# American Institute of Mathematical Sciences

September  2015, 4(3): 355-372. doi: 10.3934/eect.2015.4.355

## A note on global well-posedness and blow-up of some semilinear evolution equations

 1 University Tunis El Manar, Faculty of Sciences of Tunis, Department of Mathematics, 2092, Tunis

Received  January 2015 Revised  July 2015 Published  September 2015

We investigate the initial value problems for some semilinear wave, heat and Schrödinger equations in two space dimensions, with exponential nonlinearities. Using the potential well method based on the concepts of invariant sets, we prove either global well-posedness or finite time blow-up.
Citation: Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations & Control Theory, 2015, 4 (3) : 355-372. doi: 10.3934/eect.2015.4.355
##### References:
 [1] S. Adachi and K. Tanaka, Trudinger type inequalities in $\mathbbR^N$ and their best exponent,, Proc. Amer. Math. Society., 128 (1999), 2051. doi: 10.1090/S0002-9939-99-05180-1. Google Scholar [2] D. R. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar [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. Google Scholar [4] H. Bahouri, S. Ibrahim and G. Perleman, Scattering for the critical 2-$D$ NLS with exponential growth,, Diff. Int. Eq., 27 (2014), 233. Google Scholar [5] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations,, Textos de Metodos Matematicos, (1996). Google Scholar [6] J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions,, J. Hyperbolic Differ. Equ., 6 (2009), 549. doi: 10.1142/S0219891609001927. Google Scholar [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. doi: 10.1016/0022-1236(79)90077-6. Google Scholar [8] J. Ginibre and G. Velo, The Global Cauchy problem for nonlinear Klein-Gordon equation,, Math. Z, 189 (1985), 487. doi: 10.1007/BF01168155. Google Scholar [9] E. Gross, Hydrodynamics of a superfluid condensate,, J. Math. Phys., 4 (1963), 195. doi: 10.1063/1.1703944. Google Scholar [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. doi: 10.1002/cpa.20127. Google Scholar [11] S. Ibrahim, M. Majdoub and N. Masmoudi, Instability of $H^1$-supercritical waves,, C. R. Acad. Sci. Paris, 345 (2007), 133. doi: 10.1016/j.crma.2007.06.008. Google Scholar [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. Google Scholar [13] N. Ioku, The Cauchy problem for heat equations with exponential nonlinearity,, J. D. E., 251 (2011), 1172. doi: 10.1016/j.jde.2011.02.015. Google Scholar [14] J. F. Lam, B. Lippman and F. Trappert, Self trapped laser beams in plasma,, Phys. Fluid, 20 (1977), 1176. doi: 10.1063/1.861679. Google Scholar [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. Google Scholar [16] S. Le Coz, A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations,, Adv. Nonlinear Stud., 8 (2008), 455. Google Scholar [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. Google Scholar [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. doi: 10.4208/jpde.v24.n4.7. Google Scholar [19] C. Miao and B. Zhang, The Cauchy problem for semilinear parabolic equations in Besov spaces,, Houston J. Math., 30 (2004), 829. Google Scholar [20] J. Moser, A sharp form of an inequality of N. Trudinger,, Ind. Univ. Math. J., 20 (1971), 1077. Google Scholar [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. doi: 10.1007/PL00004737. Google Scholar [22] M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order,, Journal of Functional Analysis, 155 (1998), 364. doi: 10.1006/jfan.1997.3236. Google Scholar [23] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel Journal of Mathematics, 22 (1975), 273. doi: 10.1007/BF02761595. Google Scholar [24] L. P. Pitaevski, Vortex lines in an imperfect Bose gas,, J. Experimental Theoret. Phys., 13 (1961), 451. Google Scholar [25] B. Ruf, A sharp Moser-Trudinger type inequality for unbounded domains in $\mathbbR^2$,, J. Funct. Anal, 219 (2005), 340. doi: 10.1016/j.jfa.2004.06.013. Google Scholar [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. Google Scholar [27] T. Saanouni, Decay of solutions to a $2D$ Schrödinger equation with exponential growth,, J. P. D. E., 24 (2011), 37. Google Scholar [28] T. Saanouni, Remarks on the semilinear Schrödinger equation,, J. Math. Anal. Appl., 400 (2013), 331. doi: 10.1016/j.jmaa.2012.11.037. Google Scholar [29] T. Saanouni, Scattering of a $2D$ Schrödinger equation with exponential growth in the conformal space,, Math. Meth. Appl. Sci., 33 (2010), 1046. doi: 10.1002/mma.1237. Google Scholar [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. Google Scholar [31] T. Saanouni, Blowing-up semilinear wave equation with exponential nonlinearity in two space dimensions,, Proc. Indian Acad. Sci. (Math. Sci.), 123 (2013), 365. doi: 10.1007/s12044-013-0132-9. Google Scholar [32] T. Saanouni, Global well-posedness of a damped Schrödinger equation in two space dimensions,, Math. Meth. Appl. Sci., 37 (2014), 488. doi: 10.1002/mma.2804. Google Scholar [33] T. Saanouni, A blowing up wave equation with exponential type nonlinearity and arbitrary positive energy,, J. Math. Anal. Appl., 421 (2015), 444. doi: 10.1016/j.jmaa.2014.07.033. Google Scholar [34] W. Strauss, Nonlinear Wave Equations,, CBMS Regional Conference Series in Math., (1989). Google Scholar [35] M. Struwe, The critical nonlinear wave equation in $2$ space dimensions,, J. European Math. Soc., 15 (2013), 1805. doi: 10.4171/JEMS/404. Google Scholar [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. doi: 10.1007/s00208-010-0567-6. Google Scholar [37] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse,, Applied Mathematical Sciences. Vol. 139, (1999). Google Scholar [38] N. S. Trudinger, On imbedding into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473. Google Scholar [39] S. R. S. Varadhan, Lectures on Diffusion Problems and Partial Differential Equations,, Courant Institute of Mathematical Sciences, (1989). Google Scholar

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##### References:
 [1] S. Adachi and K. Tanaka, Trudinger type inequalities in $\mathbbR^N$ and their best exponent,, Proc. Amer. Math. Society., 128 (1999), 2051. doi: 10.1090/S0002-9939-99-05180-1. Google Scholar [2] D. R. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar [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. Google Scholar [4] H. Bahouri, S. Ibrahim and G. Perleman, Scattering for the critical 2-$D$ NLS with exponential growth,, Diff. Int. Eq., 27 (2014), 233. Google Scholar [5] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations,, Textos de Metodos Matematicos, (1996). Google Scholar [6] J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions,, J. Hyperbolic Differ. Equ., 6 (2009), 549. doi: 10.1142/S0219891609001927. Google Scholar [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. doi: 10.1016/0022-1236(79)90077-6. Google Scholar [8] J. Ginibre and G. Velo, The Global Cauchy problem for nonlinear Klein-Gordon equation,, Math. Z, 189 (1985), 487. doi: 10.1007/BF01168155. Google Scholar [9] E. Gross, Hydrodynamics of a superfluid condensate,, J. Math. Phys., 4 (1963), 195. doi: 10.1063/1.1703944. Google Scholar [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. doi: 10.1002/cpa.20127. Google Scholar [11] S. Ibrahim, M. Majdoub and N. Masmoudi, Instability of $H^1$-supercritical waves,, C. R. Acad. Sci. Paris, 345 (2007), 133. doi: 10.1016/j.crma.2007.06.008. Google Scholar [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. Google Scholar [13] N. Ioku, The Cauchy problem for heat equations with exponential nonlinearity,, J. D. E., 251 (2011), 1172. doi: 10.1016/j.jde.2011.02.015. Google Scholar [14] J. F. Lam, B. Lippman and F. Trappert, Self trapped laser beams in plasma,, Phys. Fluid, 20 (1977), 1176. doi: 10.1063/1.861679. Google Scholar [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. Google Scholar [16] S. Le Coz, A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations,, Adv. Nonlinear Stud., 8 (2008), 455. Google Scholar [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. Google Scholar [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. doi: 10.4208/jpde.v24.n4.7. Google Scholar [19] C. Miao and B. Zhang, The Cauchy problem for semilinear parabolic equations in Besov spaces,, Houston J. Math., 30 (2004), 829. Google Scholar [20] J. Moser, A sharp form of an inequality of N. Trudinger,, Ind. Univ. Math. J., 20 (1971), 1077. Google Scholar [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. doi: 10.1007/PL00004737. Google Scholar [22] M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order,, Journal of Functional Analysis, 155 (1998), 364. doi: 10.1006/jfan.1997.3236. Google Scholar [23] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel Journal of Mathematics, 22 (1975), 273. doi: 10.1007/BF02761595. Google Scholar [24] L. P. Pitaevski, Vortex lines in an imperfect Bose gas,, J. Experimental Theoret. Phys., 13 (1961), 451. Google Scholar [25] B. Ruf, A sharp Moser-Trudinger type inequality for unbounded domains in $\mathbbR^2$,, J. Funct. Anal, 219 (2005), 340. doi: 10.1016/j.jfa.2004.06.013. Google Scholar [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. Google Scholar [27] T. Saanouni, Decay of solutions to a $2D$ Schrödinger equation with exponential growth,, J. P. D. E., 24 (2011), 37. Google Scholar [28] T. Saanouni, Remarks on the semilinear Schrödinger equation,, J. Math. Anal. Appl., 400 (2013), 331. doi: 10.1016/j.jmaa.2012.11.037. Google Scholar [29] T. Saanouni, Scattering of a $2D$ Schrödinger equation with exponential growth in the conformal space,, Math. Meth. Appl. Sci., 33 (2010), 1046. doi: 10.1002/mma.1237. Google Scholar [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. Google Scholar [31] T. Saanouni, Blowing-up semilinear wave equation with exponential nonlinearity in two space dimensions,, Proc. Indian Acad. Sci. (Math. Sci.), 123 (2013), 365. doi: 10.1007/s12044-013-0132-9. Google Scholar [32] T. Saanouni, Global well-posedness of a damped Schrödinger equation in two space dimensions,, Math. Meth. Appl. Sci., 37 (2014), 488. doi: 10.1002/mma.2804. Google Scholar [33] T. Saanouni, A blowing up wave equation with exponential type nonlinearity and arbitrary positive energy,, J. Math. Anal. Appl., 421 (2015), 444. doi: 10.1016/j.jmaa.2014.07.033. Google Scholar [34] W. Strauss, Nonlinear Wave Equations,, CBMS Regional Conference Series in Math., (1989). Google Scholar [35] M. Struwe, The critical nonlinear wave equation in $2$ space dimensions,, J. European Math. Soc., 15 (2013), 1805. doi: 10.4171/JEMS/404. Google Scholar [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. doi: 10.1007/s00208-010-0567-6. Google Scholar [37] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse,, Applied Mathematical Sciences. Vol. 139, (1999). Google Scholar [38] N. S. Trudinger, On imbedding into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473. Google Scholar [39] S. R. S. Varadhan, Lectures on Diffusion Problems and Partial Differential Equations,, Courant Institute of Mathematical Sciences, (1989). Google Scholar
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