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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Scattering for the cubic Klein-Gordon equation in two space dimensions


Authors: Rowan Killip, Betsy Stovall and Monica Visan
Journal: Trans. Amer. Math. Soc. 364 (2012), 1571-1631
MSC (2010): Primary 35L71; Secondary 35Q40
Published electronically: September 15, 2011
MathSciNet review: 2869186
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Abstract: We consider both the defocusing and focusing cubic nonlinear Klein-Gordon equations

$\displaystyle u_{tt} - \Delta u + u \pm u^3 =0 $

in two space dimensions for real-valued initial data $ u(0)\in H^1_x$ and $ u_t(0)\in L^2_x$. We show that in the defocusing case, solutions are global and have finite global $ L^4_{t,x}$ spacetime bounds. In the focusing case, we characterize the dichotomy between this behaviour and blowup for initial data with energy less than that of the ground state.

These results rely on analogous statements for the two-dimensional cubic nonlinear Schrödinger equation, which are known in the defocusing case and for spherically-symmetric initial data in the focusing case. Thus, our results are mostly unconditional.

It was previously shown by Nakanishi that spacetime bounds for Klein-Gordon equations imply the same for nonlinear Schrödinger equations.


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Additional Information

Rowan Killip
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555

Betsy Stovall
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555

Monica Visan
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05536-4
PII: S 0002-9947(2011)05536-4
Received by editor(s): August 20, 2010
Received by editor(s) in revised form: December 27, 2010
Published electronically: September 15, 2011
Article copyright: © Copyright 2011 American Mathematical Society