Ann. Henri Poincar´ e Online First c  2016 Springer International Publishing DOI 10.1007/s00023-016-0512-7 Annales Henri Poincar´ e Fourth-Order Damped Wave Equation with Exponential Growth Nonlinearity Tarek Saanouni Abstract. The initial boundary value problem for a damped fourth-order wave equation with exponential growth nonlinearity is investigated in four space dimensions. In the defocusing case, global well-posedness, scattering and exponential decay are established. In the focusing sign, existence of ground state, instability of standing waves and blow-up results are obtained. 1. Introduction This paper is concerned with the initial boundary value problem for a damped nonlinear wave equation ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ¨ u +Δ 2 u + u + ωΔ 2 ˙ u + μ ˙ u = ǫf (u); (u, ˙ u) |t=0 =(u 0 ,u 1 ); u |∂Ω = |∇u |∂Ω | =0. (1.1) Here and hereafter (μ, ω) ∈ R 2 + , ǫ ∈ {±1},Ω ⊂ R 4 is a bounded smooth domain and u(t, x): R + × Ω → R. The nonlinearity f is a regular function satisfying f (0) = 0 with an exponential growth to fix later. Equations of type (1.1) describe a class of essential nonlinear evolution equations appearing in the elasto-plastic-microstructure models. They describe the longitudinal motion of an elasto-plastic bar and the anti-plane shearing [4]. Any solution to (1.1) formally satisfies decay of the energy ˙ E(t)= −2  ω‖Δ˙ u‖ 2 L 2 + μ‖ ˙ u‖ 2 L 2  ; E(t) := E(u(t), ˙ u(t)) :=  Ω  ˙ u 2 (t)+ |Δu(t)| 2 + u 2 (t) − 2ǫF (u(t))  dx, T. Saanouni is grateful to the Laboratory of PDE and Applications at the Faculty of Sciences of Tunis.