Nonlinear small data scattering for the wave and Klein-Gordon equation. (English) Zbl 0538.35063

Pecher, Hartmut

Math. Z. 185, 261-270 (1984).

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

Keywords:

existence; uniqueness; scattering operator; Cauchy problem; nonlinear wave equation; nonlinear Klein-Gordon equation

References:

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