Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data
Author:
Xu Runzhang
Journal:
Quart. Appl. Math. 68 (2010), 459-468
MSC (2000):
Primary 35L05, 35K05
DOI:
https://doi.org/10.1090/S0033-569X-2010-01197-0
Published electronically:
June 4, 2010
MathSciNet review:
2676971
Full-text PDF Free Access
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Abstract: We study the initial boundary value problem of semilinear hyperbolic equations $u_{tt}-\Delta u=f(u)$ and semilinear parabolic equations $u_{t}-\Delta u=f(u)$ with critical initial data $E(0)=d$ (or $J(u_0)=d$), $I(u_0)<0$, and prove that there exist non-global solutions under classical conditions on $f$.
References
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References
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- H. A. Levine, Instability and non-existence of global solutions to nonlinear wave equations of the form $Pu_{tt} = -Au + F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1–21. MR 0344697 (49:9436)
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- H. A. Levine, J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341–361. MR 1463799 (99b:34110)
- H. A. Levine, G. Todorova, Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy, Proc. Amer. Math. Soc., 129 (2001), 793–805. MR 1792187 (2001k:35212)
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- R. T. Glassey, Blow-up theorems for nonlinear wave equations, Math. Z., 32 (1973), 183–203. MR 0340799 (49:5549)
- F. Gazzola, M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185–207. MR 2201151 (2007c:35118)
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- E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155–182. MR 1719145 (2000k:35205)
- Liu Yacheng, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155–169. MR 1987088 (2004h:35151)
- Liu Yacheng, Zhao Junsheng, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Analysis, 64 (2006), 2665–2687. MR 2218541 (2007a:35108)
- Liu Yacheng, Xu Runzhang, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete and Continuous Dynamical System-Series B, 7 (2007), 171–189. MR 2257457 (2007h:35232)
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Additional Information
Xu Runzhang
Affiliation:
College of Science, Harbin Engineering University, 150001, People’s Republic of China
Email:
xurunzh@yahoo.com.cn
Keywords:
Semilinear hyperbolic equations,
semilinear parabolic equation,
critical initial data,
potential wells,
global nonexistence
Received by editor(s):
November 18, 2008
Published electronically:
June 4, 2010
Article copyright:
© Copyright 2010
Brown University