection matrix for singularly perturbed hyperbolic equations

Maria C. Carbinatto, Krzysztof P. Rybakowski (2007)

Fundamenta Mathematicae

Controllability of three-dimensional Navier–Stokes equations and applications

Armen Shirikyan (2005/2006)

Séminaire Équations aux dérivées partielles

We formulate two results on controllability properties of the 3D Navier–Stokes (NS) system. They concern the approximate controllability and exact controllability in finite-dimensional projections of the problem in question. As a consequence, we obtain the existence of a strong solution of the Cauchy problem for the 3D NS system with an arbitrary initial function and a large class of right-hand sides. We also discuss some qualitative properties of admissible weak solutions for randomly forced NS...