Global attractors for the wave equation with nonlinear damping

By Chunyou Sun, Meiling Yang, Chengkui Zhong. 2006

68 citations recorded by the source corpus.

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+|s|p), where 1⩽p 0) is arbitrary; the nonlinear term f is supposed to satisfy the growth condition |f′(s)|⩽C4(1+|s|q), 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].

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5394611df43d148fd2137547e003eeef44382eab

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Citations in the corpus

Global attractors for wave equations with nonlinear interior damping and critical exponents (2006, methodology): “[1,2] A function w ∈ C([0, T ];H 1 0 ) ∩ C1([0, T ];H) possessing the properties w(0, ·) = w0 and wt(0, ·) = w1 is said to be (S) strong solution to problem (1.”
Asymptotic Behavior of Stochastic Wave Equations with Critical Exponents on R^3 (2008, background): “Particularly, the existence of these attractors was proved in [2, 3, 4, 11, 18, 22, 32, 33] for the deterministic equation defined in bounded domains with critical exponents, and in [16, 17, 27, 28, 29] for the equation defined on unbounded domains with critical or supercritical exponents.”
Kernel sections and uniform attractors of multi-valued semiprocesses☆ (2007, methodology): “Similar to what u done in [16,28], let u = v + w, we decompose Eq.”
Long-time Behavior of Second Order Evolution Equations With Nonlinear Damping (2008, background): “We also mention the recent paper [99] where subcritical (p < 5) case was considered.”