Upper Bound Of Fractal Dimension Research Articles

Fractal dimension of random attractors for nonautonomous stochastic strongly damped wave equations on ℝN

In this article, we investigate the dynamics of a nonautonomous stochastic strongly damped wave equation defined on . We first use the energy equation and tail‐estimates to prove the asymptotic compactness of the solutions and obtain the existence of a unique pullback random attractor for the equation with critical nonlinearity. Then, we give an upper bound of fractal dimension of the random attractor when the nonlinearity is of subcritical growth. The unboundedness of the physical space will impose difficulties on the estimation for the upper bound of fractal dimension of the random attractor.

Exponential attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity

<abstract><p>In this paper, the dynamical behavior of the nonclassical diffusion equation is investigated. First, using the asymptotic regularity of the solution, we prove that the semigroup $ \{S(t)\}_{t\geq 0} $ corresponding to this equation satisfies the global exponentially $ \kappa- $dissipative. And then we estimate the upper bound of fractal dimension for the global attractors $ \mathscr{A} $ for this equation and $ \mathscr{A}\subset H^1_0(\Omega)\cap H^2(\Omega) $. Finally, we confirm the existence of exponential attractors $ \mathscr{M} $ by validated differentiability of the semigroup $ \{S(t)\}_{t\geq 0} $. It is worth mentioning that the nonlinearity $ f $ satisfies the polynomial growth of arbitrary order.</p></abstract>

Finite Fractal Dimensionality of Compact Kernel Sections for Dissipative Non-Autonomous Klein-Gordon-Schrödinger Lattice Systems

In this paper, an upper bound of fractal dimension of the compact kernel sections for the dissipative non-autonomous Klein-Gordon-Schrödinger lattice system is obtained, by applying a criterion for estimating fractal dimension of a family of compact subsets of a separable Hilbert space.

Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation

In this paper, we first prove the existence of a random attractor for stochastic non-autonomous strongly damped wave equations with additive white noise. Then we apply a criteria to obtain an upper bound of fractal dimension of the random attractor of considered system.

Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises

In this paper, we study the long-term dynamical behavior of stochastic Boisso nade systems with time-dependent deterministic forces, additive white noise and multiplicative white noise. We first prove the existence of random attrac tor for the considered systems. And then we establish the upper semi-continui ty of random attractors for the systems as the coefficient of quadratic term tends to zero and intensities of the noises approach zero, respectively. At last, we obtain an upper bound of fractal dimension of the random attractors for both systems without quadratic term.

Random attractor for stochastic non-autonomous damped wave equation with critical exponent

In this paper, we prove the existence of random attractor and obtainan upper bound of fractal dimension of random attractor forstochastic non-autonomous damped wave equation with criticalexponent and additive white noise. We first prove the existence of arandom attractor by carefully splitting the positivity of the linearoperator in the corresponding random evolution equation of the firstorder in time and by carefully decomposing the solutions of systemthrough two different modes, and we show the boundedness of randomattractor in a higher regular space by a recurrence method. Then weestablish a criterion to bound the fractal dimension of a randominvariant set for a cocycle and applied these conditions to get anupper bound of fractal dimension of the random attractor ofconsidered system.

Finite Dimensionality and Upper Semicontinuity of Kernel Sections for the Discrete Zakharov Equations

In this paper, the authors first prove an upper bound of fractal dimension of kernel sections for the discrete Zakharov equations for plasmas with a quantum correction. Then they establish the upper semicontinuity of the kernel sections when the infinite lattice system is approximated by finite ordinary differential equations.

Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation

In this paper, we first establish a criteria for estimating the upper bound of fractal dimension of a random compact invariant set of a non-autonomous random dynamical system on a separable Banach space and give some special conditions which can be easily verified to some stochastic partial differential equations for bounding the fractal dimension of random attractors. Then we apply these conditions to obtain an upper bound of fractal dimension of random attractor for stochastic damped wave equation with additive white noise.

Global attractors for the discrete Klein–Gordon–Schrödinger type equations

The purpose of this work is to investigate the asymptotic behaviours of solutions for the discrete Klein–Gordon–Schrödinger type equations in one-dimensional lattice. We first establish the global existence and uniqueness of solutions for the corresponding Cauchy problem. According to the solution's estimate, it is shown that the semi-group generated by the solution is continuous and possesses an absorbing set. Using truncation technique, we show that there exists a global attractor for the semi-group. Finally, we extend the criteria of Zhou et al. [S. Zhou, C. Zhao, and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Discrete Contin. Dyn. Syst. A 21 (2008), pp. 1259–1277.] for finite fractal dimension of a family of compact subsets in a Hilbert space to obtain an upper bound of fractal dimension for the global attractor.

Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

We first present some sufficient conditions for the existence of a pullback exponential attractor for continuous process on the product space of the weighted spaces of infinite sequences. Then we prove the existence and continuity of a pullback exponential attractor for second order lattice system with time-dependent coupled coefficients in the weighted space of infinite sequences. Moreover, we obtain the upper bound of fractal dimension and attracting rate for the attractor.

Compact kernel sections for dissipative non-autonomous Zakharov equation on infinite lattices

In this paper, we first prove the existence of compact kernel sections for dissipative non-autonomous Zakharov lattice dynamical system, then we obtain an upper bound of fractal dimension of the compact kernel sections, and finally we establish an upper semicontinuity of the compact kernel sections.

Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems

In this paper, we first establish a criteria for finite fractal dimensionality of a family of compact subsets of a Hilbert space, and apply it to obtain an upper bound of fractal dimension of compact kernel sections to first order non-autonomous lattice systems. Then we consider the upper semicontinuity of kernel sections of general first order non-autonomous lattice systems and give an application.

Attractors for a second order nonautonomous lattice dynamical system with nonlinear damping

In this Letter, the existence of compact pullback attractors is considered for a second order nonautonomous lattice dynamical system with nonlinear damping arising from spatial discretization of wave equations in Rk. And the finite-dimensional approximations of the attractors are studied. Finally, an upper bound of fractal dimension of the attractors is obtained.