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1.
Lu Yang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(12):3876-3883
In this paper, we study the long-time behavior of the reaction-diffusion equation with dynamical boundary condition, where the nonlinear terms f and g satisfy the polynomial growth condition of arbitrary order. Some asymptotic regularity of the solution has been proved. As an application of the asymptotic regularity results, we can not only obtain the existence of a global attractor A in (H1(Ω)∩Lp(Ω))×Lq(Γ) immediately, but also can show further that A attracts every L2(Ω)×L2(Γ)-bounded subset with (H1(Ω)∩Lp+δ(Ω))×Lq+κ(Γ)-norm for any δ,κ∈[0,∞). 相似文献
2.
Based on a new a priori estimate method, so-called asymptotic a priori estimate, the existence of a global attractor is proved for the wave equation utt+kg(ut)−Δu+f(u)=0 on a bounded domain Ω⊂R3 with Dirichlet boundary conditions. The nonlinear damping term g is supposed to satisfy the growth condition C1(|s|−C2)?|g(s)|?C3(1+p|s|), where 1?p<5; the damping parameter is arbitrary; the nonlinear term f is supposed to satisfy the growth condition |f′(s)|?C4(1+q|s|), where q?2. It is remarkable that when 2<p<5, we positively answer an open problem in Chueshov and Lasiecka [I. Chueshov, I. Lasiecka, Long-time behavior of second evolution equations with nonlinear damping, Math. Scuola Norm. Sup. (2004)] and improve the corresponding results in Feireisl [E. Feireisl, Global attractors for damped wave equations with supercritical exponent, J. Differential Equations 116 (1995) 431-447]. 相似文献
3.
Jürgen Sprekels 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(6):3028-3048
We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-called Wentzell boundary condition. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms f,g are real analytic. Moreover, we provide an estimate for the convergence rate. 相似文献
4.
The existence of a pullback attractor is established for a stochastic reaction-diffusion equation on all n-dimensional space. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise. The reaction-diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far-field values of solutions. 相似文献
5.
N. I. Karachalios N. B. Zographopoulos 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(1):11-30
We study a real Ginzburg-Landau equation, in a bounded domain of
with a variable, generally non-smooth diffusion coefficient having a finite number of zeroes. By using the compactness of the embeddings of the weighted Sobolev spaces involved in the functional formulation of the problem, and the associated energy equation, we show the existence of a global attractor. The extension of the main result in the case of an unbounded domain is also discussed, where in addition, the diffusion coefficient has to be unbounded. Some remarks for the case of a complex Ginzburg-Landau equation are given.Received: May 6, 2002; revised: October 3, 2002 相似文献
6.
Nikos I. Karachalios Athanasios N. Yannacopoulos 《Journal of Differential Equations》2005,217(1):88-123
We study the asymptotic behavior of solutions of discrete nonlinear Schrödinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions. Similarities and differences with the continuous counterpart (NLS-partial differential equation) are pointed out. For a dissipative system we prove existence of a global attractor and its stability under finite-dimensional approximations. Similar questions are treated in a weighted phase space. Finally, we propose possible extensions for various types of DNLS equations. 相似文献
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H.G. Rotstein et al. proposed a nonconserved phase-field system characterized by the presence of memory terms both in the heat conduction and
in the order parameter dynamics. These hereditary effects are represented by time convolution integrals whose relaxation kernels
k and h are nonnegative, smooth and decreasing. Rescaling k and h properly, we obtain a system of coupled partial integrodifferential equations depending on two relaxation times ɛ and σ.
When ɛ and σ tend to 0, the formal limiting system is the well-known nonconserved phase-field model proposed by G. Caginalp.
Assuming the exponential decay of the relaxation kernels, the rescaled system, endowed with homogeneous Neumann boundary conditions,
generates a dissipative strongly continuous semigroup Sɛ, σ(t) on a suitable phase space, which accounts for the past histories of the temperature as well as of the order parameter. Our
main result consists in proving the existence of a family of exponential attractors
for Sɛ, σ(t), with ɛ, σ ∈ [0, 1], whose symmetric Hausdorff distance from
tends to 0 in an explicitly controlled way. 相似文献
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First we establish some necessary and sufficient conditions for the existence of exponential attractors by using ω-limit compactness and a measure of non-compactness. Then we provide a new method for proving the existence of exponential attractors. We prove the existence of exponential attractors for reaction–diffusion equations and 2D Navier–Stokes equations as simple applications. 相似文献
12.
Songsong Lu 《Journal of Differential Equations》2006,230(1):196-212
First, the existence and structure of uniform attractors in H is proved for nonautonomous 2D Navier-Stokes equations on bounded domain with a new class of distribution forces, termed normal in (see Definition 3.1), which are translation bounded but not translation compact in . Then, the properties of the kernel section are investigated. Last, the fractal dimension is estimated for the kernel sections of the uniform attractors obtained. 相似文献
13.
In this work we show, for a class of dissipative semilinear parabolic problems, that the global compact attractor varies continuously with respect to parameters in the equations. Applications to a parabolic problem with nonlinear boundary conditions are also obtained. 相似文献
14.
A new approach is established to show that the semigroup {S(t)}_(t≤0) generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L~q(?)(2 ≤q ∞) with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space. 相似文献
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Ahmed Y. Abdallah 《Journal of Differential Equations》2011,251(6):1489-1504
In Abdallah (2008, 2009) [2] and [3], we have investigated the existence of exponential attractors for first and second order autonomous lattice dynamical systems. Within this work, in l2, we carefully study the existence of a uniform exponential attractor for the family of processes associated with an abstract family of first order non-autonomous lattice dynamical systems with quasiperiodic symbols acting on a closed bounded set. 相似文献
17.
Ricardo M.S. Rosa 《Journal of Differential Equations》2006,229(1):257-269
The asymptotic behavior of solutions of the three-dimensional Navier-Stokes equations is considered on bounded smooth domains with no-slip boundary conditions and on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray-Hopf weak solution to its weak ω-limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields with the appropriate boundary conditions) are achieved also in the strong topology. It is proved that the weak ω-limit set is strongly compact and strongly attracts the corresponding solution if and only if all the solutions in the weak ω-limit set are continuous in the strong topology of H. Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented. In this case, it is proved that the weak global attractor is strongly compact and strongly attracts the weak solutions, uniformly with respect to uniformly bounded sets of weak solutions, if and only if all the global weak solutions in the weak global attractor are strongly continuous in H. 相似文献
18.
Taishan Yi 《Journal of Differential Equations》2008,245(11):3376-3388
In this paper, we establish the global attractivity of the positive steady state of the diffusive Nicholson's equation with homogeneous Neumann boundary value under a condition that makes the equation a non-monotone dynamical system. To achieve this, we develop a novel method: combining a dynamical systems argument with maximum principle and some subtle inequalities. 相似文献
19.
In this paper we prove existence of global solutions and (L2(Ω)×L2(Γ),(H1(Ω)∩Lp(Ω))×Lp(Γ))-global attractors for semilinear parabolic equations with dynamic boundary conditions in bounded domains with a smooth boundary, where there is no other restriction on p(≥2). 相似文献
20.
The uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered. The existence of both weak and strong solutions to the system is proven by the Galerkin method. The exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the Riemannian manifold. This equivalent energy function shows particularly that the system is essentially a dissipative system. This result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in [A. Guesmia, A new approach of stabilization of nondissipative distributed systems, SIAM J. Control Optim. 42 (2003) 24-52] where the system is claimed to be nondissipative. 相似文献