Attractors for wave equations with degenerate memory

作者：Cavalcanti M M; Fatori L H; Ma T F^*

来源：Journal of Differential Equations, 2016, 260(1): 56-83.

DOI：10.1016/j.jde.2015.08.050

• Global attractor
• Degenerate viscoelasticity
• Past history memory
• Nonlinear wave equation
• Critical exponent

This paper is concerned with the long-time dynamics of a semilinear wave equation with degenerate is viscoelasticity
utt - Delta u + integral(infinity)(0) g(s)div[a(x)del u(t - s)]ds + f(u) = h(x),
defined in a bounded domain Omega of R-3, with Dirichlet boundary condition and nonlinear forcing f (u) with critical growth. The problem is degenerate in the sense that the function a(x) >= 0 in the memory term is allowed to vanish in a part of Omega. When a (x) does not degenerate and g decays exponentially it is well-known that the corresponding dynamical system has a global attractor without any extra dissipation. In the present work we consider the degenerate case by adding a complementary frictional damping b(x)u(t), which is in a certain sense arbitrarily small, such that a + b > 0 in Omega. Despite that the dissipation is given by two partial damping terms of different nature, none of them necessarily satisfying a geometric control condition, we establish the existence of a global attractor with finite-fractal dimension.

• 出版日期2016-1-5