Neumann problem for the nonlinear Klein-Gordon equation

The Neumann initial-boundary value problem for the nonlinear Klein-Gordon equation (0.1){vtt+v−vxx=μv3,(t,x)∈R+×R+,v(0,x)=v0(x),vt(0,x)=v1(x),x∈R+,(∂xv)(t,0)=h(t),t∈R+, for real μ, v0(x),v1(x) and h(t), is considered. We prove the global well-posedness for the initial-boundary value problem (0.1) and we present a sharp time decay estimate of the solution in the uniform norm. Also we study the asymptotic behavior of the solution to (0.1). We show that the cubic nonlinearity in the Neumann initial-boundary value problem (0.1) is scattering-critical.