# American Institute of Mathematical Sciences

• Previous Article
On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds
• CPAA Home
• This Issue
• Next Article
Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay
September  2015, 14(5): 1705-1741. doi: 10.3934/cpaa.2015.14.1705

## Optimal polynomial blow up range for critical wave maps

 1 Bâtiment des Mathématiques, EPFL, Station 8, CH-1015 Lausanne, Switzerland

Received  April 2014 Revised  March 2015 Published  June 2015

We prove that the critical Wave Maps equation with target $S^2$ and origin $R^{2+1}$ admits energy class blow up solutions of the form \begin{eqnarray} u(t, r) = Q(\lambda(t)r) + \varepsilon(t, r) \end{eqnarray} where $Q:R^2\rightarrow S^2$ is the ground state harmonic map and $\lambda(t) = t^{-1-\nu}$ for any $\nu>0$. This extends the work [14], where such solutions were constructed under the assumption $\nu>\frac{1}{2}$. In light of a result of Struwe [23], our result is optimal for polynomial blow up rates.
Citation: Can Gao, Joachim Krieger. Optimal polynomial blow up range for critical wave maps. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1705-1741. doi: 10.3934/cpaa.2015.14.1705
##### References:
 [1] P. d'Ancona and V. Georgiev, On the continuity of the solution operator to the wave map system,, \emph{Comm. Pure Appl. Math.}, 57 (2004), 357. doi: 10.1002/cpa.3043. Google Scholar [2] P. Bizoń, T. Chmaj and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity,, \emph{Nonlinearity}, 17 (2004), 2187. doi: 10.1088/0951-7715/17/6/009. Google Scholar [3] F. Gesztesy and M. Zinchenko, On spectral theory for Schrodinger operators with strongly singular potentials,, \emph{Math. Nachr.}, 279 (2006), 1041. doi: 10.1002/mana.200510410. Google Scholar [4] R. Cote, C. Kenig and F. Merle, Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system,, \emph{Communications in Mathematical Physics}, 284 (2008), 203. doi: 10.1007/s00220-008-0604-4. Google Scholar [5] R. Cote, C. Kenig, A. Lawrie and W. Schlag, Characterization of large energy solutions of the equivariant wave map problem: I,, preprint., (). doi: 10.1353/ajm.2015.0002. Google Scholar [6] R. Cote, C. Kenig, A. Lawrie and W. Schlag, Characterization of large energy solutions of the equivariant wave map problem: II,, preprint., (). doi: 10.1353/ajm.2015.0003. Google Scholar [7] R. Donninger and J. Krieger, Nonscattering solutions and blow up at infinity for the critical wave equation,, preprint, (1201). doi: 10.1007/s00208-013-0898-1. Google Scholar [8] T. Duyckaerts, C. Kenig and F. Merle, Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation,, \emph{J. Eur. Math. Soc.}, 13 (2011), 533. doi: 10.4171/JEMS/261. Google Scholar [9] T. Duyckaerts, C. Kenig and F. Merle, Universality of the blow-up profile for small type II blow-up solutions of energy-critical wave equation: the non-radial case,, preprint, (). doi: 10.4171/JEMS/336. Google Scholar [10] T. Duyckaerts, C. Kenig and F. Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation,, preprint, (). doi: 10.1007/s00039-012-0174-7. Google Scholar [11] T. Duyckaerts, C. Kenig and F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation,, preprint, (). doi: 10.4310/CJM.2013.v1.n1.a3. Google Scholar [12] S. Klainerman, M. Machedon, Smoothing estimates for null forms and applications. A celebration of John F. Nash, Jr,, \emph{Duke Math. J.}, 81 (1995), 99. doi: 10.1215/S0012-7094-95-08109-5. Google Scholar [13] J. Krieger and W. Schlag, Full range of blow up exponents for the quintic wave equation in three dimensions,, \emph{Journal de Mathematiques Pures et Appliquees}, (). doi: 10.1016/j.matpur.2013.10.008. Google Scholar [14] J. Krieger, W. Schlag and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps,, \emph{Invent. Math.}, 171 (2008), 543. doi: 10.1007/s00222-007-0089-3. Google Scholar [15] J. Krieger, W. Schlag and D. Tataru, Slow blow-up solutions for the $H^1(\R^3)$ critical focusing semilinear wave equation,, \emph{Duke Math. J.}, 147 (2009), 1. doi: 10.1215/00127094-2009-005. Google Scholar [16] F. Merle, P. Raphael and I. Rodnianski, Blowup dynamics for smooth data equivariant solutions to the critical Schröinger map problem,, \emph{Invent. Math.}, 193 (2013), 249. doi: 10.1007/s00222-012-0427-y. Google Scholar [17] P. Raphael and I. Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems,, \emph{Publ. Math. Inst. Hautes tudes Sci.}, 115 (2012), 1. doi: 10.1007/s10240-011-0037-z. Google Scholar [18] I. Rodnianski and J. Sterbenz, On the Formation of Singularities in the Critical $O(3)$ Sigma-Model,, \emph{Annals of Math.}, 172 (2010), 187. doi: 10.4007/annals.2010.172.187. Google Scholar [19] J. Shatah and S. Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps,, \emph{Comm. Pure Appl. Math.}, 47 (1994), 719. doi: 10.1002/cpa.3160470507. Google Scholar [20] J. Sterbenz and D. Tataru, Regularity of Wave-Maps in dimension $2+1$,, Preprint 2009., (2009). doi: 10.1007/s00220-010-1062-3. Google Scholar [21] J. Sterbenz and D. Tataru, Energy dispersed large data wave maps in $2+1$ dimensions,, Preprint 2009., (2009). doi: 10.1007/s00220-010-1061-4. Google Scholar [22] M. Struwe, Variational methods, Applications to Nonlinear PDEs and Hamiltonian Systems,, Second edition, (1996). doi: 10.1007/978-3-662-03212-1. Google Scholar [23] M. Struwe, Equivariant wave maps in two space dimensions,, \emph{Comm. Pure Appl. Math.}, 56 (2003), 815. doi: 10.1002/cpa.10074. Google Scholar [24] T. Tao, Global regularity of wave maps II. Small energy in two dimensions,, \emph{Comm. Math. Phys.}, 224 (2001), 443. doi: 10.1007/PL00005588. Google Scholar [25] T. Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension,, \emph{Internat. Math. Res. Notices}, (2001), 299. doi: 10.1155/S1073792801000150. Google Scholar [26] D. Tataru, Rough solutions for the wave maps equation,, \emph{Amer. J. Math.}, 127 (2005), 293. Google Scholar

show all references

##### References:
 [1] P. d'Ancona and V. Georgiev, On the continuity of the solution operator to the wave map system,, \emph{Comm. Pure Appl. Math.}, 57 (2004), 357. doi: 10.1002/cpa.3043. Google Scholar [2] P. Bizoń, T. Chmaj and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity,, \emph{Nonlinearity}, 17 (2004), 2187. doi: 10.1088/0951-7715/17/6/009. Google Scholar [3] F. Gesztesy and M. Zinchenko, On spectral theory for Schrodinger operators with strongly singular potentials,, \emph{Math. Nachr.}, 279 (2006), 1041. doi: 10.1002/mana.200510410. Google Scholar [4] R. Cote, C. Kenig and F. Merle, Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system,, \emph{Communications in Mathematical Physics}, 284 (2008), 203. doi: 10.1007/s00220-008-0604-4. Google Scholar [5] R. Cote, C. Kenig, A. Lawrie and W. Schlag, Characterization of large energy solutions of the equivariant wave map problem: I,, preprint., (). doi: 10.1353/ajm.2015.0002. Google Scholar [6] R. Cote, C. Kenig, A. Lawrie and W. Schlag, Characterization of large energy solutions of the equivariant wave map problem: II,, preprint., (). doi: 10.1353/ajm.2015.0003. Google Scholar [7] R. Donninger and J. Krieger, Nonscattering solutions and blow up at infinity for the critical wave equation,, preprint, (1201). doi: 10.1007/s00208-013-0898-1. Google Scholar [8] T. Duyckaerts, C. Kenig and F. Merle, Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation,, \emph{J. Eur. Math. Soc.}, 13 (2011), 533. doi: 10.4171/JEMS/261. Google Scholar [9] T. Duyckaerts, C. Kenig and F. Merle, Universality of the blow-up profile for small type II blow-up solutions of energy-critical wave equation: the non-radial case,, preprint, (). doi: 10.4171/JEMS/336. Google Scholar [10] T. Duyckaerts, C. Kenig and F. Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation,, preprint, (). doi: 10.1007/s00039-012-0174-7. Google Scholar [11] T. Duyckaerts, C. Kenig and F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation,, preprint, (). doi: 10.4310/CJM.2013.v1.n1.a3. Google Scholar [12] S. Klainerman, M. Machedon, Smoothing estimates for null forms and applications. A celebration of John F. Nash, Jr,, \emph{Duke Math. J.}, 81 (1995), 99. doi: 10.1215/S0012-7094-95-08109-5. Google Scholar [13] J. Krieger and W. Schlag, Full range of blow up exponents for the quintic wave equation in three dimensions,, \emph{Journal de Mathematiques Pures et Appliquees}, (). doi: 10.1016/j.matpur.2013.10.008. Google Scholar [14] J. Krieger, W. Schlag and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps,, \emph{Invent. Math.}, 171 (2008), 543. doi: 10.1007/s00222-007-0089-3. Google Scholar [15] J. Krieger, W. Schlag and D. Tataru, Slow blow-up solutions for the $H^1(\R^3)$ critical focusing semilinear wave equation,, \emph{Duke Math. J.}, 147 (2009), 1. doi: 10.1215/00127094-2009-005. Google Scholar [16] F. Merle, P. Raphael and I. Rodnianski, Blowup dynamics for smooth data equivariant solutions to the critical Schröinger map problem,, \emph{Invent. Math.}, 193 (2013), 249. doi: 10.1007/s00222-012-0427-y. Google Scholar [17] P. Raphael and I. Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems,, \emph{Publ. Math. Inst. Hautes tudes Sci.}, 115 (2012), 1. doi: 10.1007/s10240-011-0037-z. Google Scholar [18] I. Rodnianski and J. Sterbenz, On the Formation of Singularities in the Critical $O(3)$ Sigma-Model,, \emph{Annals of Math.}, 172 (2010), 187. doi: 10.4007/annals.2010.172.187. Google Scholar [19] J. Shatah and S. Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps,, \emph{Comm. Pure Appl. Math.}, 47 (1994), 719. doi: 10.1002/cpa.3160470507. Google Scholar [20] J. Sterbenz and D. Tataru, Regularity of Wave-Maps in dimension $2+1$,, Preprint 2009., (2009). doi: 10.1007/s00220-010-1062-3. Google Scholar [21] J. Sterbenz and D. Tataru, Energy dispersed large data wave maps in $2+1$ dimensions,, Preprint 2009., (2009). doi: 10.1007/s00220-010-1061-4. Google Scholar [22] M. Struwe, Variational methods, Applications to Nonlinear PDEs and Hamiltonian Systems,, Second edition, (1996). doi: 10.1007/978-3-662-03212-1. Google Scholar [23] M. Struwe, Equivariant wave maps in two space dimensions,, \emph{Comm. Pure Appl. Math.}, 56 (2003), 815. doi: 10.1002/cpa.10074. Google Scholar [24] T. Tao, Global regularity of wave maps II. Small energy in two dimensions,, \emph{Comm. Math. Phys.}, 224 (2001), 443. doi: 10.1007/PL00005588. Google Scholar [25] T. Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension,, \emph{Internat. Math. Res. Notices}, (2001), 299. doi: 10.1155/S1073792801000150. Google Scholar [26] D. Tataru, Rough solutions for the wave maps equation,, \emph{Amer. J. Math.}, 127 (2005), 293. Google Scholar
 [1] Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108 [2] Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459 [3] Joachim Krieger, Kenji Nakanishi, Wilhelm Schlag. Global dynamics of the nonradial energy-critical wave equation above the ground state energy. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2423-2450. doi: 10.3934/dcds.2013.33.2423 [4] Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807 [5] Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076 [6] Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351 [7] Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261 [8] Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735 [9] George Osipenko. Linearization near a locally nonunique invariant manifold. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 189-205. doi: 10.3934/dcds.1997.3.189 [10] Antonios Zagaris, Christophe Vandekerckhove, C. William Gear, Tasso J. Kaper, Ioannis G. Kevrekidis. Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2759-2803. doi: 10.3934/dcds.2012.32.2759 [11] Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839 [12] Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 [13] Boris Hasselblatt. Critical regularity of invariant foliations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 931-937. doi: 10.3934/dcds.2002.8.931 [14] Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921 [15] Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307 [16] Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285 [17] Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309 [18] Matti Lassas, Teemu Saksala, Hanming Zhou. Reconstruction of a compact manifold from the scattering data of internal sources. Inverse Problems & Imaging, 2018, 12 (4) : 993-1031. doi: 10.3934/ipi.2018042 [19] Emile Franc Doungmo Goufo, Abdon Atangana. Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 645-662. doi: 10.3934/dcdss.2020035 [20] Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil. Opinion Dynamics on a General Compact Riemannian Manifold. Networks & Heterogeneous Media, 2017, 12 (3) : 489-523. doi: 10.3934/nhm.2017021

2018 Impact Factor: 0.925

## Metrics

• PDF downloads (6)
• HTML views (0)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]