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Nonlinear small data scattering for the wave and Klein-Gordon equation. (English) Zbl 0538.35063
The author considers the pair of partial differential equations $$(1)\quad u_{tt}+Au+f(u)=0,\quad(2)\quad u_{tt}+Au=0$$ and arbitrarily given initial data $$(\phi^-,\psi^-)$$. Here A denotes the operator $$m^ 2- \sum^{n}_{j=1}\partial^ 2/\partial x^ 2_ j$$ and $$f(u)=\lambda | u|^{\rho -1}u$$ with m, $$\lambda\in {\mathbb{R}}$$ and $$\rho>1$$. The main objective is to study the existence and uniqueness problem associated with the operator S (scattering operator) which map the initial data $$(\phi^-,\psi^-)$$ into another initial data $$(\phi^+,\psi^+)$$ having the following properties: Let $$u^-_ 0$$ and $$u^+_ 0$$ denote the solution of the Cauchy problem for (2) with data $$(\phi^-,\psi^-)$$ and $$(\phi^+,\psi^+)$$, respectively. Then, there is a solution u of (1), such that $$\| u-u^-_ 0\|_ e\to 0$$ as $$t\to -\infty$$ while $$\| u-u^+_ 0\|_ e\to 0$$ as $$t\to +\infty$$. Here $$\|.\|_ e$$ denotes the energy norm, defined by $$\| v\|^ 2_ e=frac{1}{2}\{\| A^{\frac{1}{2}}v\|^ 2+\| v_ t\|^ 2\}.$$ He gives sufficient conditions under which S exists and is unique. The results related to the cases $$m=0$$ (nonlinear wave equation) and $$m\neq 0$$ (nonlinear Klein-Gordon equation) are stated in different theorems.
We remark that some factors (2$$\pi)$$ an (-1) are omitted in some places. For example, the signs before the integrals cited in theorems 2, 3 and 4 should be (-) in order that u(t) and $$u^+_ 0(t)$$ could satisfy (1) or (2).
Reviewer: M.Idemen

##### MSC:
 35P25 Scattering theory for PDEs 35L70 Second-order nonlinear hyperbolic equations 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) 58D25 Equations in function spaces; evolution equations
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##### References:
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