The attractors for the nonhomogeneous nonautonomous Navier–Stokes equations

AbstractIn this paper, we consider the attractors for the two-dimensional nonautonomous Navier–Stokes equations in nonsmooth bounded domain Ω with nonhomogeneous boundary condition u=φ on ∂Ω. Assuming f=f(x,t)∈Lloc2((0,T);D(Aα4)), which is translation compact and φ∈L∞(∂Ω), we establish the existence of the uniform attractor in L2(Ω) and D(A14)

Dynamics of non-autonomous reaction-diffusion equations in locally uniform spaces

In this paper, we first prove the well-posedness for thenon-autonomous reaction-diffusion equations on the entire space $\R^N$ in thesetting of locally uniform spaces with singular initial data. Thenwe study the asymptotic behavior of solutions of such equation andshow the existence of$(H^1,q_U(\R^N),H^1,q_\phi(\R^N))$-uniform(w.r.t.g\in\mcH_L^q_U(\R^N)(g_0)) attractor\mcA_\mcH_L^q_U(\R^N)(g_0) with locally uniform externalforces being translation uniform bounded but not translation compactin $L_b^p(\R;L^q_U(\R^N))$. We also obtain the uniform attracting propertyin the stronger topology

Well-posedness and strong attractors for a beam model with degenerate nonlocal strong damping

This paper is devoted to initial-boundary value problem of an extensible beam equation with degenerate nonlocal energy damping in $\Omega\subset\mathbb{R}^n$: $u_{tt}-\kappa\Delta u+\Delta^2u-\gamma(\Vert \Delta u\Vert^2+\Vert u_t\Vert^2)^q\Delta u_t+f(u)=0$. We prove the global existence and uniqueness of weak solutions, which gives a positive answer to an open question in [24]. Moreover, we establish the existence of a strong attractor for the corresponding weak solution semigroup, where the ``strong" means that the compactness and attractiveness of the attractor are in the topology of a stronger space $\mathcal{H}_{\frac{1}{q}}$.Comment: 27 page

Global attractors for p-Laplacian equation

AbstractThe existence of a (L2(Ω),W01,p(Ω)∩Lq(Ω))-global attractor is proved for the p-Laplacian equation ut−div(|∇u|p−2∇u)+f(u)=g on a bounded domain Ω⊂Rn (n⩾3) with Dirichlet boundary condition, where p⩾2. The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c1|u|q−k⩽f(u)u⩽c2|u|q+k and f′(u)⩾−l, where q⩾2 is arbitrary. There is no other restriction on p and q. The asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method, called asymptotic a priori estimate

On the dynamics of a class of nonclassical parabolic equations

AbstractWe consider the first initial and boundary value problem of nonclassical parabolic equations ut−μΔut−Δu+g(u)=f(x) on a bounded domain Ω, where μ∈[0,1]. First, we establish some uniform decay estimates for the solutions of the problem which are independent of the parameter μ. Then we prove the continuity of solutions as μ→0. Finally we show that the problem has a unique global attractor Aμ in V2=H2(Ω)∩H01(Ω) in the topology of H2(Ω); moreover, Aμ→A0 in the sense of Hausdorff semidistance in H01(Ω) as μ goes to 0

Existence of φ\varphi-attractor and estimate of their attractive velocity for infinite-dimensional dynamical systems

This paper is devoted to the quantitative study of the attractive velocity of generalized attractors for infinite-dimensional dynamical systems. We introduce the notion of~$\varphi$-attractor whose attractive speed is characterized by a general non-negative decay function~$\varphi$, and prove that~$\varphi$-decay with respect to noncompactness measure is a sufficient condition for a dissipitive system to have a~$\varphi$-attractor. Furthermore, several criteria for~$\varphi$-decay with respect to noncompactness measure are provided. Finally, as an application, we establish the existence of a generalized exponential attractor and the specific estimate of its attractive velocity for a semilinear wave equation with a critical nonlinearity.Comment: arXiv admin note: substantial text overlap with arXiv:2108.0741

Finite-dimensionality of attractors for wave equations with degenerate nonlocal damping

In this paper we study the fractal dimension of global attractors for a class of wave equations with (single-point) degenerate nonlocal damping. Both the equation and its linearization degenerate into linear wave equations at the degenerate point and the usual approaches to bound the dimension of the entirety of attractors do not work directly. Instead, we develop a new process concerning the dimension near the degenerate point individually and show the finite dimensionality of the attractor.Comment: 33 page

Burst phase distribution of SGR J1935+2154 based on Insight-HXMT

On April 27, 2020, the soft gamma ray repeater SGR J1935+2154 entered its intense outburst episode again. Insight-HXMT carried out about one month observation of the source. A total number of 75 bursts were detected during this activity episode by Insight-HXMT, and persistent emission data were also accumulated. We report on the spin period search result and the phase distribution of burst start times and burst photon arrival times of the Insight-HXMT high energy detectors and Fermi Gamma-ray Burst Monitor (GBM). We find that the distribution of burst start times is uniform within its spin phase for both Insight-HXMT and Fermi-GBM observations, whereas the phase distribution of burst photons is related to the type of a burst's energy spectrum. The bursts with the same spectrum have different distribution characteristics in the initial and decay episodes for the activity of magnetar SGR J1935+2154.Comment: 12 pages, 9 figure