Wave operators to a quadratic nonlinear Klein–Gordon equation in two space dimensions revisited

Autor: Satoshi Tonegawa, Nakao Hayashi, Pavel I. Naumkin
Rok vydání: 2011
Předmět:
Zdroj: Zeitschrift für angewandte Mathematik und Physik. 63:655-673
ISSN: 1420-9039
0044-2275
DOI: 10.1007/s00033-011-0183-7
Popis: We continue to study the existence of the wave operators for the nonlinear Klein–Gordon equation with quadratic nonlinearity in two space dimensions \({\left(\partial_{t}^{2}-\Delta+m^{2}\right) u=\lambda u^{2},\left( t,x\right) \in\mathbf{R}\times\mathbf{R}^{2}}\). We prove that if $$u_{1}^{+}\in\mathbf{H}^{\frac{3}{2}+3\gamma,1}\left( \mathbf{R}^{2}\right),\text{ }u_{2}^{+}\in\mathbf{H}^{\frac{1}{2}+3\gamma,1}\left( \mathbf{R} ^{2}\right),$$ where \({\gamma\in\left( 0,\frac{1}{4}\right)}\) and the norm \({\left\Vert u_{1}^{+}\right\Vert_{\mathbf{H}_{1}^{\frac{3}{2}+\gamma}}+\left\Vert u_{2}^{+}\right\Vert_{\mathbf{H}_{1}^{\frac{1}{2}+\gamma}}\leq\rho,}\) then there exist ρ > 0 and T > 1 such that the nonlinear Klein–Gordon equation has a unique global solution \({u\in\mathbf{C}\left( \left[ T,\infty\right) ;\mathbf{H}^{\frac{1}{2}}\left( \mathbf{R}^{2}\right) \right) }\) satisfying the asymptotics $$\left\Vert u\left( t\right) -u_{0} \left( t\right) \right\Vert _{\mathbf{H}^{\frac{1}{2}}} \leq Ct^{-\frac{1}{2}-\gamma}$$ for all t > T, where u0 denotes the solution of the free Klein–Gordon equation.
Databáze: OpenAIRE
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