Let $T_{\varepsilon}$ be the lifespan of solutions to the initial value problem for the one dimensional, derivative nonlinear Schrödinger equations with small initial data of size $O(\varepsilon)$. If the nonlinear term is cubic and gauge invariant, it is known that $\liminf_{\varepsilon \to +0} \varepsilon^{2} \log T_\varepsilon$ is positive. In this paper we obtain a sharp estimate of this lower limit, which is explicitly computed from the initial data and the nonlinear term.
References
S. Alinhac: Blowup for Nonlinear Hyperbolic Equations, Birkhäuser, Boston, 1995. MR1339762 0820.35001 S. Alinhac: Blowup for Nonlinear Hyperbolic Equations, Birkhäuser, Boston, 1995. MR1339762 0820.35001
S. Alinhac: Blowup of small data solutions for a quasilinear wave equations in two space dimensions, Ann. of Math. 149 (1999), 97--127. MR1680539 10.2307/121020 euclid.annm/1126626722
S. Alinhac: Blowup of small data solutions for a quasilinear wave equations in two space dimensions, Ann. of Math. 149 (1999), 97--127. MR1680539 10.2307/121020 euclid.annm/1126626722
S. Alinhac: Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions, II, Acta Math. 182 (1999), 1--23. MR1687180 10.1007/BF02392822 0973.35135 S. Alinhac: Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions, II, Acta Math. 182 (1999), 1--23. MR1687180 10.1007/BF02392822 0973.35135
S. Alinhac: The null condition for quasilinear wave equations in two space dimensions, I, Invent. Math. 145 (2001), 597--618. MR1856402 10.1007/s002220100165 1112.35341 S. Alinhac: The null condition for quasilinear wave equations in two space dimensions, I, Invent. Math. 145 (2001), 597--618. MR1856402 10.1007/s002220100165 1112.35341
S. Alinhac: The null condition for quasilinear wave equations in two space dimensions, II, Amer. J. Math. 123 (2001), 1071--1101. MR1867312 10.1353/ajm.2001.0037 1112.35342 S. Alinhac: The null condition for quasilinear wave equations in two space dimensions, II, Amer. J. Math. 123 (2001), 1071--1101. MR1867312 10.1353/ajm.2001.0037 1112.35342
H. Chihara: The initial value problem for cubic semilinear Schrödinger equations, Publ. RIMS. 32 (1996), 445--471. MR1409797 10.2977/prims/1195162851 H. Chihara: The initial value problem for cubic semilinear Schrödinger equations, Publ. RIMS. 32 (1996), 445--471. MR1409797 10.2977/prims/1195162851
D. Christodoulou: Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), 267--282. MR820070 10.1002/cpa.3160390205 0612.35090 D. Christodoulou: Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), 267--282. MR820070 10.1002/cpa.3160390205 0612.35090
J.-M. Delort: Minoration du temps d'existence pour l'équation de Klein-Gordon non-linéaire en dimension $1$ d'espace, Ann. Inst. Henri Poincaré (Analyse non linéaire) 16 (1999), 563--591. MR1712572 10.1016/S0294-1449(99)80028-6 J.-M. Delort: Minoration du temps d'existence pour l'équation de Klein-Gordon non-linéaire en dimension $1$ d'espace, Ann. Inst. Henri Poincaré (Analyse non linéaire) 16 (1999), 563--591. MR1712572 10.1016/S0294-1449(99)80028-6
J.-M. Delort: Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire à données petites en dimension $1$, Ann. Sci. École Norm. Sup. (4) 34 (2001), 1--61. MR1833089 10.1016/S0012-9593(00)01059-4 J.-M. Delort: Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire à données petites en dimension $1$, Ann. Sci. École Norm. Sup. (4) 34 (2001), 1--61. MR1833089 10.1016/S0012-9593(00)01059-4
P. Godin: Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations 18 (1993), 895--916. MR1218523 10.1080/03605309308820955 0813.35055 P. Godin: Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations 18 (1993), 895--916. MR1218523 10.1080/03605309308820955 0813.35055
N. Hayashi: Global and almost global solutions to quadratic nonlinear Schrödinger equations with small initial data, Dynamics of Continuous, Discrete and Impulsive Systems 2 (1996), 109--129. MR1380418 N. Hayashi: Global and almost global solutions to quadratic nonlinear Schrödinger equations with small initial data, Dynamics of Continuous, Discrete and Impulsive Systems 2 (1996), 109--129. MR1380418
N. Hayashi and P.I. Naumkin: Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation, Ann. Inst. Henri Poincaré (Phys. Théor.) 68 (1998), 159--177. MR1618664 N. Hayashi and P.I. Naumkin: Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation, Ann. Inst. Henri Poincaré (Phys. Théor.) 68 (1998), 159--177. MR1618664
N. Hayashi and P.I. Naumkin: Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations without a self-conjugate property, Funkcial. Ekvac. 42 (1999), 311--324. MR1718755 N. Hayashi and P.I. Naumkin: Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations without a self-conjugate property, Funkcial. Ekvac. 42 (1999), 311--324. MR1718755
N. Hayashi and P.I. Naumkin: Asymptotics of small solutions to nonlinear Schrodinger equations with cubic nonlinearities, Int. J. Pure Appl. Math. 3 (2002), 255--273. MR1938962 N. Hayashi and P.I. Naumkin: Asymptotics of small solutions to nonlinear Schrodinger equations with cubic nonlinearities, Int. J. Pure Appl. Math. 3 (2002), 255--273. MR1938962
N. Hayashi, P.I. Naumkin and H. Uchida: Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations, Publ. RIMS 35 (1999), 501--513. MR1710752 10.2977/prims/1195143611 N. Hayashi, P.I. Naumkin and H. Uchida: Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations, Publ. RIMS 35 (1999), 501--513. MR1710752 10.2977/prims/1195143611
N. Hayashi and T. Ozawa: Modified wave operators for the derivative nonlinear Schrödinger equation, Math. Ann. 298 (1994), 557--576. MR1262776 10.1007/BF01459751 N. Hayashi and T. Ozawa: Modified wave operators for the derivative nonlinear Schrödinger equation, Math. Ann. 298 (1994), 557--576. MR1262776 10.1007/BF01459751
N. Hayashi and T. Ozawa: Remarks on nonlinear Schrödinger equations in one space dimension, Differential Integral Equations 7 (1994), 453--461. MR1255899 N. Hayashi and T. Ozawa: Remarks on nonlinear Schrödinger equations in one space dimension, Differential Integral Equations 7 (1994), 453--461. MR1255899
A. Hoshiga: The initial value problems for quasi-linear wave equations in two space dimensions with small data, Adv. Math. Sci. Appl. 5 (1995), 67--89. MR1325960 A. Hoshiga: The initial value problems for quasi-linear wave equations in two space dimensions with small data, Adv. Math. Sci. Appl. 5 (1995), 67--89. MR1325960
F. John: Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data, Comm. Pure Appl. Math. 40 (1987), 79--109. MR865358 10.1002/cpa.3160400104 0662.35070 F. John: Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data, Comm. Pure Appl. Math. 40 (1987), 79--109. MR865358 10.1002/cpa.3160400104 0662.35070
F. John: Nonlinear wave equations, formation of singularities; in Pitcher Lectures in the Mathematical Sciences, Lehigh University, American Mathematical Society, Providence, RI, 1990. MR1066694 0716.35043 F. John: Nonlinear wave equations, formation of singularities; in Pitcher Lectures in the Mathematical Sciences, Lehigh University, American Mathematical Society, Providence, RI, 1990. MR1066694 0716.35043
S. Katayama: Global existence for systems of nonlinear wave equations in two space dimensions, II, Publ. RIMS 31 (1995), 645--665. MR1371789 10.2977/prims/1195163919 0864.35075 S. Katayama: Global existence for systems of nonlinear wave equations in two space dimensions, II, Publ. RIMS 31 (1995), 645--665. MR1371789 10.2977/prims/1195163919 0864.35075
S. Katayama and Y. Tsutsumi: Global existence of solutions for nonlinear Schrödinger equations in one space dimension, Comm. Partial Differential Equations 19 (1994), 1971--1997. MR1301179 10.1080/03605309408821079 S. Katayama and Y. Tsutsumi: Global existence of solutions for nonlinear Schrödinger equations in one space dimension, Comm. Partial Differential Equations 19 (1994), 1971--1997. MR1301179 10.1080/03605309408821079
C.E. Kenig, G. Ponce and L. Vega: Small solutions to nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré (Analyse non linéaire) 10 (1993), 255--288. MR1230709 C.E. Kenig, G. Ponce and L. Vega: Small solutions to nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré (Analyse non linéaire) 10 (1993), 255--288. MR1230709
S. Klainerman: The null condition and global existence to nonlinear wave equations, Lectures in Appl. Math., 23 (1986), 293--326. MR837683 0599.35105 S. Klainerman: The null condition and global existence to nonlinear wave equations, Lectures in Appl. Math., 23 (1986), 293--326. MR837683 0599.35105
T. Ozawa: Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys. 139 (1991), 479--493. MR1121130 10.1007/BF02101876 euclid.cmp/1104203467
T. Ozawa: Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys. 139 (1991), 479--493. MR1121130 10.1007/BF02101876 euclid.cmp/1104203467
T. Ozawa: On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J. 45 (1996), 137--163. MR1406687 10.1512/iumj.1996.45.1962 T. Ozawa: On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J. 45 (1996), 137--163. MR1406687 10.1512/iumj.1996.45.1962
T. Ozawa and Y. Yamazaki: Life-span of smooth solutions to the complex Ginzburg-Landau type equation on a torus, Nonlinearity 16 (2003), 2029--2034. MR2012854 1038.35124 10.1088/0951-7715/16/6/309 T. Ozawa and Y. Yamazaki: Life-span of smooth solutions to the complex Ginzburg-Landau type equation on a torus, Nonlinearity 16 (2003), 2029--2034. MR2012854 1038.35124 10.1088/0951-7715/16/6/309
A. Shimomura: Asymptotic behavior of solutions for Schrödinger equation with dissipative nonlinearities, Comm. Partial Differential Equations 31 (2006), 1407--1423 MR2254620 10.1080/03605300600910316 A. Shimomura: Asymptotic behavior of solutions for Schrödinger equation with dissipative nonlinearities, Comm. Partial Differential Equations 31 (2006), 1407--1423 MR2254620 10.1080/03605300600910316
H. Sunagawa: Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan 58 (2006), 379--400. MR2228565 10.2969/jmsj/1149166781 euclid.jmsj/1149166781
1107.35087 H. Sunagawa: Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan 58 (2006), 379--400. MR2228565 10.2969/jmsj/1149166781 euclid.jmsj/1149166781
1107.35087
Y. Tsutsumi: The null gauge condition and the one dimensional nonlinear Schrödinger equation with cubic nonlinearity, Indiana Univ. Math. J. 43 (1994), 241--254. MR1275461 10.1512/iumj.1994.43.43012 Y. Tsutsumi: The null gauge condition and the one dimensional nonlinear Schrödinger equation with cubic nonlinearity, Indiana Univ. Math. J. 43 (1994), 241--254. MR1275461 10.1512/iumj.1994.43.43012