In this paper we study the structure of the global attractor for a reaction- di§usion equation in which uniqueness of the Cauchy problem is not guarantied. We prove that the global attractor can be characterized using either the unstable manifold of the set of stationary points or the stable one but considering in this last case only solutions in the set of bounded complete trajectories.
In the article, we aim to investigate the well-posedness of solution and the regularity of the global attractor for the couple stress fluid in saturated porous media with the local thermal non-equilibrium effect. To be more specific, we firstly show the existence and uniqueness of global weak solution to the model by making use of the standard Galerkin method. Second, relying on verifying the uniformly compact condition required, we prove the existence of the global attractor of the model in the space where the weak solution resides. Finally, we improve the regularity of global attractor by un…
This paper is concerned with the existence and regularity of global attractor $\mathcal A$ for a Kirchhoff wave equation with strong damping and memory in the weighted time-dependent spaces $\mathcal H$ and $\mathcal H^{1}$, respectively. In order to obtain the existence of $\mathcal A$, we mainly use the energy method in the priori estimations, and then verify the asymptotic compactness of the semigroup by the method of contraction function. Finally, by decomposing the weak solutions into two parts and some elaborate calculations, we prove the regularity of $\mathcal A$.
In this paper, we study the asymptotic behavior of solution to a non-autonomous diffusion equations with delay containing some hereditary characteristics and nonlocal diffusion in time-dependent space $C_{\mathcal{H}_{t}(Ω)}$. When the nonlinear function $f$ satisfies the polynomial growth of arbitrary order $p-1$ $(p \ge 2)$ and the external force $h \in L_{l o c}^{2}\left(\mathbb{R} ; H^{-1}(Ω)\right)$, we establish the existence and regularity of the time-dependent pullback attractors.
In this paper we consider the long time behavior of the weakly damped, forced Korteweg-de Vries equation in the Sololev spaces of the negative indices in the periodic case. We prove that the solutions are uniformly bounded in $\dot{H}^s(\T)$ for $s>-\dfrac{1}{2}$. Moreover, we show that the solution-map possesses a global attractor in $\dot{H}^s(\T)$ for $s>-\dfrac{1}{2}$, which is a compact set in $H^{s+3}(\T)$.
In this paper, we mainly study the regularity of pullback $\mathcal{D}$-attractors for a nonautonomous nonclassical diffusion equation with delay term $b(t,u_t)$ which contains some hereditary characteristics. Under a critical nonlinearity $f$, a time-dependent force $g(t,x)$ with exponential growth and a delayed force term $b(t,u_t)$, we prove that there exists a pullback $\mathcal{D}$-attractor $\mathcal{A}=\{A(t):t \in \mathbb{R}\}$ in $\mathbb{K}^1=H_0^1(Ω) \times L^2((-h,0);L^2(Ω))$ to problem \eqref{ine01} and for each $t \in \mathbb{R}$, $A(t)$ is bounded in $\mathbb{K}^2=H^2(Ω) \cap H_…
In this paper, we prove the existences of pullback attractors in $L^{p}(\mathbb{R}^N)\times L^{2}(\mathbb{R}^N)$ for stochastic Fitzhugh-Nagumo system driven by both additive noises and deterministic non-autonomous forcings. The nonlinearity is polynomial like growth with exponent $p-1$. The asymptotic compactness for the cocycle in $L^{p}(\mathbb{R}^N)\times L^{2}(\mathbb{R}^N)$ is proved by using asymptotic a priori method, where the plus and minus signs of the nonlinearity at large value are not required.
This note is focused on a novel technique in order to establish the boundedness in more regular spaces for global attractors of dissipative dynamical systems, without appealing to uniform-in-time estimates. As an application of the abstract result, the semigroup generated by the strongly damped wave equation $$u_{tt}-Δu_t-Δu+ϕ(u)=f$$ with critical nonlinearity is considered, whose attractor is shown to possess the optimal regularity.
We estimate numerically the regularities of a family of Strange Non--Chaotic Attractors related with one of the models studied in C. Grebogy et al. (1984) (see also G. Keller (1996)). To estimate these regularities we use wavelet analysis in the spirit of R. de la Llave et al. (2002) together with some ad-hoc techniques that we develop to overcome the theoretical difficulties that arise in the application of the method to the particular family that we consider. These difficulties are mainly due to the facts that we do not have an explicit formula for the attractor and it is discontinuous almos…
We investigate the long term behavior in terms of global attractors, as time goes to infinity, of solutions to a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. We consider the aggregation equation with both degenerate and non-degenerate diffusion in a bounded domain subject to various boundary conditions. In the degenerate case, we prove the existence of the global attractor and derive some optimal regularity results. Furthermore, in the non-degenerate case we give a complete structural characterization of the…
This work is focused on the dissipative system describing the dynamics of an extensible thermoelastic beam, where the dissipation is entirely contributed by the second equation ruling the evolution of the temperature. Under natural boundary conditions, we prove the existence of bounded absorbing sets. When both the external body force and the heat source are time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity for all values of the external axial load. The same result holds true when the rotational inertia is taken into considerati…
We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose density are regular. The maps we consider are given by $T(x,y) = (E (x), C(y) + f(x) )$, where $E$ is a linear expanding map of $\mathbb{T}$, $C$ is a linear contracting map of $\mathbb{R}^d$, $f$ is in $C^r(\mathbb{T}^u,\mathbb{R}^d)$ and $r \geq 2$. We prove that if $|(\det C)(\det E)| \|C^{-1}\|^{-2s}>1$ for some $s<r-(\frac{u+d}{2}+1)$ and $T$ satisfies a certain transversality condition, then the density of the SRB measure of $T$ is contained in the Sobolev space $…
We discuss various issues related to the finite-dimensionality of the asymptotic dynamics of solutions of parabolic equations. In particular, we study the regularity of the vector field on the global attractor associated with these equations. We show that certain dissipative partial differential equations possess a linear term that is log-Lipschitz continuous on the attractor. We then prove that this property implies that the associated global attractor $\mathcal A$ lies within a small neighbourhood of a smooth manifold, given as a Lipschitz graph over a finite number of Fourier modes. Consequ…
The long-term behaviour of solutions to a model for acoustic-structure interactions is addressed; the system is comprised of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are: existence of a global attractor for the dynamics generated by this composite system, as well as dimensionality and regularity of the attractor. A distinct and challenging feature of the problem is the geometrically restricted dissipation on the wave component of the system. It is shown that the existence of a global attractor of finite fractal dime…
A doubly nonlinear parabolic equation of the form $α(u_t)-Δu+W'(u)= f$, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions, is addressed. The two nonlinearities are given by the maximal monotone function $α$ and by the derivative $W'$ of a smooth but possibly nonconvex potential $W$; $f$ is a known source. After defining a proper notion of solution and recalling a related existence result, we show that from any initial datum emanates at least one solution which gains further regularity for $t>0$. Such "regularizing solutions" constitute a semiflow $S$ for…
In this work we consider the non local evolution equation with time-dependent terms which arises in models of phase separation in $\mathbb{R}^N$ \[ \partial_t u=- u + g \left(β(J*u) +βh(t,u)\right) \] under some restrictions on $h$, growth restrictions on the nonlinear term $g$ and $β>1$. We prove, under suitable assumptions, existence, regularity and upper-semicontinuity of pullback attractors with respect to functional parameter $h(t)$ in some weighted spaces.
We investigate the long-term behavior, as a certain regularization parameter vanishes, of the three-dimensional Navier-Stokes-Voigt model of a viscoelastic incompressible fluid. We prove the existence of global and exponential attractors of optimal regularity. We then derive explicit upper bounds for the dimension of these attractors in terms of the three-dimensional Grashof number and the regularization parameter. Finally, we also prove convergence of the (strong) global attractor of the 3D Navier-Stokes-Voigt model to the (weak) global attractor of the 3D Navier-Stokes equation. Our analysis…
Under consideration is the hyperbolic relaxation of a semilinear reaction-diffusion equation on a bounded domain, subject to a dynamic boundary condition. We also consider the limit parabolic problem with the same dynamic boundary condition. Each problem is well-posed in a suitable phase space where the global weak solutions generate a Lipschitz continuous semiflow which admits a bounded absorbing set. We prove the existence of a family of global attractors of optimal regularity. After fitting both problems into a common framework, a proof of the upper-semicontinuity of the family of global at…
This paper concerns the tempered pullback dynamics of 2D incompressible non-autonomous Navier-Stokes equation with non-homogeneous boundary condition on Lipschitz-like domain. With the presence of a time-dependent external force f(t) which only needs to be pullback translation bounded, we establish the existence of a minimal pullback attractor with respect to a universe of tempered sets for the corresponding non-autonomous dynamical system. We then give estimate on the finite fractal dimension of the attractor based on trace formula. Under the additional assumption that the external force is t…
We study the surface regularity of compact sets $G \subset R^n$ which is equal to the supremum of numbers $s\ge 0$ such that the measure of the set $G_{\varepsilon}\setminus G$ does not exceed $C\varepsilon^{s}, \varepsilon > 0$, where $G_{\varepsilon}$ denotes the $\varepsilon$-neighbourhood of~$G$. The surface dimension is by definition the difference between~$n$ and the surface regularity. Those values provide a natural characterisation of regularity for sets of positive measure. We show that for self-affine attractors and tiles those characteristics are explicitly computable and find them …
Under fairly general assumptions, we prove that every compact invariant subset $\mathcal I$ of the semiflow generated by the semilinear damped wave equation εu_{tt}+u_t+β(x)u-\sum_{ij}(a_{ij} (x)u_{x_j})_{x_i}&=f(x,u),&& (t,x)\in[0,+\infty[\timesΩ, u&=0,&&(t,x)\in[0,+\infty[\times\partialΩin $H^1_0(Ω)\times L^2(Ω)$ is in fact bounded in $D(\mathbf A)\times H^1_0(Ω)$. Here $Ω$ is an arbitrary, possibly unbounded, domain in $\R^3$, $\mathbf A u=β(x)u-\sum_{ij}(a_{ij}(x)u_{x_j})_{x_i}$ is a positive selfadjoint elliptic operator and $f(x,u)$ is a nonlinearity of critical growth. The nonlinearity …
This paper is concerned with the long-time behavior of solutions for the three dimensional globally modified Navier-Stokes equations in a three-dimensional bounded domain. We prove the existence of a global attractor $\mathcal{A}_0$ in $H$ and investigate the regularity of the global attractors by proving that $\mathcal{A}_0=\mathcal{A}$ established in \cite{ct}, which implies the asymptotic smoothing effect of solutions for the three dimensional globally modified Navier-Stokes equations in the sense that the solutions will eventually become more regular than the initial data. Furthermore, we …
Under consideration is the damped semilinear wave equation \[ u_{tt}+u_t-Δu + u + f(u)=0 \] on a bounded domain $Ω$ in $\mathbb{R}^3$ with a perturbation parameter $\varepsilon>0$ occurring in an acoustic boundary condition, limiting ($\varepsilon=0$) to a Robin boundary condition. With minimal assumptions on the nonlinear term $f$, the existence and uniqueness of global weak solutions is shown for each $\varepsilon\in[0,1]$. Also, the existence of a family of global attractors is shown to exist. After proving a general result concerning the upper-semicontinuity of a one-parameter family of se…
We consider a family of semilinear parabolic problems with nonlinear boundary conditions \[ \left\{ \begin{aligned} u_t(x,t) &=Δu(x,t) -au(x,t) + f(u(x,t)),\ x \in Ω_ε\mbox{ and } t>0\,,\\ \displaystyle\frac{\partial u}{\partial N}(x,t) &=g(u(x,t)),\ x \in \partialΩ_ε\mbox{ and } t>0\,, \end{aligned} \right. \] where $Ω_0 \subset \mathbb{R}^n$ is a smooth (at least $\mathcal{C}^2$) domain , $Ω_ε = h_ε(Ω_0)$ and $h_ε$ is a family of diffeomorphisms converging to the identity in the $\mathcal{C}^1$-norm. Assuming suitable regularity and dissipative conditions for the nonlinearites, we show that …
The occurrence of strange non-chaotic attractors (SNA) in quasiperiodically forced systems has attracted considerable interest over the last two decades, in particular since it provides a rich class of examples for the possibility of complicated dynamics in the absence of chaos. Their existence was discovered in the early 1980's, independently by Herman for quasiperiodic SL(2,R)-cocycles and by Grebogi et al for so-called 'pinched skew products'. However, except for these two particular classes there are still hardly any rigorous results on the topic, despite a large number of numerical studie…
We consider an initial boundary value problem for a quantum version of the Zakharov system arising in plasma physics. We prove the global well-posedness of this problem in some Sobolev type classes and study properties of solutions. This result confirms the conclusion recently made in physical literature concerning the absence of collapse in the quantum Langmuir waves. In the dissipative case the existence of a finite dimensional global attractor is established and regularity properties of this attractor are studied. For this we use the recently developed method of quasi-stability estimates. I…
Dimensions of surface repellers and attractors of non-linear planar IFSs
Fudan University（复旦大学）；University of Toronto（多伦多大学）；Peking University（北京大学）；Great Bay University and Great bay institute for advanced study（大湾区大学和大湾区高等研究院）
This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain $C^k$ lower bounds on the regul…
We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Holder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, unique…
We show that for systems that allow a Young tower construction with polynomially decaying correlations the return times to metric balls are in the limit Poisson distributed. We also provide error terms which are powers of logarithm of the radius. In order to get those uniform rates of convergence the balls centres have to avoid a set whose size is estimated to be of similar order. This result can be applied to non-uniformly hyperbolic maps and to any invariant measure that satisfies a weak regularity condition. In particular it shows that the return times to balls is Poissonian for SRB measure…
We consider a general family of regularized models for incompressible two-phase flows based on the Allen-Cahn formulation in n-dimensional compact Riemannian manifolds for n=2,3. The system we consider consists of a regularized family of Navier-Stokes equations (including the Navier-Stokes-α-like model, the Leray-α model, the Modified Leray-α model, the Simplified Bardina model, the Navier-Stokes-Voight model and the Navier-Stokes model) for the fluid velocity suitably coupled with a convective Allen-Cahn equation for the (phase) order parameter. We give a unified analysis of the entire three-…
