Longtime behavior of the semilinear wave equation with gentle dissipation

Abstract

The paper investigates the well-posedness and longtime dynamics of the semilinear wave equation with gentle dissipation utt − u + γ(−) α ut + f (u) = g(x), with α ∈ (0, 1/2). The main results are concerned with the relationships among the growth exponent p of nonlinearity f (u) and the wellposedness and longtime behavior of solutions of the equation. We show that (i) the well-posedness and longtime dynamics of the equation are of characters of parabolic equations as 1 ≤ p < p * ≡ N +4α (N −2) + ; (ii) the subclass G of limit solutions has a weak global attractor as p * ≤ p < p * * ≡ N +2 N −2 (N ≥ 3).

The existence of time-dependent attractor for wave equation with fractional damping and lower regular forcing term

“…Equation (1.1) arises as an evolutionary mathematical model in various systems for the relevant physical application including electrodynamics, quantum mechanics, nonlinear elasticity etc.. The asymptotic behavior of (1.1) has been investigated extensively by many authors (see for example [1,2,5,13,3,20,7,11,14]).…”

“…He found the critical exponent p = N +2α N −2 (N ≥ 3) of nonlinearity f (u), and established the existences of global and exponential attractors in natural energy space. Recently, in [20] the authors considered equation (1.1) as 1 ≤ p < p * ≡ N +4α N −2 (0 < α < 1 2 ) and ε(t) = 1. When ε(t) is a positive decreasing function and vanishes at positive intinity, the problem (1.1) becomes more complex and interesting, one of the reasons is that the dynamical system associated with (1.1) is still understand under the nonautonomous framework even through the forcing term is not dependent on the time t. In [10], for these problems, Plinio, Duane and Temam described the solution operators, which still was called a process, as a family of maps U (t, τ ) : X τ → X t , t ≥ τ ∈ R, acting on a time-dependent family of spaces X t , that is, the norm of the original linear space depended on the time explicitly.…”

Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness

“…[6,34]), the subclass LS of limit solutions (cf. [10,32,33]), the subclass of Shatah-Struwe solutions (cf. [22,27,28,29]), and so on (cf.…”

Attractors and their stability on Boussinesq type equations with gentle dissipation

“…[13,14,20,36]) and that of the semilinear wave equation with structural damping or gentle dissipation (cf. [24,25,34,35]). And there have been extensive studies on the stability of exponential attractors when the perturbations are some coefficients of the evolution equations (cf.…”