Elastic waves inside and on the surface of a half-space

Global attractors for a system of elasticity with small delays

“…We can easily check the equivalence between this norm and the usual one of (H 1 0 (Ω)) 3 . Moreover, it satisfies…”

“…2 For u, v ∈ L 2 (Ω), ⟨u, v⟩ denotes the usual inner-product. This same notation will be used for vectors u, v ∈ (L 2 (Ω)) 3 , that is,…”

“…Analogously, the inner-product of H 1 0 (Ω) is denoted by ⟨∇•, ∇•⟩, as well as, the inner-product of (H 1 0 (Ω)) 3 , that is,…”

“…By classical operator theory fractional powers of −Δ e are allowed since −Δ e is positive and self-adjoint in (L 2 (Ω)) 3 . In particular,…”

“…(i) Elasticity systems have been successfully used in Seismology, modeling seismic waves composed of P waves and S waves, for example, previous works [3][4][5] In addition, delay effects come naturally in these models, for example, previous works 3,6-8 (ii) In the case = 0, that is, without delay, the system (5) has been studied in, 6 where existence of global attractors was established allowing nonlinearities of critical growth. On the other hand, with a fixed delay as in (2), the problem was considered in a previous study.…”