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On almost global existence for nonrelativistic wave equations in 3D. (English) Zbl 0867.35064

This work establishes the almost global existence of solutions for three-dimensional, quadratically nonlinear wave equations, with the use of only the classical invariance of the equations under translation, rotations and changes of scale. Previous proofs utilized, in addition, either Lorentz invariance or direct estimation of the fundamental solution of the linear wave equation. The proof relies on generalized energy estimates and a new decay estimate. Consider a linear hyperbolic partial differential operator of the form \(Pu\equiv\partial^2_t u-Au\) for certain second-order linear elliptic operators \(A\). It is shown that, by manipulating differential operators, \(|Au|\), \(|\nabla\partial_tu|\) can be controlled pointwise by a decaying factor times derivatives of \(u\) with respect to the generators of the invariants plus a term involving \(Pu\).
Reviewer: L.Vazquez (Madrid)

MSC:

35L70 Second-order nonlinear hyperbolic equations
74B20 Nonlinear elasticity
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