35 research outputs found
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 in thesetting of locally uniform spaces with singular initial data. Thenwe study the asymptotic behavior of solutions of such equation andshow the existence of-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 . 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
: . 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 .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 -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~-attractor whose attractive speed is characterized by a
general non-negative decay function~, and prove that~-decay
with respect to noncompactness measure is a sufficient condition for a
dissipitive system to have a~-attractor. Furthermore, several criteria
for~-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
On Global Attractors for a Class of Reaction-Diffusion Equations on Unbounded Domains with Some Strongly Nonlinear Weighted Term
We consider the existence and properties of the global attractor for a class of reaction-diffusion equation ∂u/∂t-Δu-u+κ(x)|u|p-2u+f(u)=0,  in  Rn×R+;  u(x,0)=u0(x),  in  Rn. Under some suitable assumptions, we first prove that the problem has a global attractor A in L2(Rn). Then, by using the Z2-index theory, we verify that A is an infinite dimensional set and it contains infinite distinct pairs of equilibrium points
Existence of strong solutions and global attractors for the coupled suspension bridge equations
AbstractIn this paper, we show the existence of the strong solutions for the coupled suspension bridge equations. Furthermore, existence of the strong global attractors is investigated using a new semigroup scheme. Since the solutions of the coupled equation have no higher regularity and the semigroup associated with the solutions is not continuous in the strong Hilbert space, the results are new and appear to be optimal