Weakly Compact Uniform Attractor for the Nonautonomous Long-Short Wave Equations

and Applied Analysis 3 Remark 5. Assumption (13) holds if the system has a unique solution. For brevity, we rewrite system (2)∼(5) in the vector form by introducing W(x, t) = (u(x, t), n(x, t)) and Y(x, t) = (a(x, t), b(x, t)). We denote by E 0 = H 2 ⋂H 1 0 (Ω) × H 1 0 (Ω) the space of vector functionsW(x, t) = (u(x, t), n(x, t)) with norm E0 = {‖u‖ 2 H 2 + ‖n‖ 2 H 1} 1/2 . (14) Similarly, we denote by ∑ 0 the space of Y(x, t) with norm ‖Y‖∑ 0 = {‖a‖ 2 H 1 + ‖b‖ 2 H 1} 1/2 . (15) Then system (2)∼(5) can be considered as ∂ t W = AW + σ (t) , W| t=τ = (u τ , n τ ) = W τ ,