Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions

Merle, Frank; Zaag, Hatem

doi:10.1090/tran/6450

Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions
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Trans. Amer. Math. Soc. 368 (2016), 27-87 Request permission

Abstract:

This is the first of two papers devoted to the study of the properties of the blow-up surface for the $N$ dimensional semilinear wave equation with subconformal power nonlinearity. In a series of papers, we have clarified the situation in one space dimension. Our goal here is to extend some of the properties to higher dimension. In dimension one, an essential tool was to study the dynamics of the solution in similarity variables, near the set of non-zero equilibria, which are obtained by a Lorentz transform of the space-independent solution. As a matter of fact, the main part of this paper is to study similar objects in higher dimensions. More precisely, near that set of equilibria, we show that solutions are either non-global or go to zero or converge to some explicit equilibrium. We also show that the first case cannot occur in the characteristic case and that only the third possibility occurs in the non-characteristic case, thanks to the non-degeneracy of the blow-up limit, another new result in our paper. As a by-product of our techniques, we obtain the stability of the zero solution.

References

Serge Alinhac, Blowup for nonlinear hyperbolic equations, Progress in Nonlinear Differential Equations and their Applications, vol. 17, Birkhäuser Boston, Inc., Boston, MA, 1995. MR 1339762, DOI 10.1007/978-1-4612-2578-2
Serge Alinhac, A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations, Journées “Équations aux Dérivées Partielles” (Forges-les-Eaux, 2002) Univ. Nantes, Nantes, 2002, pp. Exp. No. I, 33. MR 1968197
Christophe Antonini and Frank Merle, Optimal bounds on positive blow-up solutions for a semilinear wave equation, Internat. Math. Res. Notices 21 (2001), 1141–1167. MR 1861514, DOI 10.1155/S107379280100054X
Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313, DOI 10.1007/BFb0064643
Piotr Bizoń, Threshold behavior for nonlinear wave equations, J. Nonlinear Math. Phys. 8 (2001), no. suppl., 35–41. Nonlinear evolution equations and dynamical systems (Kolimbary, 1999). MR 1821505, DOI 10.2991/jnmp.2001.8.s.7
P. Bizoń, P. Breitenlohner, D. Maison, and A. Wasserman, Self-similar solutions of the cubic wave equation, Nonlinearity 23 (2010), no. 2, 225–236. MR 2578477, DOI 10.1088/0951-7715/23/2/002
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
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, DOI 10.1088/0951-7715/20/9/003
Piotr Bizoń and Anıl Zenginoğlu, Universality of global dynamics for the cubic wave equation, Nonlinearity 22 (2009), no. 10, 2473–2485. MR 2539764, DOI 10.1088/0951-7715/22/10/009
Luis A. Caffarelli and Avner Friedman, Differentiability of the blow-up curve for one-dimensional nonlinear wave equations, Arch. Rational Mech. Anal. 91 (1985), no. 1, 83–98. MR 802832, DOI 10.1007/BF00280224
Luis A. Caffarelli and Avner Friedman, The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc. 297 (1986), no. 1, 223–241. MR 849476, DOI 10.1090/S0002-9947-1986-0849476-3
Raphaël Côte and Hatem Zaag, Construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension, Comm. Pure Appl. Math. 66 (2013), no. 10, 1541–1581. MR 3084698, DOI 10.1002/cpa.21452
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, DOI 10.4310/DPDE.2012.v9.n1.a3
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
Yoshikazu Giga and Robert V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), no. 6, 845–884. MR 1003437, DOI 10.1002/cpa.3160420607
J. Ginibre, A. Soffer, and G. Velo, The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal. 110 (1992), no. 1, 96–130. MR 1190421, DOI 10.1016/0022-1236(92)90044-J
Mohamed Ali Hamza and Hatem Zaag, Lyapunov functional and blow-up results for a class of perturbations of semilinear wave equations in the critical case, J. Hyperbolic Differ. Equ. 9 (2012), no. 2, 195–221. MR 2928106, DOI 10.1142/S0219891612500063
M. A. Hamza and H. Zaag, A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations, Nonlinearity 25 (2012), no. 9, 2759–2773. MR 2967123, DOI 10.1088/0951-7715/25/9/2759
Mohamed-Ali Hamza and Hatem Zaag, Blow-up results for semilinear wave equations in the superconformal case, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), no. 9, 2315–2329. MR 3108855, DOI 10.3934/dcdsb.2013.18.2315
Satyanad Kichenassamy and Walter Littman, Blow-up surfaces for nonlinear wave equations. I, Comm. Partial Differential Equations 18 (1993), no. 3-4, 431–452. MR 1214867, DOI 10.1080/03605309308820936
Satyanad Kichenassamy and Walter Littman, Blow-up surfaces for nonlinear wave equations. II, Comm. Partial Differential Equations 18 (1993), no. 11, 1869–1899. MR 1243529, DOI 10.1080/03605309308820997
Rowan Killip, Betsy Stovall, and Monica Visan, Blowup behaviour for the nonlinear Klein-Gordon equation, Math. Ann. 358 (2014), no. 1-2, 289–350. MR 3157999, DOI 10.1007/s00208-013-0960-z
R. Killip and M. Vişan, Smooth solutions to the nonlinear wave equation can blow up on Cantor sets, (2011), arXiv:1103.5257v1.
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
Hans Lindblad and Christopher D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), no. 2, 357–426. MR 1335386, DOI 10.1006/jfan.1995.1075
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, DOI 10.1353/ajm.2003.0033
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, DOI 10.1155/IMRN.2005.1127
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, DOI 10.1007/s00208-004-0587-1
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, DOI 10.1016/j.jfa.2007.03.007
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, DOI 10.1007/s00220-008-0532-3
Frank Merle and Hatem Zaag, Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension, Seminaire: Equations aux Dérivées Partielles. 2009–2010, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2012, pp. Exp. No. XI, 10. MR 3098623
Frank Merle and Hatem Zaag, Blow-up behavior outside the origin for a semilinear wave equation in the radial case, Bull. Sci. Math. 135 (2011), no. 4, 353–373. MR 2799813, DOI 10.1016/j.bulsci.2011.03.001
Frank Merle and Hatem Zaag, Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension, Amer. J. Math. 134 (2012), no. 3, 581–648. MR 2931219, DOI 10.1353/ajm.2012.0021
Frank Merle and Hatem Zaag, Isolatedness of characteristic points at blowup for a 1-dimensional semilinear wave equation, Duke Math. J. 161 (2012), no. 15, 2837–2908. MR 2999314, DOI 10.1215/00127094-1902040
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