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Focusing at a point and absorption of nonlinear oscillations


Authors: J.-L. Joly, G. Métivier and J. Rauch
Journal: Trans. Amer. Math. Soc. 347 (1995), 3921-3969
MSC: Primary 35B40; Secondary 35B05, 35C20, 35L70, 35Q99
DOI: https://doi.org/10.1090/S0002-9947-1995-1297533-8
MathSciNet review: 1297533
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Abstract: Several recent papers give rigorous justifications of weakly nonlinear geometric optics. All of them consider oscillating wave trains on domains where focusing phenomena do not exist, either because the space dimension is equal to one, or thanks to a coherence assumption on the phases. This paper is devoted to a study of some nonlinear effects of focusing. In a previous paper, the authors have given a variety of examples which show how focusing in nonlinear equations can spoil even local existence in the sense that the domain of existence shrinks to zero as the wavelength decreases to zero. On the other hand, there are many problems for which global existence is known and in those cases it is natural to ask what happens to oscillations as they pass through a focus. The main goal of this paper is to present such a study for some strongly dissipative semilinear wave equations and spherical wavefronts which focus at the origin. We show that the strongly nonlinear phenomenon which is produced is that oscillations are killed by the simultaneous action of focusing and dissipation. Our study relies on the analysis of Young measures and two-scale Young measures associated to sequences of solutions. The main step is to prove that these measures satisfy appropriate transport equations. Then, their variances are shown to satisfy differential inequalities which imply a propagation result for their support.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1297533-8
Keywords: Oscillation, focusing, dissipation, geometric optics, Young measure, transport equation, independence, absorption, nonlinear waves
Article copyright: © Copyright 1995 American Mathematical Society

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