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Scattering and the Levandosky-Strauss conjecture for fourth-order nonlinear wave equations. (English) Zbl 1145.35090

This paper studies the scattering theory approach for a certain type of nonlinear fourth order wave equation. The equation is usually referred to as the nonlinear beam equation or the Bretherton equation. The equation does not satisfy either the finite speed propagation or the mass conservation conditions which result in difficult to obtain solutions. The equation finds applications in the study of weak interactions of dispersive waves or of the motion of a clamped plate. The author employs scattering theory to study the behavior of this equation. The main result of the paper is the proof of the Levandosky-Strauss conjecture when the equation is defocussing and for certain large values of the critical exponent for the embedding of the Sobolev space of second differentiable functions into Lebesgue spaces.

MSC:

35L75 Higher-order nonlinear hyperbolic equations
35B33 Critical exponents in context of PDEs
35P25 Scattering theory for PDEs
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