Klein-Gordon equation with critical nonlinearity and inhomogeneous Dirichlet boundary conditions

Abstract

We study the initial-boundary value problem for the cubic nonlinear Klein-Gordon equation \[ \Bigg \{ \begin{array} [c]{c} v_{tt}+v-v_{xx}=F ( v ) ,\text{ } ( t,x ) \in \mathbb{R}^{+}\times\mathbb{R}^{+}\mathbf{,}\\ v ( 0,x ) =v_{0}(x),v_{t} ( 0,x ) =v_{1}(x),x\in \mathbb{R}^{+}{\mathbf{,}}\\ v ( t,0 ) =h(t),t\in\mathbb{R}^{+} \end{array} \] where \[ F ( v ) :=\sum_{\alpha+\beta+\gamma=3}C_{\alpha,\beta,\gamma } ( i\partial_{t}v ) ^{\alpha} ( -i\partial_{x}v ) ^{\beta}v^{\gamma}, \] with real constants $C_{\alpha,\beta,\gamma},$ with inhomogeneous Dirichlet boundary conditions. We prove the global in time existence of solutions of IBV problem for cubic Klein-Gordon equations with inhomogeneous Dirichlet boundary conditions. We obtain the asymptotic behavior of the solution. Our approach is based on the estimates of the integral equation in the Sobolev spaces. We propose a new method of the decomposition of the critical cubic nonlinearity, into a resonant, nonresonant and remainder terms, in order to obtain the smoothness of the solutions.