## Differential and Integral Equations

- Differential Integral Equations
- Volume 29, Number 1/2 (2016), 55-92.

### 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.

#### Article information

**Source**

Differential Integral Equations, Volume 29, Number 1/2 (2016), 55-92.

**Dates**

First available in Project Euclid: 24 November 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1448323253

**Mathematical Reviews number (MathSciNet)**

MR3450749

**Zentralblatt MATH identifier**

1349.35246

**Subjects**

Primary: 35M13: Initial-boundary value problems for equations of mixed type 35B40: Asymptotic behavior of solutions 35A01: Existence problems: global existence, local existence, non-existence 35A02: Uniqueness problems: global uniqueness, local uniqueness, non- uniqueness

#### Citation

Naumkin, I.P. Klein-Gordon equation with critical nonlinearity and inhomogeneous Dirichlet boundary conditions. Differential Integral Equations 29 (2016), no. 1/2, 55--92. https://projecteuclid.org/euclid.die/1448323253