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September  2019, 18(5): 2397-2408. doi: 10.3934/cpaa.2019108

Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation

Asma Azaiez

Université de Cergy-Pontoise, AGM, CNRS (UMR 8088), 95302, Cergy-Pontoise, France

Received  April 2018 Revised  January 2019 Published  April 2019

Fund Project: This author is supported by the ERC Advanced Grant no. 291214, BLOWDISOL.

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 show that the set of non characteristic points $ I_0 $ is open and that the blow-up curve is of class $ C^{1, \mu_0} $ and the phase $ \theta $ is $ C^{\mu_0} $ on this set. In order to prove this result, we introduce a Liouville Theorem for that equation. Our results hold also for the case of solutions with values in $ \mathbb{R}^m $ with $ m\ge 3 $, with the same proof.

Keywords: Wave equation, vector-valued PDE, Liouville Theorem, regularity.
Mathematics Subject Classification: Primary: 35L05, 35B44, 35B53; Secondary: 58J45.
Citation: Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108
References:
[1]

S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Progress in Nonlinear Differential Equations and Their Applications, 17. Birkhäuser Boston Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-2578-2.  Google Scholar

[2]

S. Alinhac, A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations, In Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002), pages Exp. No. I, 33. Univ. Nantes, Nantes, 2002.  Google Scholar

[3]

C. Antonini and F. Merle, Optimal bounds on positive blow-up solutions for a semilinear wave equation, Internat. Math. Res. Notices, 21 (2001), 1141-1167.  doi: 10.1155/S107379280100054X.  Google Scholar

[4]

A. Azaiez, Blow-up profile for the complex-valued semilinear wave equation, Trans. Amer. Math. Soc., 367 (2015), 5891-5933.  doi: 10.1090/S0002-9947-2014-06370-8.  Google Scholar

[5]

Blow-up rate for a semilinear wave equation with exponential nonlinearity in one space dimension, Proceedings of the MIMS-CIMPA Research School "Partial Differential Equations arising from Physics and Geometry", 2015. Google Scholar

[6]

A. Azaiez and H. Zaag, A modulation technique for the blow-up profile of the vector-valued semilinear wave equation, Bull. Sci. Math., 141 (2017), 312-352.  doi: 10.1016/j.bulsci.2017.04.001.  Google Scholar

[7]

L. A. Caffarelli and A. Friedman, The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc., 297 (1986), 223-241.  doi: 10.2307/2000465.  Google Scholar

[8]

R. Côte and H. Zaag, Construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension, Comm. Pure Appl. Math., 66 (2013), 1541-1581.  doi: 10.1002/cpa.21452.  Google Scholar

[9]

J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal., 110 (1992), 96-130.  doi: 10.1016/0022-1236(92)90044-J.  Google Scholar

[10]

J. Ginibre and G. Velo, Regularity of solutions of critical and subcritical nonlinear wave equations, Nonlinear Anal., 22 (1994), 1-19.  doi: 10.1016/0362-546X(94)90002-7.  Google Scholar

[11]

S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations. Ⅰ, Comm. Partial Differential Equations, 18 (1993), 431-452.  doi: 10.1080/03605309308820936.  Google Scholar

[12]

S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations. Ⅱ, Comm. Partial Differential Equations, 18 (1993), 1869-1899.  doi: 10.1080/03605309308820997.  Google Scholar

[13]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt} = -Au+{\mathcal F}(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.  Google Scholar

[14]

H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130 (1995), 357-426.  doi: 10.1006/jfan.1995.1075.  Google Scholar

[15]

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

[16]

F. Merle and H. Zaag, On growth rate near the blowup surface for semilinear wave equations, Int. Math. Res. Not., 19 (2005), 1127-1155.  doi: 10.1155/IMRN.2005.1127.  Google Scholar

[17]

F. Merle and H. Zaag, Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Anal., 253 (2007), 43-121.  doi: 10.1016/j.jfa.2007.03.007.  Google Scholar

[18]

F. Merle and H. 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), 55-86.  doi: 10.1007/s00220-008-0532-3.  Google Scholar

[19]

Points caractéristiques à l'explosion pour une équation semilinéaire des ondes, In "Séminaire X-EDP". École Polytech., Palaiseau, 2010. Google Scholar

[20]

F. Merle and H. Zaag, Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension, Amer. J. Math., 134 (2012), 581-648.  doi: 10.1353/ajm.2012.0021.  Google Scholar

[21]

F. Merle and H. Zaag, Isolatedness of characteristic points at blowup for a 1-dimensional semilinear wave equation, Duke Math. J, 161 (2012), 2837-2908.  doi: 10.1215/00127094-1902040.  Google Scholar

[22]

F. Merle and H. Zaag, On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations, Comm. Math. Phys., 333 (20153), 1529-1562.  doi: 10.1007/s00220-014-2132-8.  Google Scholar

[23]

F. Merle and H. 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), 27-87.  doi: 10.1090/tran/6450.  Google Scholar

[24]

N. Nouaili, $c^{1,\mu_0}$ regularity of the blow-up curve at non characteristic points for the one dimensional semilinear wave equation, Comm. Partial Differential Equations, 33 (2008), 1540-1548.  doi: 10.1080/03605300802234937.  Google Scholar

[25]

J. Shatah and M. Struwe, Geometric Wave Equations, volume 2 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 1998.  Google Scholar

show all references

References:
[1]

S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Progress in Nonlinear Differential Equations and Their Applications, 17. Birkhäuser Boston Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-2578-2.  Google Scholar

[2]

S. Alinhac, A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations, In Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002), pages Exp. No. I, 33. Univ. Nantes, Nantes, 2002.  Google Scholar

[3]

C. Antonini and F. Merle, Optimal bounds on positive blow-up solutions for a semilinear wave equation, Internat. Math. Res. Notices, 21 (2001), 1141-1167.  doi: 10.1155/S107379280100054X.  Google Scholar

[4]

A. Azaiez, Blow-up profile for the complex-valued semilinear wave equation, Trans. Amer. Math. Soc., 367 (2015), 5891-5933.  doi: 10.1090/S0002-9947-2014-06370-8.  Google Scholar

[5]

Blow-up rate for a semilinear wave equation with exponential nonlinearity in one space dimension, Proceedings of the MIMS-CIMPA Research School "Partial Differential Equations arising from Physics and Geometry", 2015. Google Scholar

[6]

A. Azaiez and H. Zaag, A modulation technique for the blow-up profile of the vector-valued semilinear wave equation, Bull. Sci. Math., 141 (2017), 312-352.  doi: 10.1016/j.bulsci.2017.04.001.  Google Scholar

[7]

L. A. Caffarelli and A. Friedman, The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc., 297 (1986), 223-241.  doi: 10.2307/2000465.  Google Scholar

[8]

R. Côte and H. Zaag, Construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension, Comm. Pure Appl. Math., 66 (2013), 1541-1581.  doi: 10.1002/cpa.21452.  Google Scholar

[9]

J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal., 110 (1992), 96-130.  doi: 10.1016/0022-1236(92)90044-J.  Google Scholar

[10]

J. Ginibre and G. Velo, Regularity of solutions of critical and subcritical nonlinear wave equations, Nonlinear Anal., 22 (1994), 1-19.  doi: 10.1016/0362-546X(94)90002-7.  Google Scholar

[11]

S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations. Ⅰ, Comm. Partial Differential Equations, 18 (1993), 431-452.  doi: 10.1080/03605309308820936.  Google Scholar

[12]

S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations. Ⅱ, Comm. Partial Differential Equations, 18 (1993), 1869-1899.  doi: 10.1080/03605309308820997.  Google Scholar

[13]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt} = -Au+{\mathcal F}(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.  Google Scholar

[14]

H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130 (1995), 357-426.  doi: 10.1006/jfan.1995.1075.  Google Scholar

[15]

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

[16]

F. Merle and H. Zaag, On growth rate near the blowup surface for semilinear wave equations, Int. Math. Res. Not., 19 (2005), 1127-1155.  doi: 10.1155/IMRN.2005.1127.  Google Scholar

[17]

F. Merle and H. Zaag, Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Anal., 253 (2007), 43-121.  doi: 10.1016/j.jfa.2007.03.007.  Google Scholar

[18]

F. Merle and H. 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), 55-86.  doi: 10.1007/s00220-008-0532-3.  Google Scholar

[19]

Points caractéristiques à l'explosion pour une équation semilinéaire des ondes, In "Séminaire X-EDP". École Polytech., Palaiseau, 2010. Google Scholar

[20]

F. Merle and H. Zaag, Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension, Amer. J. Math., 134 (2012), 581-648.  doi: 10.1353/ajm.2012.0021.  Google Scholar

[21]

F. Merle and H. Zaag, Isolatedness of characteristic points at blowup for a 1-dimensional semilinear wave equation, Duke Math. J, 161 (2012), 2837-2908.  doi: 10.1215/00127094-1902040.  Google Scholar

[22]

F. Merle and H. Zaag, On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations, Comm. Math. Phys., 333 (20153), 1529-1562.  doi: 10.1007/s00220-014-2132-8.  Google Scholar

[23]

F. Merle and H. 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), 27-87.  doi: 10.1090/tran/6450.  Google Scholar

[24]

N. Nouaili, $c^{1,\mu_0}$ regularity of the blow-up curve at non characteristic points for the one dimensional semilinear wave equation, Comm. Partial Differential Equations, 33 (2008), 1540-1548.  doi: 10.1080/03605300802234937.  Google Scholar

[25]

J. Shatah and M. Struwe, Geometric Wave Equations, volume 2 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 1998.  Google Scholar

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