Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Stable blow up dynamics for energy supercritical wave equations


Authors: Roland Donninger and Birgit Schörkhuber
Journal: Trans. Amer. Math. Soc. 366 (2014), 2167-2189
MSC (2010): Primary 35L05, 35B44, 35C06
DOI: https://doi.org/10.1090/S0002-9947-2013-06038-2
Published electronically: November 14, 2013
MathSciNet review: 3152726
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the semilinear wave equation

$\displaystyle \partial _t^2 \psi -\Delta \psi =\vert\psi \vert^{p-1}\psi $

for $ p > 3$ with radial data in three spatial dimensions. There exists an explicit solution which blows up at $ t=T>0$ given by

$\displaystyle \psi ^T(t,x)=c_p (T-t)^{-\frac {2}{p-1}}, $

where $ c_p$ is a suitable constant. We prove that the blow up described by $ \psi ^T$ is stable in the sense that there exists an open set (in a topology strictly stronger than the energy) of radial initial data that leads to a solution which converges to $ \psi ^T$ as $ t\to T-$ in the backward lightcone of the blow up point $ (t,r)=(T,0)$.

References [Enhancements On Off] (What's this?)

  • [1] Piotr Bizoń, Tadeusz Chmaj, and Zbisław Tabor, On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity 17 (2004), no. 6, 2187-2201. MR 2097671 (2005f:35210), https://doi.org/10.1088/0951-7715/17/6/009
  • [2] Piotr Bizoń, Dieter Maison, and Arthur Wasserman, Self-similar solutions of semilinear wave equations with a focusing nonlinearity, Nonlinearity 20 (2007), no. 9, 2061-2074. MR 2351023 (2008j:35126), https://doi.org/10.1088/0951-7715/20/9/003
  • [3] Aynur Bulut,
    The defocusing energy-supercritical cubic nonlinear wave equation in dimension five.
    Preprint arXiv:1112.0629v1, 2011.
  • [4] Raphaël Côte and Hatem Zaag,
    Construction of a multi-soliton blow-up solution to the semilinear wave equation in one space dimension.
    Preprint arXiv:1110.2512, 2011.
  • [5] Roland Donninger, On stable self-similar blowup for equivariant wave maps, Comm. Pure Appl. Math. 64 (2011), no. 8, 1095-1147. MR 2839272 (2012f:58034), https://doi.org/10.1002/cpa.20366
  • [6] Roland Donninger,
    Stable self-similar blowup in energy supercritical Yang-Mills theory.
    Preprint arXiv:1202.1389, 2012.
  • [7] Roland Donninger and Joachim Krieger,
    Nonscattering solutions and blowup at infinity for the critical wave equation.
    Preprint arXiv:1201.3258, 2012.
  • [8] Roland Donninger and Birgit Schörkhuber, Stable self-similar blow up for energy subcritical wave equations, Dyn. Partial Differ. Equ. 9 (2012), no. 1, 63-87. MR 2909934
  • [9] Roland Donninger, Birgit Schörkhuber, and Peter C. Aichelburg, On stable self-similar blow up for equivariant wave maps: the linearized problem, Ann. Henri Poincaré 13 (2012), no. 1, 103-144. MR 2881965, https://doi.org/10.1007/s00023-011-0125-0
  • [10] Thomas Duyckaerts, Carlos Kenig, and Frank Merle, 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. (JEMS) 14 (2012), no. 5, 1389-1454. MR 2966655, https://doi.org/10.4171/JEMS/336
  • [11] Thomas Duyckaerts, Carlos Kenig, and Frank Merle, Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 3, 533-599. MR 2781926 (2012e:35160), https://doi.org/10.4171/JEMS/261
  • [12] Thomas Duyckaerts, Carlos Kenig, and Frank Merle.
    Classification of radial solutions of the focusing, energy-critical wave equation.
    Preprint arXiv:1204.0031, 2012.
  • [13] Thomas Duyckaerts, Carlos Kenig, and Frank Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation, Geom. Funct. Anal. 22 (2012), no. 3, 639-698. MR 2972605, https://doi.org/10.1007/s00039-012-0174-7
  • [14] Thomas Duyckaerts and Frank Merle, Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP (2008), Art ID rpn002, 67. MR 2470571 (2009m:35329)
  • [15] Klaus-Jochen Engel and Rainer Nagel.
    One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, 2000.
    With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
  • [16] B. Helffer and J. Sjöstrand.
    From resolvent bounds to semigroup bounds.
    Preprint arXiv:1001.4171, 2010.
  • [17] Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452 (96a:47025)
  • [18] Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math. 201 (2008), no. 2, 147-212. MR 2461508 (2011a:35344), https://doi.org/10.1007/s11511-008-0031-6
  • [19] Carlos E. Kenig and Frank Merle, Radial solutions to energy supercritical wave equations in odd dimensions, Discrete Contin. Dyn. Syst. 31 (2011), no. 4, 1365-1381. MR 2836357 (2012j:35256), https://doi.org/10.3934/dcds.2011.31.1365
  • [20] Rowan Killip, Betsy Stovall, and Monica Visan.
    Blowup behaviour for the nonlinear Klein-Gordon equation.
    Preprint arXiv:1203.4886, 2012.
  • [21] Rowan Killip and Monica Visan, The defocusing energy-supercritical nonlinear wave equation in three space dimensions, Trans. Amer. Math. Soc. 363 (2011), no. 7, 3893-3934. MR 2775831 (2012e:35161), https://doi.org/10.1090/S0002-9947-2011-05400-0
  • [22] J. Krieger and W. Schlag, On the focusing critical semi-linear wave equation, Amer. J. Math. 129 (2007), no. 3, 843-913. MR 2325106 (2009f:35231), https://doi.org/10.1353/ajm.2007.0021
  • [23] Joachim Krieger, Kenji Nakanishi, and Wilhelm Schlag,
    Global dynamics away from the ground state for the energy-critical nonlinear wave equation.
    Preprint arXiv:1010.3799, 2010.
  • [24] Joachim Krieger, Kenji Nakanishi, and Wilhelm Schlag,
    Global dynamics of the nonradial energy-critical wave equation above the ground state energy.
    Preprint arXiv:1112.5663, 2011.
  • [25] Joachim Krieger, Wilhelm Schlag, and Daniel Tataru, Slow blow-up solutions for the $ H^1(\mathbb{R}^3)$ critical focusing semilinear wave equation, Duke Math. J. 147 (2009), no. 1, 1-53. MR 2494455 (2010h:58045), https://doi.org/10.1215/00127094-2009-005
  • [26] Howard A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $ Pu_{tt}=-Au+{\mathcal {F}}(u)$, Trans. Amer. Math. Soc. 192 (1974), 1-21. MR 0344697 (49 #9436)
  • [27] Frank Merle and Hatem Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math. 125 (2003), no. 5, 1147-1164. MR 2004432 (2004g:35163)
  • [28] Frank Merle and Hatem Zaag, Determination of the blow-up rate for a critical semilinear wave equation, Math. Ann. 331 (2005), no. 2, 395-416. MR 2115461 (2005k:35286), https://doi.org/10.1007/s00208-004-0587-1
  • [29] Frank Merle and Hatem Zaag, On growth rate near the blowup surface for semilinear wave equations, Int. Math. Res. Not. 19 (2005), 1127-1155. MR 2147056 (2006h:35183), https://doi.org/10.1155/IMRN.2005.1127
  • [30] Frank Merle and Hatem Zaag, Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Anal. 253 (2007), no. 1, 43-121. MR 2362418 (2009b:35288), https://doi.org/10.1016/j.jfa.2007.03.007
  • [31] Frank Merle and Hatem Zaag, Openness of the set of non-characteristic points and regularity of the blow-up curve for the 1 D semilinear wave equation, Comm. Math. Phys. 282 (2008), no. 1, 55-86. MR 2415473 (2010e:35193), https://doi.org/10.1007/s00220-008-0532-3
  • [32] Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark, editors.
    NIST handbook of mathematical functions.
    U.S. Department of Commerce National Institute of Standards and Technology, Washington, DC, 2010.
    With 1 CD-ROM (Windows, Macintosh and UNIX).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35L05, 35B44, 35C06

Retrieve articles in all journals with MSC (2010): 35L05, 35B44, 35C06


Additional Information

Roland Donninger
Affiliation: Department of Mathematics, École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland
Email: roland.donninger@epfl.ch

Birgit Schörkhuber
Affiliation: Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria
Email: birgit.schoerkhuber@tuwien.ac.at

DOI: https://doi.org/10.1090/S0002-9947-2013-06038-2
Received by editor(s): August 20, 2012
Published electronically: November 14, 2013
Additional Notes: The second author acknowledges partial support from the Austrian Science Fund (FWF), grants P22108, P23598, P24304, and I395; the Austrian-French Project of the Austrian Exchange Service (ÖAD); and the Innovative Ideas Program of Vienna University of Technology.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society