Blow-up profile for the complex-valued semilinear wave equation
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Abstract:
In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power non-linearity in one space dimension. We first characterize all the solutions of the associated stationary problem as a two-parameter family. Then, we use a dynamical system formulation to show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic case. This gives the blow-up profile for the original equation in the non-characteristic case. Our analysis is not just a simple adaptation of the already handled real case. In particular, there is one more neutral-direction in our problem, which we control thanks to a modulation technique.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
- 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
- R. Donninger, M. Huang, J. Krieger and W. Schlag, Exotic blowup solutions for the $u^5$ focusing wave equation in $\mathbb {R}^3$, preprint, arXiv:1212.4718, 2012.
- 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, 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
- 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, 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, 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
- T. Duyckaerts, C. Kenig, and F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation, Cambridge J. Math. 1 (2013), 75–144.
- 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
- Stathis Filippas and Frank Merle, Modulation theory for the blowup of vector-valued nonlinear heat equations, J. Differential Equations 116 (1995), no. 1, 119–148. MR 1317705, DOI 10.1006/jdeq.1995.1031
- 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
- J. Ginibre and G. Velo, Regularity of solutions of critical and subcritical nonlinear wave equations, Nonlinear Anal. 22 (1994), no. 1, 1–19. MR 1256167, DOI 10.1016/0362-546X(94)90002-7
- M. A. Hamza and H. Zaag, Blow-up behavior for the Klein-Gordon and other perturbed semilinear wave equations, Bull. Sci. Math. 137 (2013), no. 8, 1087–1109. MR 3130348, DOI 10.1016/j.bulsci.2013.05.004
- 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
- S. Ibrahim, N. Masmoudi and K. Nakanishi, Threshold solutions in the case of mass-shift for the critical Klein–Gordon equation, Trans. Amer. Math. Soc., to appear, 2014.
- 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
- Carlos E. Kenig and Frank Merle, Radial solutions to energy supercritical wave equations in odd dimensions, Discrete Contin. Dyn. Syst. 31 (2011), no. 4, 1365–1381. MR 2836357, DOI 10.3934/dcds.2011.31.1365
- 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
- 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, 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
- 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
- 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, 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, 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
- F. Merle and H. Zaag. Isolatedness of characteristic points for a semilinear for a semilinear wave equation in one space dimension. Points caractéristiques à l’explosion pour une équation semilinéaire des ondes. In “Séminaire sur les Équations aux Dérivées Partielles, 2009-2010”, pages Exp. No. 11, 10p. École Polytech., Palaiseau, 2010.
- 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
- F. Merle and H. Zaag, Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions, Submitted. arXiv:1309.7756, 2013
- F. Merle and H. Zaag, On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations, Submitted. arXiv:1309.7760, 2013
- 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
- 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
Additional Information
- Asma Azaiez
- Affiliation: Institut Galilée, Laboratoire Analyse Géometrie et Applications, CNRS-UMR 7539, Université Paris 13, 99 avenue J.B. Clément 93430, Villetaneuse, France
- Email: azaiez@math.univ-paris13.fr
- Received by editor(s): June 19, 2013
- Received by editor(s) in revised form: October 25, 2013, and November 30, 2013
- Published electronically: August 12, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 5891-5933
- MSC (2010): Primary 35L05, 35L81, 35B44, 39B32, 35B40, 34K21, 35B35
- DOI: https://doi.org/10.1090/S0002-9947-2014-06370-8
- MathSciNet review: 3347192