We prove that every solution of the focusing energy-critical wave equation with the compactness property is global. We also give similar results for supercritical wave and Schrödinger equations.
@article{AIHPC_2016__33_6_1675_0, author = {Duyckaerts, Thomas and Kenig, Carlos and Merle, Frank}, title = {Global existence for solutions of the focusing wave equation with the compactness property}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1675--1690}, publisher = {Elsevier}, volume = {33}, number = {6}, year = {2016}, doi = {10.1016/j.anihpc.2015.08.002}, mrnumber = {3569247}, zbl = {1362.35190}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2015.08.002/} }
TY - JOUR AU - Duyckaerts, Thomas AU - Kenig, Carlos AU - Merle, Frank TI - Global existence for solutions of the focusing wave equation with the compactness property JO - Annales de l'I.H.P. Analyse non linéaire PY - 2016 SP - 1675 EP - 1690 VL - 33 IS - 6 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2015.08.002/ DO - 10.1016/j.anihpc.2015.08.002 LA - en ID - AIHPC_2016__33_6_1675_0 ER -
%0 Journal Article %A Duyckaerts, Thomas %A Kenig, Carlos %A Merle, Frank %T Global existence for solutions of the focusing wave equation with the compactness property %J Annales de l'I.H.P. Analyse non linéaire %D 2016 %P 1675-1690 %V 33 %N 6 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2015.08.002/ %R 10.1016/j.anihpc.2015.08.002 %G en %F AIHPC_2016__33_6_1675_0
Duyckaerts, Thomas; Kenig, Carlos; Merle, Frank. Global existence for solutions of the focusing wave equation with the compactness property. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1675-1690. doi : 10.1016/j.anihpc.2015.08.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.08.002/
[1] Hardy's inequalities revisited, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 25 (1997) no. 1–2, pp. 217–237 (1998), dedicated to Ennio De Giorgi | Numdam | MR | Zbl
[2] Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation, J. Eur. Math. Soc., Volume 13 (2011) no. 3, pp. 533–599 | MR | Zbl
[3] Scattering for radial, bounded solutions of focusing supercritical wave equations, Int. Math. Res. Not. (2012) | MR | Zbl
[4] Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case, J. Eur. Math. Soc., Volume 14 (2012) no. 5, pp. 1389–1454 | DOI | MR | Zbl
[5] Solutions of the focusing, energy-critical wave equation with the compactness property, 2014 (preprint) | arXiv
[6] Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations, Commun. Pure Appl. Anal., Volume 14 (2015) no. 4, pp. 1275–1326 | MR | Zbl
[7] Nondegeneracy of blowup for semilinear heat equations, Commun. Pure Appl. Math., Volume 42 (1989) no. 6, pp. 845–884 | DOI | MR | Zbl
[8] Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., Volume 166 (2006) no. 3, pp. 645–675 | DOI | MR | Zbl
[9] Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., Volume 201 (2008) no. 2, pp. 147–212 | DOI | MR | Zbl
[10] Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications, Am. J. Math., Volume 133 (2011) no. 4, pp. 1029–1065 | DOI | MR | Zbl
[11] Energy-supercritical NLS: critical -bounds imply scattering, Commun. Partial Differ. Equ., Volume 35 (2010) no. 6, pp. 945–987 | DOI | MR | Zbl
[12] The defocusing energy-supercritical nonlinear wave equation in three space dimensions, Trans. Am. Math. Soc., Volume 363 (2011) no. 7, pp. 3893–3934 | DOI | MR | Zbl
[13] Determination of the blow-up rate for the semilinear wave equation, Am. J. Math., Volume 125 (2003) no. 5, pp. 1147–1164 | DOI | MR | Zbl
[14] A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations, Dyn. Partial Differ. Equ., Volume 4 (2007) no. 1, pp. 1–53 | MR | Zbl
Cité par Sources :