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Type II Blow Up Solutions with Optimal Stability Properties for the Critical Focussing Nonlinear Wave Equation on $\mathbb {R}^{3+1}$
About this Title
Stefano Burzio and Joachim Krieger
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 278, Number 1369
ISBNs: 978-1-4704-5346-6 (print); 978-1-4704-7169-9 (online)
DOI: https://doi.org/10.1090/memo/1369
Published electronically: May 23, 2022
Keywords: critical wave equation,
blowup
Table of Contents
Chapters
- 1. Introduction
- 2. The main theorem and outline of the proof
- 3. Construction of a two parameter family of approximate blow up solutions
- 4. Modulation theory; determination of the parameters $\gamma _{1,2}$.
- 5. Iterative construction of blow up solution almost matching the perturbed initial data
- 6. Proof of Theorem
- 7. Outlook
Abstract
We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation \[ \Box u = -u^5 \] on $\mathbb {R}^{3+1}$ constructed in Krieger, Schlag, and Tartaru (“Slow blow-up solutions for the $H^1(\mathbb {R}^3)$ critical focusing semilinear wave equation”, 2009) and Krieger and Schlag (“Full range of blow up exponents for the quintic wave equation in three dimensions”, 2014) are stable along a co-dimension one Lipschitz manifold of data perturbations in a suitable topology, provided the scaling parameter $\lambda (t) = t^{-1-\nu }$ is sufficiently close to the self-similar rate, i. e., $\nu >0$ is sufficiently small. This result is qualitatively optimal in light of the result of Krieger, Nakamishi, and Schlag (“Center-stable manifold of the ground state in the energy space for the critical wave equation”, 2015). The paper builds on the analysis of Krieger and Wong (“On type I blow-up formation for the critical NLW”, 2014).- Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. MR 431287
- Hajer Bahouri and Patrick Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), no. 1, 131–175. MR 1705001
- Peter W. Bates and Christopher K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics reported, Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 1–38. MR 1000974
- 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, DOI 10.1088/0951-7715/17/6/009
- Cătălin I. Cârstea, A construction of blow up solutions for co-rotational wave maps, Comm. Math. Phys. 300 (2010), no. 2, 487–528. MR 2728732, DOI 10.1007/s00220-010-1118-4
- Athanasios Chatzikaleas and Roland Donninger, Stable blowup for the cubic wave equation in higher dimensions, J. Differential Equations 266 (2019), no. 10, 6809–6865. MR 3926085, DOI 10.1016/j.jde.2018.11.016
- Manuel del Pino, Monica Musso, and Juncheng Wei, Infinite-time blow-up for the 3-dimensional energy-critical heat equation, Anal. PDE 13 (2020), no. 1, 215–274. MR 4047646, DOI 10.2140/apde.2020.13.215
- Roland Donninger, On stable self-similar blowup for equivariant wave maps, Comm. Pure Appl. Math. 64 (2011), no. 8, 1095–1147. MR 2839272, DOI 10.1002/cpa.20366
- Roland Donninger and Joachim Krieger, Nonscattering solutions and blowup at infinity for the critical wave equation, Math. Ann. 357 (2013), no. 1, 89–163. MR 3084344, DOI 10.1007/s00208-013-0898-1
- Roland Donninger and Birgit Schörkhuber, Stable blow up dynamics for energy supercritical wave equations, Trans. Amer. Math. Soc. 366 (2014), no. 4, 2167–2189. MR 3152726, DOI 10.1090/S0002-9947-2013-06038-2
- Roland Donninger and Birgit Schörkhuber, On blowup in supercritical wave equations, Comm. Math. Phys. 346 (2016), no. 3, 907–943. MR 3537340, DOI 10.1007/s00220-016-2610-2
- Roland Donninger, Min Huang, Joachim Krieger, and Wilhelm Schlag, Exotic blowup solutions for the $u^5$ focusing wave equation in $\Bbb {R}^3$, Michigan Math. J. 63 (2014), no. 3, 451–501. MR 3255688, DOI 10.1307/mmj/1409932630
- 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, DOI 10.4171/JEMS/261
- 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, DOI 10.4171/JEMS/336
- 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, DOI 10.1007/s00039-012-0174-7
- Thomas Duyckaerts, Carlos Kenig, and Frank Merle, Classification of radial solutions of the focusing, energy-critical wave equation, Camb. J. Math. 1 (2013), no. 1, 75–144. MR 3272053, DOI 10.4310/CJM.2013.v1.n1.a3
- Thomas Duyckaerts and Frank Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal. 18 (2009), no. 6, 1787–1840. MR 2491692, DOI 10.1007/s00039-009-0707-x
- Thomas Duyckaerts and Frank Merle, Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP , posted on (2008), Art ID rpn002, 67. MR 2470571, DOI 10.1093/imrp/rpn002
- Can Gao and Joachim Krieger, Optimal polynomial blow up range for critical wave maps, Commun. Pure Appl. Anal. 14 (2015), no. 5, 1705–1741. MR 3359541, DOI 10.3934/cpaa.2015.14.1705
- Matthieu Hillairet and Pierre Raphaël, Smooth type II blow-up solutions to the four-dimensional energy-critical wave equation, Anal. PDE 5 (2012), no. 4, 777–829. MR 3006642, DOI 10.2140/apde.2012.5.777
- Slim Ibrahim, Nader Masmoudi, and Kenji Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE 4 (2011), no. 3, 405–460. MR 2872122, DOI 10.2140/apde.2011.4.405
- Konrad Jörgens, Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77 (1961), 295–308 (German). MR 130462, DOI 10.1007/BF01180181
- Paschalis Karageorgis and Walter A. Strauss, Instability of steady states for nonlinear wave and heat equations, J. Differential Equations 241 (2007), no. 1, 184–205. MR 2356215, DOI 10.1016/j.jde.2007.06.006
- Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645–675. MR 2257393, DOI 10.1007/s00222-006-0011-4
- 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, DOI 10.1007/s11511-008-0031-6
- Joachim Krieger, On stability of type II blow up for the critical nonlinear wave equation on $\Bbb R^{3+1}$, Mem. Amer. Math. Soc. 267 (2020), no. 1301, v + 129. MR 4194893, DOI 10.1090/memo/1301
- Joachim Krieger, Kenji Nakanishi, and Wilhelm Schlag, Global dynamics away from the ground state for the energy-critical nonlinear wave equation, Amer. J. Math. 135 (2013), no. 4, 935–965. MR 3086065, DOI 10.1353/ajm.2013.0034
- Joachim Krieger, Kenji Nakanishi, and Wilhelm Schlag, Global dynamics of the nonradial energy-critical wave equation above the ground state energy, Discrete Contin. Dyn. Syst. 33 (2013), no. 6, 2423–2450. MR 3007693, DOI 10.3934/dcds.2013.33.2423
- Joachim Krieger, Kenji Nakanishi, and Wilhelm Schlag, Threshold phenomenon for the quintic wave equation in three dimensions, Comm. Math. Phys. 327 (2014), no. 1, 309–332. MR 3177940, DOI 10.1007/s00220-014-1900-9
- Joachim Krieger, Kenji Nakanishi, and Wilhelm Schlag, Center-stable manifold of the ground state in the energy space for the critical wave equation, Math. Ann. 361 (2015), no. 1-2, 1–50. MR 3302610, DOI 10.1007/s00208-014-1059-x
- J. Krieger and W. Schlag, On the focusing critical semi-linear wave equation, Amer. J. Math. 129 (2007), no. 3, 843–913. MR 2325106, DOI 10.1353/ajm.2007.0021
- Joachim Krieger and Wilhelm Schlag, Full range of blow up exponents for the quintic wave equation in three dimensions, J. Math. Pures Appl. (9) 101 (2014), no. 6, 873–900. MR 3205646, DOI 10.1016/j.matpur.2013.10.008
- Joachim Krieger, Wilhelm Schlag, and Daniel Tataru, Slow blow-up solutions for the $H^1(\Bbb R^3)$ critical focusing semilinear wave equation, Duke Math. J. 147 (2009), no. 1, 1–53. MR 2494455, DOI 10.1215/00127094-2009-005
- J. Krieger, W. Schlag, and D. Tataru, Renormalization and blow up for the critical Yang-Mills problem, Adv. Math. 221 (2009), no. 5, 1445–1521. MR 2522426, DOI 10.1016/j.aim.2009.02.017
- J. Krieger, W. Schlag, and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math. 171 (2008), no. 3, 543–615. MR 2372807, DOI 10.1007/s00222-007-0089-3
- Joachim Krieger and Willie Wong, On type I blow-up formation for the critical NLW, Comm. Partial Differential Equations 39 (2014), no. 9, 1718–1728. MR 3246040, DOI 10.1080/03605302.2013.861847
- Howard A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+{\cal F}(u)$, Trans. Amer. Math. Soc. 192 (1974), 1–21. MR 344697, DOI 10.1090/S0002-9947-1974-0344697-2
- Vladimir A. Marchenko, Sturm-Liouville operators and applications, Operator Theory: Advances and Applications, vol. 22, Birkhäuser Verlag, Basel, 1986. Translated from the Russian by A. Iacob. MR 897106, DOI 10.1007/978-3-0348-5485-6
- Yvan Martel, Frank Merle, and Pierre Raphaël, Blow up for the critical gKdV equation III: exotic regimes, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (2015), no. 2, 575–631. MR 3410473
- Pierre Raphaël and Igor Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 1–122. MR 2929728, DOI 10.1007/s10240-011-0037-z
- Frank Merle, Pierre Raphaël, and Jeremie Szeftel, The instability of Bourgain-Wang solutions for the $L^2$ critical NLS, Amer. J. Math. 135 (2013), no. 4, 967–1017. MR 3086066, DOI 10.1353/ajm.2013.0033
- Frank Merle and Hatem Zaag, On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations, Comm. Math. Phys. 333 (2015), no. 3, 1529–1562. MR 3302641, DOI 10.1007/s00220-014-2132-8
- Frank Merle and Hatem Zaag, Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions, Trans. Amer. Math. Soc. 368 (2016), no. 1, 27–87. MR 3413856, DOI 10.1090/tran/6450
- K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, J. Differential Equations 250 (2011), no. 5, 2299–2333. MR 2756065, DOI 10.1016/j.jde.2010.10.027
- K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations 44 (2012), no. 1-2, 1–45. MR 2898769, DOI 10.1007/s00526-011-0424-9
- K. Nakanishi and W. Schlag, Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption, Arch. Ration. Mech. Anal. 203 (2012), no. 3, 809–851. MR 2928134, DOI 10.1007/s00205-011-0462-7
- Kenji Nakanishi and Wilhelm Schlag, Invariant manifolds and dispersive Hamiltonian evolution equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2011. MR 2847755, DOI 10.4171/095
- C. Ortoleva and G. Perelman, Nondispersive vanishing and blow up at infinity for the energy critical nonlinear Schrödinger equation in $\Bbb R^3$, Algebra i Analiz 25 (2013), no. 2, 162–192; English transl., St. Petersburg Math. J. 25 (2014), no. 2, 271–294. MR 3114854, DOI 10.1090/S1061-0022-2014-01290-3
- Kenneth J. Palmer, Linearization near an integral manifold, J. Math. Anal. Appl. 51 (1975), 243–255. MR 374564, DOI 10.1016/0022-247X(75)90156-0
- L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), no. 3-4, 273–303. MR 402291, DOI 10.1007/BF02761595
- Pierre Raphaël and Igor Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 1–122. MR 2929728, DOI 10.1007/s10240-011-0037-z
- Pierre Raphaël and Remi Schweyer, Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow, Comm. Pure Appl. Math. 66 (2013), no. 3, 414–480. MR 3008229, DOI 10.1002/cpa.21435
- Pierre Raphaël and Remi Schweyer, Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow, Anal. PDE 7 (2014), no. 8, 1713–1805. MR 3318739, DOI 10.2140/apde.2014.7.1713
- Wilhelm Schlag, Spectral theory and nonlinear partial differential equations: a survey, Discrete Contin. Dyn. Syst. 15 (2006), no. 3, 703–723. MR 2220744, DOI 10.3934/dcds.2006.15.703
- Jalal Shatah and Michael Struwe, Geometric wave equations, Courant Lecture Notes in Mathematics, vol. 2, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1998. MR 1674843
- A. N. Šošitaĭšvili, Bifurcations of topological type of singular points of vector fields that depend on parameters, Funkcional. Anal. i Priložen. 6 (1972), no. 2, 97–98 (Russian). MR 0296977
- A. N. Šošitaĭšvili, The bifurcation of the topological type of the singular points of vector fields that depend on parameters, Trudy Sem. Petrovsk. Vyp. 1 (1975), 279–309 (Russian). MR 0478239
- Christopher D. Sogge, Lectures on nonlinear wave equations, Monographs in Analysis, II, International Press, Boston, MA, 1995. MR 1715192
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Walter A. Strauss, Nonlinear wave equations, CBMS Regional Conference Series in Mathematics, vol. 73, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1989. MR 1032250
- Michael Struwe, Globally regular solutions to the $u^5$ Klein-Gordon equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 3, 495–513 (1989). MR 1015805
- Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, DOI 10.1007/BF02418013
- Michael E. Taylor, Tools for PDE, Mathematical Surveys and Monographs, vol. 81, American Mathematical Society, Providence, RI, 2000. Pseudodifferential operators, paradifferential operators, and layer potentials. MR 1766415, DOI 10.1090/surv/081