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Cauchy problem for the Kuznetsov equation. (English) Zbl 1406.35194

Summary: We consider the Cauchy problem for a model of non-linear acoustic, named the Kuznetsov equation, describing a sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation, for which we prove the global existence in time of regular solutions for sufficiently small initial data, the size of which is specified, and give the corresponding energy estimates. In the inviscid case, we update the known results of John for quasi-linear wave equations, obtaining the well-posedness results for less regular initial data. We obtain, using a priori estimates and a Klainerman inequality, the estimations of the maximal existence time, depending on the space dimension, which are optimal, thanks to the blow-up results of Alinhac. Alinhac’s blow-up results are also confirmed by a \(L^2\)-stability estimate, obtained between a regular and a less regular solutions.

MSC:

35L71 Second-order semilinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35B45 A priori estimates in context of PDEs
35B44 Blow-up in context of PDEs
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