
Previous Article
Robustly expansive homoclinic classes are generically hyperbolic
 DCDS Home
 This Issue

Next Article
Reflection of highly oscillatory waves with continuous oscillatory spectra for semilinear hyperbolic systems
Adapted linearnonlinear decomposition and global wellposedness for solutions to the defocusing cubic wave equation on $\mathbb{R}^{3}$
1.  UCLA Mathematics Department, Box 951555 Los Angeles, CA 900951555, United States 
∂_{tt} $u  \Delta u = u^{3} $
$u(0,x) = u_{0}(x) $
$\partial_{t} u(0,x) = u_{1}(x)$
with data $( u_{0}, u_{1} ) \in H^{s} \times H^{s1}$, $1 > s > \frac{13}{18} $≈ 0.722. The main task is to estimate the variation of an almost conserved quantity on an arbitrary long time interval. We divide it into subintervals. On each of these subintervals we write the solution as the sum of its linear part adapted to the subinterval and its corresponding nonlinear part. Some terms resulting from this decomposition have a controlled global variation and other terms have a slow local variation.
[1] 
Benjamin Dodson. Global wellposedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linearnonlinear decomposition. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 19051926. doi: 10.3934/dcds.2013.33.1905 
[2] 
Tayeb Hadj Kaddour, Michael Reissig. Global wellposedness for effectively damped wave models with nonlinear memory. Communications on Pure and Applied Analysis, 2021, 20 (5) : 20392064. doi: 10.3934/cpaa.2021057 
[3] 
JeanDaniel Djida, Arran Fernandez, Iván Area. Wellposedness results for fractional semilinear wave equations. Discrete and Continuous Dynamical Systems  B, 2020, 25 (2) : 569597. doi: 10.3934/dcdsb.2019255 
[4] 
Luc Molinet, Francis Ribaud. On global wellposedness for a class of nonlocal dispersive wave equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 657668. doi: 10.3934/dcds.2006.15.657 
[5] 
DanAndrei Geba, Kenji Nakanishi, Sarada G. Rajeev. Global wellposedness and scattering for Skyrme wave maps. Communications on Pure and Applied Analysis, 2012, 11 (5) : 19231933. doi: 10.3934/cpaa.2012.11.1923 
[6] 
Takafumi Akahori. Low regularity global wellposedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure and Applied Analysis, 2010, 9 (2) : 261280. doi: 10.3934/cpaa.2010.9.261 
[7] 
Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global wellposedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 3765. doi: 10.3934/dcds.2007.19.37 
[8] 
Zihua Guo, Yifei Wu. Global wellposedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 257264. doi: 10.3934/dcds.2017010 
[9] 
Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global wellposedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure and Applied Analysis, 2007, 6 (4) : 10231041. doi: 10.3934/cpaa.2007.6.1023 
[10] 
Chao Yang. Sharp condition of global wellposedness for inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems  S, 2021, 14 (12) : 46314642. doi: 10.3934/dcdss.2021136 
[11] 
Jerry Bona, Hongqiu Chen. Wellposedness for regularized nonlinear dispersive wave equations. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 12531275. doi: 10.3934/dcds.2009.23.1253 
[12] 
Tadahiro Oh, Yuzhao Wang. On global wellposedness of the modified KdV equation in modulation spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 29712992. doi: 10.3934/dcds.2020393 
[13] 
Takamori Kato. Global wellposedness for the Kawahara equation with low regularity. Communications on Pure and Applied Analysis, 2013, 12 (3) : 13211339. doi: 10.3934/cpaa.2013.12.1321 
[14] 
Zhaoyang Yin. Wellposedness, blowup, and global existence for an integrable shallow water equation. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 393411. doi: 10.3934/dcds.2004.11.393 
[15] 
SeungYeal Ha, Jinyeong Park, Xiongtao Zhang. A global wellposedness and asymptotic dynamics of the kinetic Winfree equation. Discrete and Continuous Dynamical Systems  B, 2020, 25 (4) : 13171344. doi: 10.3934/dcdsb.2019229 
[16] 
Hideo Takaoka. Global wellposedness for the KadomtsevPetviashvili II equation. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 483499. doi: 10.3934/dcds.2000.6.483 
[17] 
Hartmut Pecher. Local wellposedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure and Applied Analysis, 2014, 13 (2) : 673685. doi: 10.3934/cpaa.2014.13.673 
[18] 
Zhaohui Huo, Boling Guo. The wellposedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 387402. doi: 10.3934/dcds.2005.12.387 
[19] 
Lassaad Aloui, Slim Tayachi. Local wellposedness for the inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 54095437. doi: 10.3934/dcds.2021082 
[20] 
Tarek Saanouni. Global wellposedness of some highorder semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure and Applied Analysis, 2014, 13 (1) : 273291. doi: 10.3934/cpaa.2014.13.273 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]