Geodesic
On blow-up for the supercritical defocusing nonlinear wave equation
Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e15

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In this paper, we consider the defocusing nonlinear wave equation $-\partial _t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb {R}\times \mathbb {R}^d$. Building on our companion work (Self-similar imploding solutions of the relativistic Euler equations, arXiv:2403.11471), we prove that for $d=4, p\geq 29$ and $d\geq 5, p\geq 17$, there exists a smooth complex-valued solution that blows up in finite time. 
Shao, Feng; Wei, Dongyi; Zhang, Zhifei. On blow-up for the supercritical defocusing nonlinear wave equation. Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e15. doi: 10.1017/fmp.2025.7
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[1] Alinhac, S., Blowup for nonlinear hyperbolic equations (Progr. Nonlinear Differential Equations Appl.) vol. (Birkhäuser Boston, Inc., Boston, MA, 1995). Google Scholar

[2] Bourgain, J., ‘Problems in Hamitonian PDE’s’, Gemo. Funct. Anal. Special Volume, Part I (2000), 32–56. Google Scholar

[3] Brenner, P. and Von Wahl, W., ‘Global classical solutions of nonlinear wave equations’, Math. Z. 176(1) (1981), 87–121. Google Scholar | DOI

[4] Bulut, A., ‘Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation’, J. Funct. Anal. 263(6) (2012), 1609–1660. Google Scholar

[5] Bulut, A., ‘The defocusing energy-supercritical cubic nonlinear wave equation in dimension five’, Trans. Amer. Math. Soc. 367(9) (2015), 6017–6061. Google Scholar

[6] Christ, M., Colliander, J. and Tao, T., ‘Ill-posedness for nonlinear Schrodinger and wave equations’, Preprint, 2003, . Google Scholar | arXiv

[7] Collot, C., Raphaël, P. and Szeftel, J., ‘On the stability of type I blow up for the energy super critical heat equation’, Mem. Amer. Math. Soc. 260(1255) (2019), v+97 pp. Google Scholar

[8] Collot, C., Raphaël, P. and Szeftel, J., ‘Strongly anisotropic type II blow up at an isolated point’, J. Amer. Math. Soc. 33(2) (2020), 527–607. Google Scholar

[9] Cortázar, C., Del Pino, M. and Musso, M., ‘Green’s function and infinite-time bubbling in the critical nonlinear heat equation’, J. Eur. Math. Soc. (JEMS) 22(1) (2020), 283–344. Google Scholar

[10] Dávila, J., Del Pino, M. and Wei, J., ‘Singularity formation for the two-dimensional harmonic map flow into ’, Invent. Math. 219(2) (2020), 345–466. Google Scholar

[11] Del Pino, M., Musso, M. and Wei, J., ‘Infinite-time blow-up for the 3-dimensional energy-critical heat equation’, Anal. PDE 13(1) (2020), 215–274. Google Scholar | DOI

[12] Donninger, R., ‘Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation’, Duke Math. J. 166(9) (2017), 1627–1683. Google Scholar | DOI

[13] Duyckaerts, T., Jia, H., Kenig, C. E. and Merle, F., ‘Soliton resolution along a sequence of times for the focusing energy critical wave equation’, Geom. Funct. Anal. 27(4) (2017), 798–862. Google Scholar

[14] Duyckaerts, T., Jia, H., Kenig, C. E. and Merle, F., ‘Universality of blow up profile for small blow up solutions to the energy critical wave map equation’, Int. Math. Res. Not. IMRN 2018 (22) (2018), 6961–7025. Google Scholar

[15] Duyckaerts, T., Kenig, C. E. and Merle, F., ‘Profiles of bounded radial solutions of the focusing, energy-critical wave equation’, Geom. Funct. Anal. 22(3) (2012), 639–698. Google Scholar

[16] Duyckaerts, T., Kenig, C. E. and Merle, F., ‘Classification of radial solutions of the focusing, energy-critical wave equation’, Camb. J. Math. 1(1) (2013), 75–144. Google Scholar

[17] Duyckaerts, T., Kenig, C. E. and Merle, F., ‘Soliton resolution for the radial critical wave equation in all odd space dimensions’, Acta Math. 230(1) (2023), 1–92. Google Scholar

[18] Duyckaerts, T. and Yang, J., ‘Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations’, Anal. PDE 11(4) (2018), 983–1028. Google Scholar | DOI

[19] Ginibre, J. and Velo, G., ‘The global Cauchy problem for the nonlinear Klein-Gordon equation’, Math. Z. 189(4) (1985), 487–505. Google Scholar

[20] Ginibre, J. and Velo, G., ‘The global Cauchy problem for the nonlinear Klein-Gordon equation. II’, Ann. Inst. H. Poincaré Anal. Non Linéaire 6(1) (1989), 15–35. Google Scholar

[21] Grillakis, M., ‘Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity’, Ann. of Math. 132(3) (1990), 485–509. Google Scholar

[22] Grillakis, M., ‘Regularity for the wave equation with a critical nonlinearity’, Comm. Pure Appl. Math. 45(6) (1992), 749–774. Google Scholar | DOI

[23] Harada, J., ‘A type II blowup for the six dimensional energy critical heat equation’, Ann. PDE 6(2) (2020), Paper No. 13, 63 pp. Google Scholar

[24] Hardy, M., ‘Combinatorics of partial derivatives’, Electron. J. Combin. 13(1) (2006), Research Paper 1, 13 pp. Google Scholar

[25] Hörmander, L., Lectures on nonlinear hyperbolic differential equations (Mathématiques & Applications (Berlin)) vol. (Springer-Verlag, New York, 1997). Google Scholar

[26] Ibrahim, S., Majdoub, M. and Masmoudi, N., ‘Well- and ill-posedness issues for energy supercritical waves’, Anal. PDE 4(2) (2011), 341–367. Google Scholar

[27] Jendrej, J., ‘Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5’, J. Funct. Anal. 272(3) (2017), 866–917. Google Scholar | DOI

[28] Jendrej, J. and Lawrie, A., ‘Bubble decomposition for the harmonic map heat flow in the equivariant case’, Calc. Var. Partial Differential Equations (9) (2023), Paper No. 264, 36 pp. Google Scholar

[29] Jendrej, J. and Lawrie, A., ‘Soliton resolution for the energy-critical nonlinear wave equation in the radial case’, Ann. PDE (2) (2023), Paper No. 18, 117 pp. Google Scholar

[30] John, F., ‘Blow-up of solutions of nonlinear wave equations in three space dimensions’, Manuscripta Math. 28(1-3) (1979), 235–268. Google Scholar

[31] Jörgens, K., ‘Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen’, Math. Z. 77 (1961), 295–308. Google Scholar

[32] Kenig, C. E., Lectures on the energy critical nonlinear wave equation (CBMS Reg. Conf. Ser. Math.) vol. (Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2015). Google Scholar | DOI

[33] Kenig, C. E. and Merle, F., ‘Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case’, Invent. Math. 166(3) (2006), 645–675. Google Scholar

[34] Kenig, C. E. and Merle, F., ‘Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation’, Acta Math. 201(2) (2008), 147–212. Google Scholar

[35] Killip, R. and Visan, M., ‘The defocusing energy-supercritical nonlinear wave equation in three space dimensions’, Trans. Amer. Math. Soc. 363(7) (2011), 3893–3934. Google Scholar

[36] Killip, R. and Visan, M., ‘The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions’, Proc. Amer. Math. Soc. 139(5) (2011), 1805–1817. Google Scholar

[37] Kim, K. and Merle, F., ‘On classification of global dynamics for energy-critical equivariant harmonic map heat flows and radial nonlinear heat equation’, Preprint, 2024, . Google Scholar | arXiv

[38] Krieger, J. and Miao, S., ‘On the stability of blowup solutions for the critical corotational wave-map problem’, Duke Math. J. 169(3) (2020), 435–532. Google Scholar

[39] Krieger, J., Miao, S. and Schlag, W., ‘A stability theory beyond the co-rotational setting for critical Wave Maps blow up’, Preprint, 2024, . Google Scholar | arXiv

[40] Krieger, J. and Schlag, W., ‘Full range of blow up exponents for the quintic wave equation in three dimensions’, J. Math. Pures Appl. (9) 101(6) (2014), 873–900. Google Scholar

[41] Krieger, J., Schlag, W. and Tataru, D., ‘Renormalization and blow up for charge one equivariant critical wave maps’, Invent. Math. 171(3) (2008), 543–615. Google Scholar

[42] Krieger, J., Schlag, W. and Tataru, D., ‘Slow blow-up solutions for the critical focusing semilinear wave equation’, Duke Math. J. 147(1) (2009), 1–53. Google Scholar

[43] Levine, H. A., ‘Instability and nonexistence of global solutions to nonlinear wave equations of the form ’, Trans. Amer. Math. Soc. 192 (1974), 1–21. Google Scholar

[44] Lindblad, H. and Sogge, C. D., ‘Long-time existence for small amplitude semilinear wave equations’, Amer. J. Math. 118(5) (1996), 1047–1135. Google Scholar

[45] Luk, J., Introduction to nonlinear wave equations, https://web.stanford.edu/~jluk/NWnotes.pdf. Google Scholar

[46] Martel, Y., Yvan, and Merle, F., ‘Inelasticity of soliton collisions for the 5D energy critical wave equation’, Invent. Math. 214(3) (2018), 1267–1363. Google Scholar | DOI

[47] Matano, H. and Merle, F., ‘On nonexistence of type II blowup for a supercritical nonlinear heat equation’, Comm. Pure Appl. Math. 57(11) (2004), 1494–1541. Google Scholar | DOI

[48] Merle, F. and Raphaël, P., ‘On a sharp lower bound on the blow-up rate for the critical nonlinear Schrödinger equation’, J. Amer. Math. Soc. 19(1) (2006), 37–90. Google Scholar | DOI

[49] Merle, F., Raphaël, P. and Rodnianski, I., ‘Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem’, Invent. Math. 193(2) (2013), 249–365. Google Scholar

[50] Merle, F., Raphaël, P. and Rodnianski, I., ‘Type II blow up for the energy supercritical NLS’, Camb. J. Math. 3(4) (2015), 439–617. Google Scholar

[51] Merle, F., Raphaël, P., Rodnianski, I. and Szeftel, J., On blow up for the energy super critical defocusing nonlinear Schrödinger equations, Invent. Math. 227(1) (2022), 247–413. Google Scholar

[52] Merle, F., Raphaël, P., Rodnianski, I. and Szeftel, J., ‘On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles’, Ann. of Math. (2) 196(2) (2022), 567–778. Google Scholar | DOI

[53] Merle, F., Raphaël, P., Rodnianski, I. and Szeftel, J., ‘On the implosion of a compressible fluid II: Singularity formation’, Ann. of Math. (2) 196(2) (2022), 779–889. Google Scholar

[54] Merle, F., Raphaël, P. and Szeftel, J., ‘Stable self-similar blow-up dynamics for slightly super-critical NLS equations’, Geom. Funct. Anal. 20(4) (2010), 1028–1071. Google Scholar | DOI

[55] Merle, F., Raphaël, P. and Szeftel, J., ‘On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation’, Duke Math. J. 163(2) (2014), 369–431. Google Scholar | DOI

[56] Merle, F., Raphaël, P. and Szeftel, J., ‘On strongly anisotropic type I blowup’, Int. Math. Res. Not. IMRN 2020 (2) (2020), 541–606. Google Scholar

[57] Merle, F. and Zaag, H., ‘Stability of the blow-up profile for equations of the type ’, Duke Math. J. 86(1) (1997), 143–195. Google Scholar

[58] Merle, F. and Zaag, H., ‘Determination of the blow-up rate for the semilinear wave equation’, Amer. J. Math. 125(5) (2003), 1147–1164. Google Scholar

[59] Perelman, G., ‘On the formation of singularities in solutions of the critical nonlinear Schrödinger equation’, Ann. Henri Poincaré 2(4) (2001), 605–673. Google Scholar | DOI

[60] Perelman, G., ‘Blow up dynamics for equivariant critical Schrödinger maps’, Comm. Math. Phys. 330(1) (2014), 69–105. Google Scholar

[61] Perelman, G., ‘Formation of singularities in nonlinear dispersive PDEs’, in ICM—International Congress of Mathematicians. Vol. 5. Sections 9-11 (EMS Press, Berlin, 2023), 3854–3879. Google Scholar

[62] Raphaël, P. and Rodnianski, I., ‘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. Google Scholar | DOI

[63] Raphaël, P. and Schweyer, R., ‘Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow’, Comm. Pure Appl. Math. 66(3) (2013), 414–480. Google Scholar

[64] Rodnianski, I. and Sterbenz, J., ‘On the formation of singularities in the critical -model’, Ann. of Math. (2) 172(1) (2010), 187–242. Google Scholar | DOI

[65] Shao, F., Wei, D. and Zhang, Z., ‘Self-similar imploding solutions of the relativistic Euler equations’, Preprint, 2024, . Google Scholar | arXiv

[66] Shatah, J. and Struwe, M., ‘Regularity results for nonlinear wave equations’, Ann. of Math. (2) 138(3) (1993), 503–518. Google Scholar | DOI

[67] Shatah, J. and Struwe, M., ‘Well-posedness in the energy space for semilinear wave equations with critical growth’, Int. Math. Res. Not. IMRN 1994 (7) (1994), 303–309. Google Scholar | DOI

[68] Sogge, C., Lectures on Nonlinear Wave Equations (Monographs in Analysis, II. International Press, Boston, MA, 1995). Google Scholar

[69] Stein, E. M. and Shakarchi, R., Complex analysis (Princeton Lectures in Analysis) vol. (Princeton University Press, Princeton, NJ, 2003). Google Scholar

[70] Strauss, W. A., Nonlinear wave equations (CBMS Regional Conference Series in Mathematics) vol. (American Mathematical Society, Providence, RI, 1989). Google Scholar

[71] Struwe, M., ‘Globally regular solutions to the Klein-Gordon equation’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15(3) (1988), 495–513. Google Scholar

[72] Tao, T., ‘Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data’, J. Hyperbolic Differ. Equ. 4(2) (2007), 259–265. Google Scholar

[73] Tao, T., ‘Finite-time blowup for a supercritical defocusing nonlinear wave system’, Anal. PDE 9(8) (2016), 1999–2030. Google Scholar | DOI

[74] Tao, T., ‘Finite time blowup for a supercritical defocusing nonlinear Schrödinger system’, Anal. PDE 11(2) (2018), 383–438. Google Scholar | DOI

[75] Yang, S., ‘Global behaviors of defocusing semilinear wave equations’, Ann. Sci. Éc. Norm. Supér. (4), 55(2) (2022), 405–428. Google Scholar | DOI

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