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Evolution Equations and Control Theory (EECT)
 

A note on global well-posedness and blow-up of some semilinear evolution equations
Pages: 355 - 372, Issue 3, September 2015

doi:10.3934/eect.2015.4.355      Abstract        References        Full text (504.7K)           Related Articles

Tarek Saanouni - University Tunis El Manar, Faculty of Sciences of Tunis, Department of Mathematics, 2092, Tunis, Tunisia (email)

1 S. Adachi and K. Tanaka, Trudinger type inequalities in $\mathbbR^N$ and their best exponent, Proc. Amer. Math. Society., 128 (1999), 2051-2057.       
2 D. R. Adams, Sobolev Spaces, Academic Press, New York, 1975.       
3 A. Atallah Baraket, Local existence and estimations for a semilinear wave equation in two dimension space, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 7 (2004), 1-21.       
4 H. Bahouri, S. Ibrahim and G. Perleman, Scattering for the critical 2-$D$ NLS with exponential growth, Diff. Int. Eq., 27 (2014), 233-268.       
5 T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Metodos Matematicos, 26, Instituto de Matematica UFRJ, 1996.
6 J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions, J. Hyperbolic Differ. Equ., 6 (2009), 549-575.       
7 J. Ginibre and G. Velo, On a class of a nonlinear Schrödinger equations. II: Scattering theory, general case, J. Funct. Anal., 32 (1979), 33-71.       
8 J. Ginibre and G. Velo, The Global Cauchy problem for nonlinear Klein-Gordon equation, Math. Z, 189 (1985), 487-505.       
9 E. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys., 4 (1963), 195-207.
10 S. Ibrahim, M. Majdoub and N. Masmoudi, Global solutions for a semilinear $2D$ Klein-Gordon equation with exponential type nonlinearity, Comm. Pure App. Math., 59 (2006), 1639-1658.       
11 S. Ibrahim, M. Majdoub and N. Masmoudi, Instability of $H^1$-supercritical waves, C. R. Acad. Sci. Paris, ser. I, 345 (2007), 133-138.       
12 S. Ibrahim, M. Majdoub, R. Jrad and T. Saanouni, Global well posedness of a $2D$ semilinear heat equation, Bull. Belg. Math. Soc., 21 (2014), 535-551.       
13 N. Ioku, The Cauchy problem for heat equations with exponential nonlinearity, J. D. E., 251 (2011), 1172-1194.       
14 J. F. Lam, B. Lippman and F. Trappert, Self trapped laser beams in plasma, Phys. Fluid, 20 (1977), 1176-1179.
15 H. A. Levine, Some nonexistence and stability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+F(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.       
16 S. Le Coz, A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations, Adv. Nonlinear Stud., 8 (2008), 455-463.       
17 O. Mahouachi and T. Saanouni, Global well posedness and linearization of a semilinear wave equation with exponential growth, Georgian Math. J., 17 (2010), 543-562.       
18 O. Mahouachi and T. Saanouni, Well and ill-posedness issues for a class of $2D$ wave equation with exponential growth, J. Partial. Diff. Eqs., 24 (2011), 361-384.       
19 C. Miao and B. Zhang, The Cauchy problem for semilinear parabolic equations in Besov spaces, Houston J. Math., 30 (2004), 829-878.       
20 J. Moser, A sharp form of an inequality of N. Trudinger, Ind. Univ. Math. J., 20 (1971), 1077-1092.       
21 M. Nakamura and T. Ozawa, Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth, Math. Z., 231 (1999), 479-487.       
22 M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order, Journal of Functional Analysis, 155 (1998), 364-380.       
23 L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics, 22 (1975), 273-303.       
24 L. P. Pitaevski, Vortex lines in an imperfect Bose gas, J. Experimental Theoret. Phys., 13 (1961), p.646; Translation in Soviet Phys. JETP, 40 (1961), 451-454.
25 B. Ruf, A sharp Moser-Trudinger type inequality for unbounded domains in $\mathbbR^2$, J. Funct. Anal, 219 (2005), 340-367.       
26 T. Saanouni, Global well-posedness and scattering of a $2D$ Schrödinger equation with exponential growth, Bull. Belg. Math. Soc. Simon Stevin., 17 (2010), 441-462.       
27 T. Saanouni, Decay of solutions to a $2D$ Schrödinger equation with exponential growth, J. P. D. E., 24 (2011), 37-54.       
28 T. Saanouni, Remarks on the semilinear Schrödinger equation, J. Math. Anal. Appl., 400 (2013), 331-344.       
29 T. Saanouni, Scattering of a $2D$ Schrödinger equation with exponential growth in the conformal space, Math. Meth. Appl. Sci., 33 (2010), 1046-1058.       
30 T. Saanouni, Global well-posedness and instability of a $2D$ Schrödinger equation with harmonic potential in the conformal space, Journal of Abstract Differential Equations and Applications, 4 (2013), 23-42.       
31 T. Saanouni, Blowing-up semilinear wave equation with exponential nonlinearity in two space dimensions, Proc. Indian Acad. Sci. (Math. Sci.), 123 (2013), 365-372.       
32 T. Saanouni, Global well-posedness of a damped Schrödinger equation in two space dimensions, Math. Meth. Appl. Sci., 37 (2014), 488-495.       
33 T. Saanouni, A blowing up wave equation with exponential type nonlinearity and arbitrary positive energy, J. Math. Anal. Appl., 421 (2015), 444-452.       
34 W. Strauss, Nonlinear Wave Equations, CBMS Regional Conference Series in Math., 73, Amer. Math. Soc., Providence, RI, 1989.       
35 M. Struwe, The critical nonlinear wave equation in $2$ space dimensions, J. European Math. Soc., 15 (2013), 1805-1823.       
36 M. Struwe, Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions, Math. Ann. Vol., 350 (2011), 707-719.       
37 C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse, Applied Mathematical Sciences. Vol. 139, Springer-Verlag, New York, 1999.       
38 N. S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.       
39 S. R. S. Varadhan, Lectures on Diffusion Problems and Partial Differential Equations, Courant Institute of Mathematical Sciences, New York, 1989.

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