• Previous Article
    On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data
  • DCDS Home
  • This Issue
  • Next Article
    Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence
October  1995, 1(4): 503-520. doi: 10.3934/dcds.1995.1.503

Breakdown of solutions to $\square u+u_t=|u|^{1+\alpha}$

1. 

School of Mathematical Sciences, Fudan University, Han Dan Road 220, Shanghai 200433

2. 

Department of Mathematics, Fudan University, Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education of China, Shanghai 200433, China

Received  November 1994 Published  August 1995

In the paper we give an upper bound for the life-span of the mild solution to the Cauchy problem for semilinear equations $\square u+u_t=|u|^{1+\alpha}$ ($\alpha >0,$ constant) with certain small initial data. This shows the sharpness of the lower bound obtained in [2] on the life-span of classical solutions to the Cauchy problem for fully nonlinear wave equations with linear dissipation with small initial data.
Citation: Tatsien Li, Yi Zhou. Breakdown of solutions to $\square u+u_t=|u|^{1+\alpha}$. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 503-520. doi: 10.3934/dcds.1995.1.503
[1]

Meng-Rong Li. Estimates for the life-span of the solutions for some semilinear wave equations. Communications on Pure & Applied Analysis, 2008, 7 (2) : 417-432. doi: 10.3934/cpaa.2008.7.417

[2]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065

[3]

Lingwei Ma, Zhong Bo Fang. A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1697-1706. doi: 10.3934/cpaa.2017081

[4]

Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669

[5]

Binbin Shi, Weike Wang. Existence and blow up of solutions to the $ 2D $ Burgers equation with supercritical dissipation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1169-1192. doi: 10.3934/dcdsb.2019215

[6]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[7]

Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639

[8]

Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183

[9]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843

[10]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

[11]

Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583

[12]

Vural Bayrak, Emil Novruzov, Ibrahim Ozkol. Local-in-space blow-up criteria for two-component nonlinear dispersive wave system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6023-6037. doi: 10.3934/dcds.2019263

[13]

Moez Daoulatli, Irena Lasiecka, Daniel Toundykov. Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 67-94. doi: 10.3934/dcdss.2009.2.67

[14]

Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731

[15]

Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280

[16]

Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831

[17]

Ning-An Lai, Yi Zhou. Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1499-1510. doi: 10.3934/cpaa.2018072

[18]

Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108

[19]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006

[20]

Kangsheng Liu, Xu Liu, Bopeng Rao. Eventual regularity of a wave equation with boundary dissipation. Mathematical Control & Related Fields, 2012, 2 (1) : 17-28. doi: 10.3934/mcrf.2012.2.17

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (50)

Other articles
by authors

[Back to Top]