Forced oscillations for some nonlinear, weakly dissipative wave equations

Periodic solutions and asymptotic behavior are studied for the equationutt − Δu + β(ut) ε f(t, x)on ℝ+ × Ω;u = 0on ℝ+ × ∂Ω. In Section I, we takeΩ = |0,π|. For some maximal monotone β such thatD(β) and β(R) ≠R, we study the existence of 2π-time periodic solutions. In Section II, we take β = sgn. Then for somef such that|f(t, x)| ⩽ 1 − ɛon ℝ+ × Ω (ɛ > 0), we prove that all solutionsu(t, x)converge whent → ∞to functions independent oftsatisfying stability conditions