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The goal of this paper is to establish the Lipschitz and W2,∞ estimates for a second-order parabolic PDE ∂tu(t,x)=12Δu(t,x)+f(t,x) on Rd with zero initial data and f satisfying a Ladyzhenskaya–Prod...

In this paper we consider an initial value problem for the two-dimensional liquid crystal equations. We establish the global regularity of solutions to the equations with a weak velocity dissipation and obtain the global smooth solutions to the 2D liquid crystal equations with damping for small initial data.

In this paper we consider the global existence for the 3D generalized Hall-MHD equations with fractional dissipative terms ( − Δ ) α u and ( − Δ ) α b under small initial data in the setting of Sobolev norms with lower regularity. For the global existence we enlarge the range of dissipative exponents α = β from ( 1 , 7 6 ] to ( 1 , 3 2 ) , which established in a recent work. In addition, the long time behavior and rates of decay for both the solutions and higher derivatives in different Sobolev spaces are obtained by using the Fourier splitting method, which extends the previous work by Chae and Schonbek.

We study the Cauchy problem of the nonlinear stochastic fractional heat equation ∂ t u = − ν 2 ( − ∂ x x ) α 2 u + σ ( u ) W ˙ ( t , x ) on real line R driven by space-time white noise with bounded initial data. We analyze the large- | x | fixed- t behavior of the solution u t ( x ) for Lipschitz continuous function σ : R → R under three cases. (1) σ is bounded below away from 0. (2) σ is uniformly bounded away from 0 and ∞ . (3) σ ( x ) = c x (the parabolic Anderson model). From the sensitivity to the initial data of stochastic fractional heat equation, we describe that the solution to the Cauchy problem of stochastic fractional heat equation exhibits chaotic behavior at fixed time before the onset of intermittency.

The singular support of the global attractor is introduced. It is shown that the singular support of the global attractor for a damped BBM equation equals to the singular support of the force term. This gives a delicate description of the local regularity, which roughly says that the attractor is smooth exactly where the force is smooth.

In this paper, we consider the well-posedness of a one-dimensional transport equation with nonlocal velocity in the Lei–Lin space Xs(R). We first modify the product estimate and then establish the global existence of solutions to the Cauchy problem with small enough initial data. Finally, we discuss the stability of the global solution. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we first establish the local well-posedness of strong solutions to the Cauchy problem of the incompressible viscous resistive Hall-MHD equations in $$H^s(\mathbb {R}^3)$$ $$(\frac{3}{2}< s\le \frac{5}{2})$$ , and then we prove that the local solution is global when the initial data is small enough.

For one-dimensional damped Benjamin–Bona–Mahony equation, a recent result claimed that there exists a global attractor in , in fact the attractor is smoother and it belongs to for every . In this paper, by some techniques of harmonic analysis, we first show that the solution operator of one-dimensional damped BBM equation is asymptotically compact, then we give the existence of global attractor in . Finally, we obtain the higher regularity of the global attractor, it is in fact bounded in .

The long time behavior of solutions of 2D quasi-geostrophic equations with damping on the whole space is studied. The existence of the global attractor in L p is proved. New difficulties come from nonlocal terms and the Banach space L p . We overcome them by using commutator estimates and positivity inequalities associated with fractional Laplacian.

In this paper, we study the random dynamical system generated by a stochastic reaction-diffusion equation with multiplicative noise and prove the existence of an -random attractor for such a random dynamical system. The nonlinearity f is supposed to satisfy some growth of arbitrary order .

We obtain a precise upper bound of the fractal dimension of the global attractor for 2D quasi-geostrophic equations. The upper bound is a decreasing function of the coefficient κ of dissipative term, which conforms to physical intuition. Moreover, the bound tends to infinity as κ → 0 and α → 1 2 , which reflects the chaotic behavior of the QG equation without dissipative effect and in critical case, respectively. Our result gives an answer to a problem posed in Ju [N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Commun. Math. Phys., 255 (2005) 161–181].

We study the random dynamical system generated by a stochastic reaction-diffusion equation with additive noise on the whole space and prove the existence of an -random attractor for such a random dynamical system. The nonlinearity is supposed to satisfy the growth of arbitrary order (). The -asymptotic compactness of the random dynamical system is obtained by using an extended version of the tail estimate method introduced by Wang (1999) and the cut-off technique.

This paper considers the statistical properties of the traveling wave fronts of the scalar FitzHugh–Nagumo equation with random perturbations by two-parameter white noise u t = u x x + f ( u ) + e W x t on the whole real line ℛ , where the traveling wave front connects two stable equilibria u = 0 and u = 1 of the reaction function f ( u ) . As well as the method of Green’s function established by Tuckwell on a bounded domain, we get the asymptotic fluctuation behavior of two stable states which are two boundaries of the traveling wave front to the Nagumo equation by the fundamental solution. That is, the perturbations about the lower (upper) stable state reveal that the mean is increased (decreased) by zero mean white noise as t → + ∞ .

In this paper, we study the global well-posedness of the 3D incompressible critical dissipative porous media equation with small initial data in the Triebel-Lizorkin space $F^{s}_{p,q}(\mathbb{R}^{3})$ . By a pointwise exponential decay estimate on the Poisson semigroup $e^{-t\nu\sqrt{-\Delta}}$ and the Fourier localization technique, we generalize the global well-posedness in the Sobolev spaces $H_{p}^{s}(\mathbb{R}^{3})=F^{s}_{p,2}(\mathbb{R}^{3})$ and $H^{s}(\mathbb {R}^{3})=F^{s}_{2,2}(\mathbb{R}^{3})$ into the general Triebel-Lizorkin spaces $F^{s}_{p,q}(\mathbb{R}^{3})$ with $s>\frac{3}{p}$ , $p, q\in (1,\infty)$ .

We are concerned with a class of reaction diffusion equations with nonlinear terms of arbitrary growth on unbounded domains. The existence of an $L^2 - L^{2p-2} \cap H^2$ global attractor is proved. This improves the results in previous references, and the proof is shorter.

In this paper we consider the long time behavior of solutions of a version of gKdV equation on the real line. The existence of the global attractor in H 2 ( R ) is proved when the force f belongs to H 2 ( R ) . Moreover, we investigate the change of the attractor according to the change of f and the coefficient γ of damping term. Finally, several numerical experiments confirm all the theoretical conclusions.

In this paper, we consider a periodic boundary value problem for a non-classical reaction–diffusion equation with memory. In other paper, we use the ω -limit compactness of the solution semigroup { S ( t ) } t ≥ 0 to get the existence of a global attractor. The main goal here is to give an estimate of the fractal dimension of the global attractor. By the fractal dimension theorem given by A.O. Celebi et al., we obtain that the fractal dimension of the global attractor for the problem is finite; this makes the results for the non-classical reaction–diffusion equations more substantial and perfect.

We consider the periodic initial value problem associated to the generalized Benjamin-Bona-Mahony equation with generalized damping on the one dimensional torus. In contrast to the classical BBM equation, the main difference is that the generalized equation contains two nonlocal operators, and the main difficulty comes from two nonlocal operators. By the fixed point theorem, we prove that the periodic initial value problem is locally well-posed. We also prove that if the solution exists globally in time, it exhibits some asymptotic behavior.

In this paper we give a rigorous bifurcation analysis for a reactor model with combination of quadratic and cubic steps. We extend the analysis of Satnoianu et al. in two directions. First, we impose Dirichlet boundary conditions instead of the semi-infinite domain in one space dimension. Second, we consider a variety of different bifurcation phenomena of the corresponding steady state system, focusing on their parametric sensitivity. We show that the system exhibits subcritical pitchfork bifurcation, supercritical pitchfork bifurcation and transcritical bifurcation in the different regimes of control parameters. To compare the decades old work by Auchmuty et al., we show that the spatial structures formed by the parameter p ∈ [0, 1] are richer than those formed by the purely quadratic step for p = 0 and the purely cubic step for p = 1 respectively.

For some species, hereditary factors have great effects on their population evolution, which can be described by the well-known Volterra model. A model developed is investigated in this article, considering the seasonal variation of the environment, where the diffusive effect of the population is also considered. The main approaches employed here are the upper-lower solution method and the monotone iteration technique. The results show that whether the species dies out or not depends on the relations among the birth rate, the death rate, the competition rate, the diffusivity and the hereditary effects. The evolution of the population may show asymptotic periodicity, provided a certain condition is satisfied for the above factors. (c) 2006 Elsevier Ltd. All rights reserved.

We consider the approximate 3D Kelvin-Voigt fluid driven by an external force depending on velocity with distributed delay. We investigate the long time behavior of solutions to Navier-Stokes-Voigt equation with a distributed delay external force depending on the velocity of fluid on a bounded domain. By a prior estimate and a contractive function, we give a sufficient condition for the existence of pullback attractor of NSV equation.

The main goal of this paper is to investigate the blow-up phenomena of solutions to a weakly dissipative generalized Camassa-Holm equation, which contains higher power nonlinear dispersion terms and a convection term. We give a sufficient condition on the initial data such that the strong solution blows up at a finite time, and then we establish an estimate of the blow-up time. Finally, we give a global existence result of the strong solution. MSC: 35A35; 35B30; 35G25; 35Q53

In this paper we investigate the effects of time delay and diffusion rate on the stability of the positive steady state for a competition diffusion system with distributed delay. We obtain the condition of instability of the positive uniform steady state. Regarding the time delay as bifurcation parameter, we reduce the original system on the center manifold and get the stability of the Hopf bifurcation periodic solutions. Finally, regarding the diffusion rate as a bifurcation parameter, we show that the onedimensional competition diffusion system with infinite delay and Dirichlet boundary condition exhibits the spatiotemporal structures near the steady state of the system.

In this paper, we consider the effects of time delay on the linear stability of the positive equilibrium of a reaction-diffusion model with time delayed nutrient recycling. We take the two delay kernels as f(s) = δ(s), g(s) = pδ(0) + (1-p)δ(r), s ∈ (0, +∞), r ∈ [0, +∞), 0 < p < 1, where δ is the Dirac delta function. When both N 1 and N 2 are space-independent, we can observe that the increasing time delay destabilizes the positive equilibrium, and stability-switches from stability to instability to stability occur. That is, for some special time delay kernels, the equilibrium is stable for small time delay, bifurcates towards stable oscillations when the time delay is increased, then regains stability for larger values. When both N 1 and N 2 are space-dependent, we are concerned with the effects of the delay kernels as well as diffusion on the dynamics of the system.

The purpose of this paper is to investigate the stability and asymptotic behavior of the time-dependent solutions to a linear parabolic equation with nonlinear boundary condition in relation to their corresponding steady state solutions. Then, the above results are extended to a semilinear parabolic equation with nonlinear boundary condition by analyzing the corresponding eigenvalue problem and using the method of upper and lower solutions.

For a periodic reaction diffusion equation with infinite delay, a sufficient condition of existence and uniqueness of periodic solution is given. By using the periodic monotone method, the asymptotic behavior of the time-dependent solutions is investigated, that is, the time-dependent solutions converge to the corresponding periodic solution in the time variable.

This paper investigates the stability and Hopf bifurcation of a delay competition diffusion system. Firstly we discuss the existence and stability of the corresponding steady state solutions. Secondly our purpose is to give more detail information about the Hopf bifurcation of this system. We derive the basis of the eigenfunction subspace and then convert the existence of periodic solutions to the study of the existence of the implicit function. Finally, we analyze the stability of the periodic solutions by reducing the original system on the center manifold.

The purpose of this paper is to study the mathematical properties of the solutions of a model for bacteria and virulent bacteriophage system in a chemostat. A general model was first proposed by Levin, Stewart and Chao [13] and then, a specific one, by Lenski and Levin [12]. The numerical simulations come from the experimental data referred in [12,13]. In our Knowledge the analysis presented herefollowing is the first mathematical attempt to analyse the model of bacteria and virulent bacteriophage and presents two fresh frontiers: 1) the modeling of delay (latency period) incorporating the realistic through time death rate in linear stability analysis brings to characteristic equations with delay dependent parameters for which only recently Beretta and Kuang [5] provided a geometric stability switch criterion which application is presented along the paper; 2) the modelling of the dynamics through three full delay stages can be reduced to two using the integral representation for the density of infected bacteria. The basic properties of the model which are investigated are the existence of equilibria, positive invariance and boundedness of solutions and permanence results. Second, using the geometric stability switch criterion in the delay differential system with delay dependent parameters, we present the local asymptotic stability of the equilibria by analyzing the corresponding characteristic equation which coefficients depend on the time delay (the latency period). Numerical simulations are presented to illustrate the results of local stability. Then, we study the global asymptotic stability of the boundary equilibria via Liapunov functional method. Finally, we give a discussion about the model.

This paper considers a stochastic nonlinear thermoelastic system coupled sine-Gordon equation driven by jump noise. We first prove the existence and uniqueness of strong probabilistic solution of an initial-boundary value problem with homogeneous Dirichlet boundary conditions. Then we give an asymptotic behavior of the solution.

The goal of this paper is to study an initial boundary value problem of stochastic viscoelastic wave equation with nonlinear damping and source terms. Under certain conditions on the initial data: the relaxation function, the indices of nonlinear damping, and source terms and the random force, we prove the local existence and uniqueness of solution by the Galerkin approximation method. Then, considering the relationship between the indices of nonlinear damping and nonlinear source, we give the necessary conditions of global existence and explosion in finite time in some sense of solutions, respectively.

We prove an abstract result on random invariant sets of finite fractal dimension. Then this result is applied to a stochastic semilinear degenerate parabolic equation and an upper bound is obtained for the random attractors of fractal dimension.

In this paper, we study the long time behavior of solutions to a weakly dissipative fractional Korteweg de Vries (KdV) equation on the real line R. The main difficulty lies in that the dissipative term is a nonlocal operator. We overcome this difficulty by the commutator estimates and product estimates associated with fractional Laplacian. The asymptotical compactness of solution semigroup is proved by the tail estimates. Finally, we conclude the existence of (H2(R), H5(R)) global attractor of the weakly dissipative fractional KdV equation.

In this paper, we consider the existence and attraction of positive periodic solution of a competition diffusion system. We first construct a pair of upper and lower solutions, then use the periodic comparison existence theorem to get a pair of T-periodic solutions (u, v) and (u, v). Finally we give a sufficient condition of (u, v) = (u, v) to answer the open question described by Ahmad and Lazer.

In this paper, we study the blow-up property of positive mild solutions to the Cauchy problem of a system of fractional reaction-diffusion equations. For the fundamental solution $P(t,x)$ of the fractional heat operator $\partial_{t}+(-\triangle)^{\frac{\beta}{2}}$ defined on the whole space $R^{N}$ , due to the properties of $P(t,x)$ established by H Yosida and some estimates of $P(t,x)$ developed by L Caffarelli and A Figalli, we first use an iteration method to establish the estimates of lower bounds of positive mild solutions; then we obtain the unboundedness of solutions for large time. Finally we give a sufficient condition that the positive mild solution to a fractional reaction-diffusion system blows up in finite time.

In this paper we consider an initial value problem of the 2D MHD equations with velocity dissipation and without magnetic diffusion. We establish a new magnetic regularity criterion in terms of the magnetic field. In contrast to the magnetic regularity criterion $\nabla b\in L^{1}(0,T; BMO(\mathbb {R}^{2}))$ , our regularity criterion $\int_{0}^{T} (\Vert b\otimes b(s)\Vert _{B_{\infty,1}^{0}(\mathbb {R}^{2})} +\Vert b\otimes b(s)\Vert _{L^{2}(\mathbb {R}^{2})} )\,ds