# American Institute of Mathematical Sciences

January  1999, 5(1): 215-231. doi: 10.3934/dcds.1999.5.215

## The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order

 1 Department of Mathematics, Hokkaido University Sapporo 060-0810, Japan 2 Department of Mathematics, Hokkaido University, Sapporo 060-0810

Received  May 1997 Revised  February 1998 Published  October 1998

We show the local in time solvability of the Cauchy problem for nonlinear wave equations in the Sobolev space of critical order with nonlinear term of exponential type.
Citation: M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215
 [1] Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102 [2] Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123 [3] Q-Heung Choi, Tacksun Jung. A nonlinear wave equation with jumping nonlinearity. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 797-802. doi: 10.3934/dcds.2000.6.797 [4] Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1221-1232. doi: 10.3934/dcdss.2021093 [5] Ahmad Z. Fino, Mokhtar Kirane. The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3625-3650. doi: 10.3934/cpaa.2020160 [6] Ying Fu. A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2011-2039. doi: 10.3934/dcds.2015.35.2011 [7] Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022039 [8] Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic and Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032 [9] Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090 [10] Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations and Control Theory, 2022, 11 (3) : 635-648. doi: 10.3934/eect.2021019 [11] Xingxing Liu, Zhijun Qiao, Zhaoyang Yin. On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1283-1304. doi: 10.3934/cpaa.2014.13.1283 [12] Miao Chen, Youyan Wan, Chang-Lin Xiang. Local uniqueness problem for a nonlinear elliptic equation. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1037-1055. doi: 10.3934/cpaa.2020048 [13] Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387 [14] Van Duong Dinh. On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1127-1143. doi: 10.3934/dcds.2018047 [15] Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations and Control Theory, 2022, 11 (3) : 837-867. doi: 10.3934/eect.2021028 [16] V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731 [17] Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012 [18] Chunyan Zhao, Chengkui Zhong, Zhijun Tang. Asymptotic behavior of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022025 [19] Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems and Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121 [20] Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967

2021 Impact Factor: 1.588