DCDS-B
Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion
Jin Li Jianhua Huang
Discrete & Continuous Dynamical Systems - B 2012, 17(7): 2483-2508 doi: 10.3934/dcdsb.2012.17.2483
A 2D Stochastic incompressible non-Newtonian fluid driven by fractional Brownian motion with Hurst index $H \in (\frac{1}{2},1)$ is studied. The Wiener-type stochastic integrals are introduced for infinite-dimensional fractional Brownian motion. Including the requirements of Nuclear and Hilbert-Schmidt operators, three kinds of condition, which ensure the existence and regularity of infinite-dimensional stochastic convolution for the corresponding additive linear stochastic equation, are summarized. Without the requirements of compact parameters, another condition is proposed for the existence and regularity of stochastic convolution. By any of the four kinds of condition, the existence and uniqueness of mild solution are obtained for the stochastic non-Newtonian fluid through a modified fixed point theorem in the selected intersection space. Existence of a random attractor is then obtained for the random dynamical system generated by non-Newtonian fluid.
keywords: random attractor. stochastic non-Newtonian fluid Infinite-dimensional fractional Brownian motion stochastic convolu- tion
DCDS
Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity
Jianhua Huang Xingfu Zou
Discrete & Continuous Dynamical Systems - A 2003, 9(4): 925-936 doi: 10.3934/dcds.2003.9.925
In this paper, we establish the existence of traveling wavefronts for delayed reaction diffusion systems without quasimonotonicity in the reaction term, by using Schauder's fixed point theorem. We show the merit of our result by applying it to the Belousov-Zhabotinskii reaction model with two delays.
keywords: Traveling wavefront reaction diffusion systems quasimonotone delay upper solution and lower solution. Schauder’s fixed point theorem
DCDS-B
Ergodicity of the stochastic coupled fractional Ginzburg-Landau equations driven by α-stable noise
Tianlong Shen Jianhua Huang
Discrete & Continuous Dynamical Systems - B 2017, 22(2): 605-625 doi: 10.3934/dcdsb.2017029

The current paper is devoted to the ergodicity of stochastic coupled fractional Ginzburg-Landau equations driven by $α$-stable noise on the Torus $\mathbb{T}$. By the maximal inequality for stochastic $α$-stable convolution and commutator estimates, the well-posedness of the mild solution for stochastic coupled fractional Ginzburg-Landau equations is established. Due to the discontinuous trajectories and non-Lipschitz nonlinear term, the existence and uniqueness of the invariant measures are obtained by the strong Feller property and the accessibility to zero.

keywords: Stochastic coupled fractional Ginzburg-Landau equations ergodicity α-stable noise invariant measure
DCDS
Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains
Jianhua Huang Wenxian Shen
Discrete & Continuous Dynamical Systems - A 2009, 24(3): 855-882 doi: 10.3934/dcds.2009.24.855
The current paper is devoted to the study of pullback attractors for general nonautonomous and random parabolic equations on non-smooth domains $D$. Mild solutions are considered for such equations. We first extend various fundamental properties for solutions of smooth parabolic equations on smooth domains to solutions of general parabolic equations on non-smooth domains, including continuous dependence on parameters, monotonicity, and compactness, which are of great importance in their own. Under certain dissipative conditions on the nonlinear terms, we prove that mild solutions with initial conditions in $L_q(D)$ exist globally for $q$ » $1$. We then show that pullback attractors for nonautonomous and random parabolic equations on non-smooth domains exist in $L_q(D)$ for $1$ « $q$ < $\infty$.
keywords: Non-smooth domains nonautonomous dynamical systems random dynamical systems top Lyapunov exponents. pullback global attractors absorbing sets random parabolic equations nonautonomous parabolic equations
DCDS-B
Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks
Zuowei Cai Jianhua Huang Lihong Huang
Discrete & Continuous Dynamical Systems - B 2017, 22(9): 3591-3614 doi: 10.3934/dcdsb.2017181

In this paper, a general class of nonlinear dynamical systems described by retarded differential equations with discontinuous right-hand sides is considered. Under the extended Filippov-framework, we investigate some basic stability problems for retarded differential inclusions (RDI) with given initial conditions by using the generalized Lyapunov-Razumikhin method. Comparing with the previous work, the main results given in this paper show that the Lyapunov-Razumikhin function is allowed to have a indefinite or positive definite derivative for almost everywhere along the solution trajectories of RDI. However, in most of the existing literature, the derivative (if it exists) of Lyapunov-Razumikhin function is required to be negative or semi-negative definite for almost everywhere. In addition, the Lyapunov-Razumikhin function in this paper is allowed to be non-smooth. To deal with the stability, we also drop the specific condition that the Lyapunov-Razumikhin function should have an infinitesimal upper limit. Finally, we apply the proposed Razumikhin techniques to handle the stability or stabilization control of discontinuous time-delayed neural networks. Meanwhile, we present two examples to demonstrate the effectiveness of the developed method.

keywords: Retarded differential inclusions(RDI) Filippov solution stability Razumikhin techniques almost indefinite Lyapunov-like function
DCDS-B
Pullback attractors of FitzHugh-Nagumo system on the time-varying domains
Zhen Zhang Jianhua Huang Xueke Pu
Discrete & Continuous Dynamical Systems - B 2017, 22(10): 3691-3706 doi: 10.3934/dcdsb.2017150

The existence and uniqueness of solutions satisfying energy equality is proved for non-autonomous FitzHugh-Nagumo system on a special time-varying domain which is a (possibly non-smooth) domain expanding with time. By constructing a suitable penalty function for the two cases respectively, we establish the existence of a pullback attractor for non-autonomous FitzHugh-Nagumo system on a special time-varying domain.

keywords: Pullback attractor FitzHugh-Nagumo equation time-varying domain penalty function
DCDS-S
Invasion traveling wave solutions in temporally discrete random-diffusion systems with delays
Hui Xue Jianhua Huang Zhixian Yu
Discrete & Continuous Dynamical Systems - S 2017, 10(5): 1107-1131 doi: 10.3934/dcdss.2017060

This paper is devoted to the invasion traveling wave solutions for a temporally discrete delayed reaction-diffusion competitive system. The existence of invasion traveling wave solutions is established by using Schauder's fixed point Theorem. Ikeharaś theorem is applied to show the asymptotic behaviors. We further investigate the monotonicity and uniqueness invasion traveling waves with the help of sliding method and strong maximum principle.

keywords: Invasion traveling wave solutions diffusive competition systems existence asymptotic behavior uniqueness
CPAA
Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system
Kun Li Jianhua Huang Xiong Li
Communications on Pure & Applied Analysis 2017, 16(1): 131-150 doi: 10.3934/cpaa.2017006

This paper is concerned with the asymptotic behavior and uniqueness of traveling wave fronts connecting two half-positive equilibria in a delayed nonlocal dispersal competitive system. We first prove the existence results by applying abstract theories. And then, we show that the traveling wave fronts decay exponentially at both infinities. At last, the strict monotonicity and uniqueness of traveling wave fronts are obtained by using the sliding method in the absent of intraspecific competitive delays. Based on the uniqueness, the exact decay rate of the stronger competitor is established under certain conditions.

keywords: Delayed nonlocal dispersal competitive system traveling wave front asymptotic behavior monotonicity uniqueness upper and lower solutions
DCDS-B
Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise
Tianlong Shen Jianhua Huang Caibin Zeng
Discrete & Continuous Dynamical Systems - B 2018, 23(4): 1523-1533 doi: 10.3934/dcdsb.2018056

We present the time-spatial regularity of the nonlocal stochastic convolution for Caputo-type time fractional nonlocal Ornstein-Ulenbeck equations, and establish the existence and uniqueness of mild solutions for time fractional and space nonlocal stochastic Boussinesq equations driven by Gaussian white noise.

keywords: Caputo-type time fractional drivative fractional Laplacian operator Mittag-Leffler functions mild solution
DCDS-B
Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay
Kun Li Jianhua Huang Xiong Li
Discrete & Continuous Dynamical Systems - B 2018, 23(6): 2091-2119 doi: 10.3934/dcdsb.2018227

This paper is concerned with a class of advection hyperbolic-parabolic systems with nonlocal delay. We prove that the wave profile is described by a hybrid system that consists of an integral transformation and an ordinary differential equation. By considering the same problem for a properly parameterized system and the continuous dependence of the wave speed on the parameter involved, we obtain the existence and uniqueness of traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay under bistable assumption. The influence of advection on the propagation speed is also considered.

keywords: Advection hyperbolic-parabolic system traveling wave solution bistable super and subsolutions nonlocal delay

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