Differential and Integral Equations

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

I.P. Naumkin

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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

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

First available in Project Euclid: 24 November 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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