A modulation technique for the blow-up profile of the vector-valued semilinear wave equation. (English) Zbl 1387.35065
Summary: We consider a vector-valued blow-up solution with values in \(\mathbb{R}^m\) for the semilinear wave equation with power nonlinearity in one space dimension (this is a system of PDEs). We first characterize all the solutions of the associated stationary problem as an \(m\)-parameter family. Then, we show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic cases. Our analysis is not just a simple adaptation of the already handled real or complex case. In particular, there is a new structure of the set of stationary solutions.
References:
| [1] | Alinhac, S., Blowup for Nonlinear Hyperbolic Equations, Progress in Nonlinear Differential Equations and Their Applications, vol. 17 (1995), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 0820.35001 |
| [2] | Alinhac, S., A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations, (Journées “Équations aux Dérivées Partielles”. Journées “Équations aux Dérivées Partielles”, Forges-les-Eaux, 2002 (2002), Univ. Nantes: Univ. Nantes Nantes), pages Exp. No. I, 33 |
| [3] | Antonini, C.; Merle, F., Optimal bounds on positive blow-up solutions for a semilinear wave equation, Int. Math. Res. Not., 21, 1141-1167 (2001) · Zbl 0989.35090 |
| [4] | Azaiez, A., Blow-up profile for the complex-valued semilinear wave equation, Trans. Am. Math. Soc., 367, 8, 5891-5933 (2015) · Zbl 1316.35200 |
| [5] | Azaiez, A.; Masmoudi, N.; Zaag, H., Blow-up rate for a semilinear wave equation with exponential nonlinearity in one space dimension, (Proceedings of the Conference in Honor of Pr. Abbas Bahri (2015)), in press |
| [6] | Caffarelli, L. A.; Friedman, A., The blow-up boundary for nonlinear wave equations, Trans. Am. Math. Soc., 297, 1, 223-241 (1986) · Zbl 0638.35053 |
| [7] | Côte, R.; Zaag, H., Construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension, Commun. Pure Appl. Math., 66, 10, 1541-1581 (2013) · Zbl 1295.35124 |
| [8] | Filippas, S.; Merle, F., Modulation theory for the blowup of vector-valued nonlinear heat equations, J. Differ. Equ., 116, 1, 119-148 (1995) · Zbl 0814.35043 |
| [9] | Ginibre, J.; Soffer, A.; Velo, G., The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal., 110, 1, 96-130 (1992) · Zbl 0813.35054 |
| [10] | Ginibre, J.; Velo, G., Regularity of solutions of critical and subcritical nonlinear wave equations, Nonlinear Anal., 22, 1, 1-19 (1994) · Zbl 0831.35108 |
| [11] | Kichenassamy, S.; Littman, W., Blow-up surfaces for nonlinear wave equations. I, Commun. Partial Differ. Equ., 18, 3-4, 431-452 (1993) · Zbl 0803.35093 |
| [12] | Kichenassamy, S.; Littman, W., Blow-up surfaces for nonlinear wave equations. II, Commun. Partial Differ. Equ., 18, 11, 1869-1899 (1993) · Zbl 0803.35094 |
| [13] | Levine, H. A., Instability and nonexistence of global solutions to nonlinear wave equations of the form \(P u_{t t} = - A u + F(u)\), Trans. Am. Math. Soc., 192, 1-21 (1974) · Zbl 0288.35003 |
| [14] | Lindblad, H.; Sogge, C. D., On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130, 2, 357-426 (1995) · Zbl 0846.35085 |
| [15] | Merle, F.; Zaag, H., Determination of the blow-up rate for the semilinear wave equation, Am. J. Math., 125, 5, 1147-1164 (2003) · Zbl 1052.35043 |
| [16] | Merle, F.; Zaag, H., Determination of the blow-up rate for a critical semilinear wave equation, Math. Ann., 331, 2, 395-416 (2005) · Zbl 1136.35055 |
| [17] | Merle, F.; Zaag, H., On growth rate near the blowup surface for semilinear wave equations, Int. Math. Res. Not., 19, 1127-1155 (2005) · Zbl 1160.35478 |
| [18] | Merle, F.; Zaag, H., Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Anal., 253, 1, 43-121 (2007) · Zbl 1133.35070 |
| [19] | Merle, F.; Zaag, H., Openness of the set of non-characteristic points and regularity of the blow-up curve for the 1 D semilinear wave equation, Commun. Math. Phys., 282, 1, 55-86 (2008) · Zbl 1159.35046 |
| [20] | Merle, F.; Zaag, H., Points caractéristiques à l’explosion pour une équation semilinéaire des ondes, (Séminaire X-EDP (2010), École Polytech: École Polytech Palaiseau) |
| [21] | Merle, F.; Zaag, H., Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension, Am. J. Math., 134, 3, 581-648 (2012) · Zbl 1252.35204 |
| [22] | Merle, F.; Zaag, H., Isolatedness of characteristic points at blowup for a 1-dimensional semilinear wave equation, Duke Math. J., 161, 15, 2837-2908 (2012) · Zbl 1270.35320 |
| [23] | Merle, F.; Zaag, H., On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations, Commun. Math. Phys., 333, 3, 1529-1562 (2015) · Zbl 1315.35134 |
| [24] | Merle, F.; Zaag, H., Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions, Trans. Am. Math. Soc., 368, 1, 27-87 (2016) · Zbl 1339.35062 |
| [25] | Shatah, J.; Struwe, M., Geometric Wave Equations, Courant Lecture Notes in Mathematics, vol. 2 (1998), New York University Courant Institute of Mathematical Sciences: New York University Courant Institute of Mathematical Sciences New York · Zbl 0993.35001 |
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