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Scattering for Radial, Semi-Linear, Super-Critical Wave Equations with Bounded Critical Norm

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Abstract

In this paper we study the focusing cubic wave equation in 1 + 5 dimensions with radial initial data as well as the one-equivariant wave maps equation in 1+3 dimensions with the model target manifolds \({\mathbb{S}^3}\) and \({\mathbb{H}^3}\). In both cases the scaling for the equation leaves the \({\dot{H}^{\frac{3}{2}} \times \dot{H}^{\frac{1}{2}}}\)-norm of the solution invariant, which means that the equation is super-critical with respect to the conserved energy. Here we prove a conditional scattering result: if the critical norm of the solution stays bounded on its maximal time of existence, then the solution is global in time and scatters to free waves as \({t \to \pm \infty}\). The methods in this paper also apply to all supercritical power-type nonlinearities for both the focusing and defocusing radial semi-linear equation in 1+5 dimensions, yielding analogous results.

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Authors and Affiliations

  1. Department of Mathematics, Johns Hopkins University, 404 Krieger Hall, 3400 N. Charles Street, Baltimore, MD, 21218, USA

    Benjamin Dodson

  2. Department of Mathematics, The University of California, Berkeley 970 Evans Hall #3840, Berkeley, CA, 94720, USA

    Andrew Lawrie

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  1. Benjamin Dodson
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  2. Andrew Lawrie
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Correspondence to Andrew Lawrie.

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Communicated by V. Šverák

Support of the National Science Foundation, DMS-1103914 for the first author, and DMS-1302782 for the second author, is gratefully acknowledged.

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Dodson, B., Lawrie, A. Scattering for Radial, Semi-Linear, Super-Critical Wave Equations with Bounded Critical Norm. Arch Rational Mech Anal 218, 1459–1529 (2015). https://doi.org/10.1007/s00205-015-0886-6

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