Long-time behavior of a semilinear wave equation with memory

Abstract

In this paper we study the long-time dynamics of the semilinear viscoelastic equationwith Dirichlet boundary condition. The functions f = f (u) and h = h(x) represent forcing terms and the kernel function μ ≥ 0 is assumed to decay exponentially. Then, by exploring only the dissipation given by the memory term, we establish the existence of a global attractor to the corresponding dynamical system. MSC: 35L71; 35B41; 74D99

Regular Global Attractors for Wave Equations with Degenerate Memory

“…(2.9) The condition (2.9) appears in many recent works on semilinear wave equations with memory (e.g. [13]) and the strongly damped wave equation (this condition refers to the subcritical setting of those problems), see for example [2,3,12,21,25,28,29]. By (2.8) we find that for some α ∈ (0, λ 1 ), there exists ρ f > 0 so that, for all s ∈ R, there hold…”

“…We mention some other works concerning semilinear wave equations with memory. On the asymptotic behavior of solutions (in the sense of global attractors) see [9,10,13,27,28,29], and on rates of decay of solutions one can also see [24,30,31].…”

On the Global Nonexistence of a Solution for Wave Equations with Nonlinear Memory Term

“…The question of the well-posedness and spatial regularity of the problem was treated owing to the the theory of evolution process and sectorial operators. Models involving a linear kernel are not new and arise in heat conduction and linear viscoelasticity theory; we mention the works [10][11][12][13][14][15][16][17] and references therein.…”