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In this note, we will review our recent work on the asymptotic behaviors of nonlinear Klein-Gordon equation with damping terms and LandauLifschitz flows from Eucliedean spaces and hyperbolic spaces. By the method of concentration-compactness attractors, we prove that the global bounded solution will decouple into a finite number of equilibrium points with different shifts from the origin. For the Landau-Lifschitz flow from Euclidean spaces, we prove that the solution with energy below 4π will converge to some constant map in the energy space. While for the Landau-Lifschitz flow from two dimensional spaces, the solution will converge to some harmonic map. 1. Damped Klein-Gordon equation There are a lot of works devoted to the study of long time behaviour for nonlinear dispersive equations which are representative of Schrödinger equation, wave equations, , Klein-Gordon equations and KdV equations, ect. It’s widely bevlieved that for “generic” large global solutions, the evolution asymptotically decouples into the superposition of divergent solitons, a free radiation term, and an error term tending to zero as time goes to infinity. This is the so called soliton resolution conjecture. For more expression and history, see A. Soffer [27]. The soliton resolution conjecture has been verified for one dimensional cubic Schrödinger equation and KdV equation because they are completely integrable. T. Duyckaerts, C. Kenig, and F. Merle [7] first made a breakthrough for non-integrable dispersive equations. For radial data to three dimensional focusing energy-critical wave equations, they proved the solution with bounded trajectory is in fact a superposition of a finite number of rescalings of the ground state plus a radiation term which is asymptotically a free wave. One of the key ingredient of their arguments is the novel tool, called “channels of energy” introduced by [7] [8]. The method developed by them has been applied to many other situations, such as [4] [5] [15] [17] [16] for wave maps, [6] [10] [13] for semilinear wave equations. By a weak version of outer energy inequality, the soliton resolution along a sequence of times was proved by R. Cote, C. Kenig, A. Lawrie and W. Schlag [6] for four dimensional critical wave equations in radial case, by R. Cote [3] for equivariant wave maps, and by H. Jia, and C. Kenig [14] for semilinear wave equations, wave maps. Since the soliton resolution is so hard for classical dispersive equations, it’s natural to consider some simpler cases: dispersive equations with damping or dissipated terms. Consider the damped Klein-Gordon equation { utt + 2αut −∆u+ u− |u|p−1u = 0, (u(0), ∂tu(0)) = (u0, u1) ∈ H, (DKG) where H = H1(Rd)× L(R), α ≥ 0 and 1 < p < d+2 d−2 . The energy is given by

Asymptotic behaviors for nonlinear dispersive equations with damping or dissipative terms