×

Slow blow-up solutions for the \(H^1(\mathbb R^3)\) critical focusing semilinear wave equation. (English) Zbl 1170.35066

Given \(\nu >1/2\) and \(\delta >0\) arbitrary, existence of an energy solution of \(u_{tt}-\Delta u-u^{5}=0\) on \(\mathbb R^{3+1},\) which blow-up exactly at \(r=t=0\) as \(t\rightarrow 0-,\) is proved. These solutions are of the form
\[ t^{-(\nu+1)/2} \Bigg(1+ \frac{1}{3} \bigg(\frac{r}{t^{\nu+1}}\bigg)^2\Bigg)^{-1/2}+\eta (r,t) \]
inside the cone \(\{r\leq t\},\) radial and their energy is bounded by \(\delta \) for small \(t>0,\) whereas the energy of radiation term \(\eta (r,t)\) tends to \(0\) as \(t\rightarrow 0.\) The proof is based on a renormalization method for stationary solutions (see also the authors’ article [Invent. Math. 171, No. 3, 543–615 (2008; Zbl 1139.35021)]).

MSC:

35L70 Second-order nonlinear hyperbolic equations
35C10 Series solutions to PDEs

Citations:

Zbl 1139.35021
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] T. Aubin, Nonlinear Analysis on Manifolds: Monge-Ampère Equations, Grundlehren Math. Wiss. 252 , Springer, New York, 1982. · Zbl 0512.53044
[2] P. Bizoń, T. Chmaj, and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity , Nonlinearity 17 (2004), 2187–2201. · Zbl 1064.74112 · doi:10.1088/0951-7715/17/6/009
[3] N. Dunford and J. I. Schwartz, Linear Operators, Part II, Wiley Classics Lib., Wiley, New York, 1988. · Zbl 0635.47002
[4] F. Gesztesy and M. Zinchenko, On spectral theory for Schrödinger operators with strongly singular potentials , Math. Nachr. 279 (2006), 1041–1082. · Zbl 1108.34063 · doi:10.1002/mana.200510410
[5] M. G. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity , Ann. of Math. (2) 132 (1990), 485–509. JSTOR: · Zbl 0736.35067 · doi:10.2307/1971427
[6] K. JöRgens, Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen , Math. Z. 77 (1961), 295–308. · Zbl 0111.09105 · doi:10.1007/BF01180181
[7] P. Karageorgis and W. A. Strauss, Instability of steady states for nonlinear wave and heat equations , J. Differential Equations 241 (2007), 184–205. · Zbl 1130.35015 · doi:10.1016/j.jde.2007.06.006
[8] C. E. Kenig and F. 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), 645–675. · Zbl 1115.35125 · doi:10.1007/s00222-006-0011-4
[9] J. Krieger and W. Schlag, On the focusing critical semi-linear wave equation , Amer. J. Math. 129 (2007), 843–913. · Zbl 1219.35144 · doi:10.1353/ajm.2007.0021
[10] J. Krieger, W. Schlag, and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps , Invent. Math. 171 (2008), 543–615. · Zbl 1139.35021 · doi:10.1007/s00222-007-0089-3
[11] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form \(Pu\sbtt=-Au+\mathcal F(u)\), Trans. Amer. Math. Soc. 192 (1974), 1–21. · Zbl 0288.35003 · doi:10.2307/1996814
[12] V. A. Marchenko, Sturm-Liouville Operators and Applications, Oper. Theory Adv. Appl. 22 , Birkhäuser, Basel, 1986. \enlargethispage2pc
[13] F. Merle and H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math. 125 (2003), 1147–1164. · Zbl 1052.35043 · doi:10.1353/ajm.2003.0033
[14] -, Determination of the blow-up rate for a critical semilinear wave equation, Math. Ann. 331 (2005), 395–416. · Zbl 1136.35055 · doi:10.1007/s00208-004-0587-1
[15] -, On growth rate near the blowup surface for semilinear wave equations, Int. Math. Res. Not. 2005 , no. 19, 1127–1155. · Zbl 1160.35478 · doi:10.1155/IMRN.2005.1127
[16] I. Rodnianski and J. Sterbenz, On the formation of singularities in the critical \(O(3)\) sigma-model , preprint.v3[math.AP] · Zbl 1213.35392
[17] W. Schlag, Spectral theory and nonlinear partial differential equations: A survey, Discrete Contin. Dyn. Syst. 15 (2006), 703–723. · Zbl 1121.35121 · doi:10.3934/dcds.2006.15.703
[18] J. Shatah and M. Struwe, Geometric Wave Equations, Courant Lect. Notes Math. 2 , Amer. Math. Soc., Providence, 1998. \enlargethispage2pc · Zbl 0993.35001
[19] C. D. Sogge, Lectures on Nonlinear Wave Equations, Monogr. Anal. II , International, Boston, 1995.
[20] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals , Princeton Math. Ser. 43 , Princeton Univ. Press, Princeton 1994. · Zbl 0821.42001
[21] W. A. Strauss, Nonlinear Wave Equations, CBMS Regional Conf. Ser. in Math. 73 , Amer. Math. Soc., Providence, 1989.
[22] W. Struwe, Globally regular solutions to the \(u^ 5\) Klein-Gordon equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), 495–513. · Zbl 0728.35072
[23] M. E. Taylor, Tools for PDE, Math. Surveys Monogr. 81 , Amer. Math. Soc., Providence, 2000. · Zbl 0963.35211
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.