Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T15:17:18.306Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on strong instability of standing waves for some semilinear wave and heat equations

Published online by Cambridge University Press:  09 March 2017

T. SAANOUNI
T. SAANOUNI*
Affiliation:
University of Tunis El Manar, Faculty of Science of Tunis, LR03ES04 partial differential Equations and applications, 2092 Tunis, Tunisia. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The initial value problems for some semilinear wave and heat equations are investigated in two space dimensions. By proving the existence of ground state, strong instability of standing waves for the associated wave and heat equations are obtained.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 1 , July 2018 , pp. 53 - 68
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Adachi, S. and Tanaka, K.. Trudinger type inequalities in ${\mathbb{R}}$N and their best exponent. Proc. Amer. Math. Soc. 128, no. 7 (1999), 2051–2057.Google Scholar
[2] Adams, D. R.. Sobolev Spaces (Academic Press, New York, 1975).Google Scholar
[3] Atallah Baraket, A.. Local existence and estimations for a semilinear wave equation in two dimension space. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat 8, no. 1 (2004), 1–21.Google Scholar
[4] Berestycki, H. and Cazenave, T.. Instabilité des tats stationnaires dans les quations de Schrödinger et de Klein–Gordon non linéaires. C. R. Acad. Sci. Paris 293, (1981), 489–492.Google Scholar
[5] Berestycki, H. and Lions, P.L.. Nonlinear scalar field equations. Arch. Rat. Mech. Anal. 82, (1983), 313–345.Google Scholar
[6] Brezis, H. and Casenave, T.. A nonlinear heat equation with singular initial data. J. Anal. math. 68, (1996), 73–90.Google Scholar
[7] Ginibre, J. and Velo, G.. The Global Cauchy problem for nonlinear Klein–Gordon equation. Math. Z. 189, (1985), 487–505.Google Scholar
[8] Grillakis, M.. Regularity and asymptotic behavior of the wave equation with a critical nonlinearity. Ann. of Math. 132, (1990), 485–509.CrossRefGoogle Scholar
[9] Haraux, A. and Weissler, F. B.. Non uniqueness for a semilinear initial value problem. Indiana Univ. Math. J. 31, (1982), 167–189.CrossRefGoogle Scholar
[10] Ibrahim, S., Majdoub, M. and Masmoudi, N.. Global solutions for a semilinear 2D Klein–Gordon equation with exponential type nonlinearity. Comm. Pure App. Math. 59, no. 11 (2006), 1639–1658.Google Scholar
[11] Ibrahim, S., Masmoudi, N. and Nakanishi, K.. Scattering threshold for the focusing nonlinear Klein–Gordon equation. Anal. PDE 4, no. 3 (2011), 405–460.CrossRefGoogle Scholar
[12] Ibrahim, S., Majdoub, M., Jrad, R. and Saanouni, T.. Global well posedness of a 2D semilinear heat equation. Bull. Belg. Math. Soc. Simon Stevin 21, no. 3 (2014), 535–551.Google Scholar
[13] Ioku, N.. The Cauchy problem for heat equations with exponential nonlinearity. J. Differential Equations 251, no. 4 (2011), 1172–1194.Google Scholar
[14] Jeanlean, L. and Le Coz, S.. Instability for standing waves on nonlinear Klein–Gordon equations via mountain pass arguments. Trans. Amer. Math. Soc. 361, no. 10 (2009), 5401–5416.Google Scholar
[15] 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, (2008), 147–212.Google Scholar
[16] Lebeau, G.. Nonlinear optics and supercritical wave equation. Bull. Soc. Roy. Sci. Liège 70, no. 4–6 (2001), 267–306.Google Scholar
[17] Lebeau, G.. Perte de régularité pour l'équation des ondes surcritique. Bull. Soc. Math. France 133, (2005), 145–157.Google Scholar
[18] Levine, H. A. Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = ċċ Au + F(u). Trans. Amer. Math. Soc. 192, (1974).Google Scholar
[19] Mahouachi, O. and Saanouni, T.. Global well posedness and linearization of a semilinear wave equation with exponential growth. Georgian Math. J. 17, no. 3 (2010), 543–562.Google Scholar
[20] Mahouachi, O. and Saanouni, T.. Well and ill-posedness issues for a class of 2D wave equation with exponential growth. J. Partial Differential Equations 24, no. 4 (2011), 361–384.Google Scholar
[21] Moser, J.. A sharp form of an inequality of N. Trudinger. Ind. Univ. Math. J. 20, (1971), 1077–1092.Google Scholar
[22] Nakamura, M. and Ozawa, T.. Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth. Math. Z 231, (1999), 479–487.CrossRefGoogle Scholar
[23] Ohta, M. and Todorova, G.. Strong instability of standing waves for nonlinear Klein–Gordon equations. Discrete Contin. Dyn. Syst. 12, no. 2 (2005), 315–322.Google Scholar
[24] Ohta, M. and Todorova, G.. Strong instability of standing waves for the nonlinear Klein–Gordon equation and the Klein–Gordon-Zakharov system. SIAM J. Math. Anal. 38, no. 6 (2007), 1912–1931.Google Scholar
[25] Payne, L. E. and Sattinger, D. H.. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 22, (1975), 273–303.Google Scholar
[26] Ruf, B.. A sharp Moser–Trudinger type inequality for unbounded domains in ${\mathbb{R}}$2. J. Funct. Anal. 219, (2004), 340–367.CrossRefGoogle Scholar
[27] Shatah, J. and Struwe, M.. Well-Posedness in the energy space for semilinear wave equation with critical growth. Ins. Math. Res. Not. 7, (1994), 303–309.Google Scholar
[28] Saanouni, T.. Global well-posedness and scattering of a 2D schrödinger equation with exponential growth. Bull. Belg. Math. Soc. Simon Stevin. 17, (2010), 441–462.CrossRefGoogle Scholar
[29] Shatah, J.. Unstable ground state of nonlinear Klein–Gordon equations. Trans. Amer. Math. Soc. 290, 2 (1985), 701–710.Google Scholar
[30] Shatah, J. and Strauss, W.. Instability of nonlinear bound states. Comm. Math. Phys. 100, no. 2 (1985), 173–190.Google Scholar
[31] Strauss, W.. Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55, no. 2 (1977), 149–162.Google Scholar
[32] Struwe, M.. Semilinear wave equations. Bull. Amer. Math. Soc, N.S. 26, (1992), 53–85.Google Scholar
[33] Struwe, M.. The critical nonlinear wave equation in two space dimensions. J. Eur. Math. Soc. 15, no. 5 (2013), 1805–1823.Google Scholar
[34] Struwe, M.. Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions. Math. Ann. 350, (2011), 707–719.Google Scholar
[35] Trudinger, N. S.. On imbedding into Orlicz spaces and some applications. J. Math. Mech. 17, (1967), 473–484.Google Scholar
[36] Weissler, F. B.. Local existence and nonexistence for a semilinear parabolic equation in Lp. Indiana Univ. Math. J. 29, (1980), 79–102.Google Scholar
[37] Weissler, F. B.. Existence and nonexistence of global solutions for a semilinear heat equation. Israel J. Math. 38, (1981), 29–40.Google Scholar