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Focusing and absorption of nonlinear oscillations. (English) Zbl 0815.35071

The Cauchy problem for the dissipative wave equation \[ u_{tt} - \Delta u + u | u |^ p = 0, \quad t > 0, \quad x \in \mathbb{R}^ d \] with fast oscillating initial functions is considered. The problem is a passage of the fast oscillating wave through a focal point, where the wave amplitude is increasing that is predicted by the geometric optics method. The main result is as follows: The oscillations which may be present in the initial data do not survive a passage through the focus, if \((d-1)p \geq 2\).
Reviewer: L.Kalyakin (Ufa)

MSC:

35L70 Second-order nonlinear hyperbolic equations
78A05 Geometric optics
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