Klein-Gordon equation with critical nonlinearity and inhomogeneous Dirichlet boundary conditions. (Q265174)

The global existence and the asymptotic behavior of the unique solution to the initial boundary value problem for the equation NEWLINE\[NEWLINEv_{tt}-v_{xx}+v=\sum_{j+k+l=3}C_{jkl}(i\partial_tv)^j(-i\partial_xv)^kv^l,\, (t,x)\in \mathbb R^{+}\times \mathbb R^{+}NEWLINE\]NEWLINE for small inhomogeneous Dirichlet boundary conditions and small initial data are proved. The author rewrites the above equation into \((\partial_t+i\left\langle \partial_x\right\rangle)u=(\left\langle \partial_x\right\rangle)^{-1}N(u)\) with \(v=\operatorname{Re} u\) and constructs the Green function for the rewritten problem. He finds asymptotics of the Green function and of its time and space derivatives as \(t\to \infty \), which helps him to find a sharp time decay of the nonlinearity. Using a decomposition of the critical cubic nonlinearity into a resonant, nonresonant and remainder terms, the author obtains smoothness of the solution.