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Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions. (English) Zbl 1173.35599

Summary: We study asymptotics around the final states of solutions to the nonlinear Klein-Gordon equations with quadratic nonlinearities in two space dimensions $\left({\partial }_{t}^{2}-{\Delta }+{m}^{2}\right)u=\lambda {u}^{2}$, $\left(t,x\right)\in ℝ×{ℝ}^{2}$, where $\lambda \in ℂ$. We prove that if the final states

$\begin{array}{cc}\hfill {u}_{1}^{+}& \in {ℍ}_{\frac{q}{q-1}}^{4-\frac{4}{q}}\left({ℝ}^{2}\right)\cap {ℍ}^{\frac{5}{2},1}\left({ℝ}^{2}\right)\cap {ℍ}_{1}^{2}\left({ℝ}^{2}\right),\hfill \\ \hfill {u}_{2}^{+}& \in {ℍ}_{\frac{q}{q-1}}^{3-\frac{4}{q}}\left({ℝ}^{2}\right)\cap {ℍ}^{\frac{3}{2},1}\left({ℝ}^{2}\right)\cap {ℍ}_{1}^{1}\left({ℝ}^{2}\right),\hfill \end{array}$

and $\parallel {u}_{1}^{+}{\parallel }_{{ℍ}_{1}^{2}}+{\parallel {u}_{2}^{+}\parallel }_{{ℍ}_{1}^{1}}$ is sufficiently small, where $4, then there exists a unique global solution $u\in ℂ\left(\left[T,\infty \right);{𝕃}^{2}\left({ℝ}^{2}\right)\right)$ to the nonlinear Klein-Gordon equations such that $u\left(t\right)$ tends as $t\to \infty$ in the ${𝕃}^{2}$ sense to the solution ${u}_{0}\left(t\right)={u}_{1}^{+}cos\left({\left(i\nabla \right)}_{m}t\right)+\left({\left(i\nabla \right)}_{m}^{-1}{u}_{2}^{+}\right)sin\left({\left(i\nabla \right)}_{m}t\right)$ of the free Klein-Gordon equation.

##### MSC:
 35L70 Nonlinear second-order hyperbolic equations 35B40 Asymptotic behavior of solutions of PDE