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

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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

Tarek Saanouni ^1,

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

Received January 2015 Revised July 2015 Published September 2015

Abstract
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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.

Keywords: ground state., global existence, nonlinear heat equation, blow-up, Nonlinear wave equation, nonlinear Schrödinger equation.

Mathematics Subject Classification: Primary: 35Q5.

Citation: Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations and 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), 2051-2057. doi: 10.1090/S0002-9939-99-05180-1.
[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. doi: 10.1142/S0219891609001927.
[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. doi: 10.1016/0022-1236(79)90077-6.
[8]	J. Ginibre and G. Velo, The Global Cauchy problem for nonlinear Klein-Gordon equation, Math. Z, 189 (1985), 487-505. doi: 10.1007/BF01168155.
[9]	E. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys., 4 (1963), 195-207. doi: 10.1063/1.1703944.
[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. doi: 10.1002/cpa.20127.
[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. doi: 10.1016/j.crma.2007.06.008.
[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. doi: 10.1016/j.jde.2011.02.015.
[14]	J. F. Lam, B. Lippman and F. Trappert, Self trapped laser beams in plasma, Phys. Fluid, 20 (1977), 1176-1179. doi: 10.1063/1.861679.
[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. doi: 10.4208/jpde.v24.n4.7.
[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. doi: 10.1007/PL00004737.
[22]	M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order, Journal of Functional Analysis, 155 (1998), 364-380. doi: 10.1006/jfan.1997.3236.
[23]	L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics, 22 (1975), 273-303. doi: 10.1007/BF02761595.
[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. doi: 10.1016/j.jfa.2004.06.013.
[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. doi: 10.1016/j.jmaa.2012.11.037.
[29]	T. Saanouni, Scattering of a $2D$ Schrödinger equation with exponential growth in the conformal space, Math. Meth. Appl. Sci., 33 (2010), 1046-1058. doi: 10.1002/mma.1237.
[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. doi: 10.1007/s12044-013-0132-9.
[32]	T. Saanouni, Global well-posedness of a damped Schrödinger equation in two space dimensions, Math. Meth. Appl. Sci., 37 (2014), 488-495. doi: 10.1002/mma.2804.
[33]	T. Saanouni, A blowing up wave equation with exponential type nonlinearity and arbitrary positive energy, J. Math. Anal. Appl., 421 (2015), 444-452. doi: 10.1016/j.jmaa.2014.07.033.
[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. doi: 10.4171/JEMS/404.
[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. doi: 10.1007/s00208-010-0567-6.
[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.

show all references

References:

[1]	S. Adachi and K. Tanaka, Trudinger type inequalities in $\mathbbR^N$ and their best exponent, Proc. Amer. Math. Society., 128 (1999), 2051-2057. doi: 10.1090/S0002-9939-99-05180-1.
[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. doi: 10.1142/S0219891609001927.
[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. doi: 10.1016/0022-1236(79)90077-6.
[8]	J. Ginibre and G. Velo, The Global Cauchy problem for nonlinear Klein-Gordon equation, Math. Z, 189 (1985), 487-505. doi: 10.1007/BF01168155.
[9]	E. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys., 4 (1963), 195-207. doi: 10.1063/1.1703944.
[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. doi: 10.1002/cpa.20127.
[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. doi: 10.1016/j.crma.2007.06.008.
[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. doi: 10.1016/j.jde.2011.02.015.
[14]	J. F. Lam, B. Lippman and F. Trappert, Self trapped laser beams in plasma, Phys. Fluid, 20 (1977), 1176-1179. doi: 10.1063/1.861679.
[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. doi: 10.4208/jpde.v24.n4.7.
[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. doi: 10.1007/PL00004737.
[22]	M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order, Journal of Functional Analysis, 155 (1998), 364-380. doi: 10.1006/jfan.1997.3236.
[23]	L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics, 22 (1975), 273-303. doi: 10.1007/BF02761595.
[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. doi: 10.1016/j.jfa.2004.06.013.
[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. doi: 10.1016/j.jmaa.2012.11.037.
[29]	T. Saanouni, Scattering of a $2D$ Schrödinger equation with exponential growth in the conformal space, Math. Meth. Appl. Sci., 33 (2010), 1046-1058. doi: 10.1002/mma.1237.
[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. doi: 10.1007/s12044-013-0132-9.
[32]	T. Saanouni, Global well-posedness of a damped Schrödinger equation in two space dimensions, Math. Meth. Appl. Sci., 37 (2014), 488-495. doi: 10.1002/mma.2804.
[33]	T. Saanouni, A blowing up wave equation with exponential type nonlinearity and arbitrary positive energy, J. Math. Anal. Appl., 421 (2015), 444-452. doi: 10.1016/j.jmaa.2014.07.033.
[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. doi: 10.4171/JEMS/404.
[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. doi: 10.1007/s00208-010-0567-6.
[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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