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Scattering theory in the energy space for a class of nonlinear wave equations. (English) Zbl 0698.35112

The authors study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the nonlinear wave equation \[ \square \phi +f(\phi)=0\quad in\quad {\mathbb{R}}^ n,\quad n\geq 3. \] They prove the existence of the wave operators, asymptotic completeness for small initial data and, for \(n\geq 4\), asymptotic completeness for arbitrarily large data. The assumptions on f cover the case where f behaves slightly better than a single power \(p=1+4/(n-2),\) both near zero and at infinity.
For instance, they cover the case where (*) \(f(\phi)=\phi g(| \phi |)\) and g is a smooth non-negative function that behaves as (**) \(g(s)=\lambda_ 1s^{p_ 1-1}\) for \(0\leq s\leq a\), \(g(s)=\lambda_ 2s^{p_ 2-1}\) for \(s\geq 1/a\) for some a, \(0<a<1\), with \(0\leq p_ 2- 1<4/(n-2)=p_ 1-1.\)
The proof of asymptotic completeness, on the other hand, requires the existence of some norm that decays integrably at infinity in time for solutions of the free equation \(\square \phi =0\). As a consequence, that proof applies only to the case of space dimension \(n\geq 4\). It requires in addition a reinforcement of the assumption on f which takes the form \[ 0\leq p_ 2-1<4/(n-2)<p_ 1-1 \] in the special case (*), (**).
Reviewer: Y.Ebihara

MSC:

35P25 Scattering theory for PDEs
35L70 Second-order nonlinear hyperbolic equations
35L05 Wave equation
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