Long-Time Behaviour of Solutions for Autonomous Evolution Hemivariational Inequality with Multidimensional “Reaction-Displacement” Law

and Applied Analysis 3 an evolution triple V ⊂ H ⊂ V ∗ see, e.g., 8 with dense and compact embeddings. We denote by 〈·, ·〉V the duality of V and its dual V ∗, by ‖ · ‖V ∗ the norm in V ∗. We have 〈u, v〉V u, v H for all u ∈ H and v ∈ V . We admit the following hypotheses. H C the linear symmetric viscosity operator C : Ω×Yd → Yd satisfies the Carathéodory condition i.e., C ·, ε is measurable onΩ for all ε ∈ Yd, and C x, · is continuous on Yd for a.e. x ∈ Ω and C x, ε : ε ≥ C2‖ε‖Yd ∀ε ∈ Yd, a.e. x ∈ Ω with C2 > 0. 1.9 H E the elasticity operator E : Ω×Yd → Yd is of the form E x, ε E x ε Hooke’s law with a symmetric elasticity tensor E ∈ L∞ Ω , that is, E gijkl , i, j, k, l 1, . . . , d with gijkl gjikl glkij ∈ L∞ Ω . Moreover, E x, ε : ε ≥ C3‖ε‖Yd ∀ε ∈ Yd, a.e. x ∈ Ω with C3 > 0. 1.10 H j j : Ω × R → R is a function such that i j ·, ξ is measurable for all ξ ∈ R and j ·, 0 ∈ L1 Ω ; ii j x, · is locally Lipschitz and regular 5 for all x ∈ Ω; iii ‖η‖ ≤ C4 1 ‖ξ‖ for all η ∈ ∂j x, ξ , x ∈ Ω with C4 > 0; iv j0 x, ξ;−ξ ≤ C5 1 ‖ξ‖ for all ξ ∈ R, x ∈ Ω, with C5 ≥ 0, where j0 x, ξ;η is the directional derivative of j x, · at the point ξ ∈ R in the direction η ∈ R. H f f1 ∈ V ∗, g0 ∈ L2 ΓN ;R , u0 ∈ V and u1 ∈ H. Next we need the spacesV L2 τ, T ;V , Ĥ L2 τ, T ;H , andW {w ∈ V : w′ ∈ V∗}, where the time derivative involved in the definition ofW is understood in the sense of vectorvalued distributions, −∞ < τ < T < ∞. Endowed with the norm ‖v‖W ‖v‖V ‖v′‖V∗ , the space W becomes a separable reflexive Banach space. We also have W ⊂ V ⊂ Ĥ ⊂ V∗. The duality for the pair V,V∗ is denoted by 〈z,w〉V ∫T τ 〈z s , w s 〉V ds. It is well known cf. 8 that the embedding W ⊂ C τ, T ;H and {w ∈ V : w′ ∈ W} ⊂ C τ, T ;V are continuous. Next we define g ∈ V ∗ by 〈 g, v 〉 V 〈 f1, v 〉 V 〈 g0, v 〉 L2 ΓN ;Rd for v ∈ V. 1.11 Taking into account the condition 1.6 , we obtain the following variational formulation of our problem: 〈 u′′ t , v 〉 V σ t , ε v H ∫ Ω j0 x, u t ;v dx ≥ g, vV ∀v ∈ V, a.e. t ∈ 0, ∞ , σ t Cεu′ t )) E ε u t for a.e. t ∈ 0, ∞ , u 0 u0, u′ 0 u1. 1.12 4 Abstract and Applied Analysis We define the operators A : V → V ∗ and B : V → V ∗ by 〈A u , v〉V C x, ε u , ε v H for u, v ∈ V, 〈Bu, v〉V E x, ε u , ε v H for u, v ∈ V. 1.13 Obviously, the bilinear forms 1.13 are symmetric, continuous and coercive. Let us introduce the functional J : L2 Ω;R → R defined by