We study estimates of lifespan and blow-up rates of solutions for the Cauchy problem of the wave equation with a time-dependent damping and a power-type nonlinearity. When the damping acts on the solutions effectively, and the nonlinearity belongs to the subcritical case, we show the sharp lifespan estimates and the blow-up rates of solutions. The upper estimates are proved by an ODE argument, and the lower estimates are given by a method of scaling variables.

This paper is concerned with the existence, stability and almost periodicity of global bounded solutions of linear and semilinear parabolic equations. We give a rather complete characterization to the linear equation u_t − Δu = λu + f(x,t), in term of the parameter λ, of whether or not the global bounded solutions exist. Also, we obtain a complete characterization on the stability of global bounded solutions.

We study conditions for the abstract periodic linear functional differential equation = Ax + F(t)x_t + f(t) to have almost periodic with the same structure of frequencies as f. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the frequencies of the forcing term f. The obtained results extend recent results on the subject. A discussion on how the results could be extended to the case when A depends on t is given.

We investigate the long-time asymptotics for the focusing integrable discrete nonlinear Schrödinger equation. Under generic assumptions on the initial value, the solution is asymptotically a sum of 1-solitons. We find different phase shift formulas in different regions. Along rays away from solitons, the behavior of the solution is decaying oscillation. This is one way of stating the soliton resolution conjecture. The proof is based on the nonlinear steepest descent method.

In this paper, we study the spectral structure of periodic Schrödinger operators on a generalization of carbon nanotubes from the point of view of the quantum graphs. Since there exist chemical double bonds between carbon atoms on a hexagonal lattice with a cylindrical structure corresponding to carbon nanotubes, we study the spectral structure of periodic Schrödinger operators on zigzag nanotubes with multiple bonds of atoms in this paper. Utilizing the Floquet-Bloch theory, the spectrum of the operator consists of the absolutely continuous spectral bands and the flat band. We study the relationship between the number of the chemical bonds and the existence of spectral gaps.