International Mathematics Research Papers

International Mathematics Research Papers (2008) Vol. 2008 : article ID rpn002, 67 pages, doi:10.1093/imrp/rpn002 published on February 11, 2008

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Dynamics of Threshold Solutions for Energy-Critical Wave Equation

Thomas Duyckaerts and Frank Merle

Département de Mathématiques, Université de Cergy-Pontoise, Site de Saint Martin, 95302 Cergy-Pontoise Cedex, France

Correspondence: Correspondence to be sent to: Thomas Duyckaerts, Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint Martin, 2 avenue Adolphe-Chauvin, 95302 Cergy-Pontoise cedex, France. e-mail: thomas.duyckaerts{at}u-cergy.fr

We consider the energy-critical nonlinear focusing wave equation in dimension N = 3, 4, 5. An explicit stationary solution, W, of this equation is known. In [8], the energy E(W, 0) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article we study the dynamics at the critical level E(u₀, u₁) = E(W, 0) and classify the corresponding solutions. We show in particular the existence of two special solutions, connecting different behaviors for negative and positive times. Our results are analogous to [3], which treats the energy-critical nonlinear focusing radial Schrödinger equation, but without any radial assumption on the data. We also refine the understanding of the dynamical behavior of the special solutions.

References

1. Aubin Thierry. Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. Journal de Mathématiques Pures et Appliquées (1976) 55(3):269–96.[Web of Science]
2. Bahouri Hajer, Gérard Patrick. High frequency approximation of solutions to critical nonlinear wave equations. American Journal of Mathematics (1999) 121(1):131–75.[CrossRef][Web of Science]
3. Duyckaerts T., Merle F. Dynamic of threshold solutions for energy-critical nls. Geometric and Functional Analysis. (forthcoming).
4. Ginibre J., Soffer A., Velo G. The global Cauchy problem for the critical nonlinear wave equation. Journal of Functional Analysis (1992) 110(1):96–130.[CrossRef][Web of Science]
5. Ginibre J., Velo G. Generalized Strichartz inequalities for the wave equation. Journal of Functional Analysis (1995) 133(1):50–68.[CrossRef][Web of Science]
6. Kapitanski Lev. Global and unique weak solutions of nonlinear wave equations. Mathematical Research Letters (1994) 1(2):211–23.
7. Kenig Carlos E., Merle Frank. Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Inventiones Mathematicae (2006) 166(3):645–75.[CrossRef][Web of Science]
8. Kenig Carlos E., Merle Frank. Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation. Acta Mathematica. (forthcoming).
9. Kenig Carlos E., Ponce Gustavo, Vega Luis. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Communications on Pure and Applied Mathematics (1993) 46(4):527–620.[CrossRef][Web of Science]
10. Krieger J., Schlag W., Tataru D. Slow blow-up solutions for the h¹(R³) critical focusing semi-linear wave equation in R³. (2007) preprint arXiv.org:math/0702033.
11. Lions P.-L. The concentration-compactness principle in the calculus of variations. The limit case. II. Revista Matematica Iberoamericana (1985) 1(2):45–121.
12. Lindblad Hans, Sogge Christopher D. On existence and scattering with minimal regularity for semilinear wave equations. Journal of Functional Analysis (1995) 130(2):357–426.[CrossRef][Web of Science]
13. Pecher Hartmut. Nonlinear small data scattering for the wave and Klein-Gordon equation. Mathematische Zeitschriftath (1984) 185(2):261–70.
14. Rey Olivier. The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent. Journal of Functional Analysis (1990) 89(1):1–52.[CrossRef][Web of Science]
15. Reed Michael, Simon Barry. Methods of Modern Mathematical Physics. IV. Analysis of Operators (1978) New York: Academic Press.
16. Schlag W., Krieger J. On the focusing critical semi-linear wave equation. American Journal of Mathematics (2007) 129(3):843–913.[Web of Science]
17. Sogge Christopher D. Lectures on Nonlinear Wave Equations (1995) Boston, MA: International Press. Monographs in Analysis 2.
18. Shatah Jalal, Struwe Michael. Well-posedness in the energy space for semilinear wave equations with critical growth. International Mathematics Research Notices (1994) 7:303–9.
19. Shatah Jalal, Struwe Michael. Geometric Wave Equations (1998) New York: New York University Courant Institute of Mathematical Sciences. Courant Lecture Notes in Mathematics 2.
20. Talenti Giorgio. Best constant in Sobolev inequality. Annali di Mathematica Pure ed Applicata (1976) 4(110):353–72.

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This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions How to cite this article Google Scholar Articles by Duyckaerts, T. Articles by Merle, F. Search for Related Content Social Bookmarking          What's this?