In this paper, we study the beam equation with weak damping in n-dimensional space. In the case of time-dependent periodic external force imposed, we prove the existence and uniqueness of time periodic solutions that have the same period as the time-dependent periodic external force in some suitable function space for all space dimensions n ≥ 1. The proof is based on the spectral analysis for the solution operators and the contraction mapping theorem. Moreover, we show the time asymptotic stability of time periodic solutions by continuous argument.

In this paper, we study the Cauchy problem for the generalized double dispersion equation with structural damping. The equation behaves as the usual diffusion phenomenon over the low frequency domain, while it admits a feature of regularity-loss on the high frequency part. The feature of regularity-loss leads to the weakly dissipative property of the equation. To overcome the weakly dissipative property, the time-weighted energy is introduced, and extra regularity on the initial data is required. Under suitable conditions on the initial data and space dimensions, we prove the global existence and time-decay rates of solutions. The proof is based on the spectral analysis for the solution operators, time-weighted energy, and the contraction mapping theorem. Moreover, we also establish the asymptotic profiles of global solutions involving the nonlinear term for n ≥ 3, ν∈(0,12) and n ≥ 4, ν∈[12,1), respectively.