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Scattering theory for the nonlinear Klein-Gordon equation with Sobolev critical power. (English) Zbl 0933.35166

The author deals with the scattering theory in the energy space for nonlinear Klein-Gordon equations of the following form: \[ (\partial^2_t - \Delta) u + m^2 u + f(u) = 0, \] where \(u = u(t,x)\), \((t,x)\in {\mathbb R}^{1+n}\) with \(n \geq 3\) and \(m>0\). The nonlinear term \(f(.)\) is of the form \(f(u)=|u|^{p-2} u\), with \(p=2^*:= {{2n}\over{n-2}}\), which is the Sobolev critical exponent. Nonradial symmetry is assumed, so that it is not predictable where the concentration may occur in the space. The considered equations have not the homogeneous character, and the finite and infinite time intervals are teated in different ways. The results achieved concern the global a priori estimates, scattering and continuous dependence on the initial data both in the strong topology and in the weak topology.

MSC:

35Q40 PDEs in connection with quantum mechanics
35P25 Scattering theory for PDEs
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