A note on the lifespan of solutions to the semilinear damped wave equation
HTML articles powered by AMS MathViewer
- by Masahiro Ikeda and Yuta Wakasugi
- Proc. Amer. Math. Soc. 143 (2015), 163-171
- DOI: https://doi.org/10.1090/S0002-9939-2014-12201-5
- Published electronically: August 19, 2014
- PDF | Request permission
Abstract:
This paper concerns estimates of the lifespan of solutions to the semilinear damped wave equation $\square u+\Phi (t,x)u_t=|u|^p$ in $(t,x)\in [0,\infty )\times \mathbf {R}^n$, where the coefficient of the damping term is $\Phi (t,x)=\langle x\rangle ^{-\alpha }(1+t)^{-\beta }$ with $\alpha \in [0,1),\ \beta \in (-1,1)$ and $\alpha \beta =0$. Our novelty is to prove an upper bound of the lifespan of solutions in subcritical cases $1<p<2/(n-\alpha )$.References
- Marcello DโAbbicco and Sandra Lucente, A modified test function method for damped wave equations, Adv. Nonlinear Stud. 13 (2013), no.ย 4, 867โ892. MR 3115143, DOI 10.1515/ans-2013-0407
- Hiroshi Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t}=\Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109โ124 (1966). MR 214914
- Nakao Hayashi, Elena I. Kaikina, and Pavel I. Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations 17 (2004), no.ย 5-6, 637โ652. MR 2054939
- Takafumi Hosono and Takayoshi Ogawa, Large time behavior and $L^p$-$L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004), no.ย 1, 82โ118. MR 2070387, DOI 10.1016/j.jde.2004.03.034
- M. Ikeda, Lifespan of solutions for the nonlinear Schrรถdinger equation without gauge invariance, arXiv:1211.6928v1.
- Ryo Ikehata, Yasuaki Miyaoka, and Takashi Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbf R^N$ with lower power nonlinearities, J. Math. Soc. Japan 56 (2004), no.ย 2, 365โ373. MR 2048464, DOI 10.2969/jmsj/1191418635
- Ryo Ikehata and Kensuke Tanizawa, Global existence of solutions for semilinear damped wave equations in $\mathbf R^N$ with noncompactly supported initial data, Nonlinear Anal. 61 (2005), no.ย 7, 1189โ1208. MR 2131649, DOI 10.1016/j.na.2005.01.097
- Ryo Ikehata, Grozdena Todorova, and Borislav Yordanov, Critical exponent for semilinear wave equations with space-dependent potential, Funkcial. Ekvac. 52 (2009), no.ย 3, 411โ435. MR 2589664, DOI 10.1619/fesi.52.411
- Hendrik J. Kuiper, Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems, Electron. J. Differential Equations (2003), No. 66, 11. MR 1993774
- Ta Tsien Li and Yi Zhou, Breakdown of solutions to $\square u+u_t=|u|^{1+\alpha }$, Discrete Contin. Dynam. Systems 1 (1995), no.ย 4, 503โ520. MR 1357291, DOI 10.3934/dcds.1995.1.503
- Jiayun Lin, Kenji Nishihara, and Jian Zhai, Critical exponent for the semilinear wave equation with time-dependent damping, Discrete Contin. Dyn. Syst. 32 (2012), no.ย 12, 4307โ4320. MR 2966748, DOI 10.3934/dcds.2012.32.4307
- Takashi Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan 56 (2004), no.ย 2, 585โ626. MR 2048476, DOI 10.2969/jmsj/1191418647
- Kenji Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003), no.ย 3, 631โ649. MR 1992029, DOI 10.1007/s00209-003-0516-0
- K. Nishihara, $L^p$-$L^q$ estimates for the 3-D damped wave equation and their application to the semilinear problem, Seminar Notes of Math. Sci., 6, Ibaraki Univ., 2003, 69-83.
- Kenji Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math. 34 (2011), no.ย 2, 327โ343. MR 2918909, DOI 10.3836/tjm/1327931389
- Kosuke Ono, Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations, Discrete Contin. Dyn. Syst. 9 (2003), no.ย 3, 651โ662. MR 1974531, DOI 10.3934/dcds.2003.9.651
- Fuqin Sun, Life span of blow-up solutions for higher-order semilinear parabolic equations, Electron. J. Differential Equations (2010), No. 17, 9. MR 2592002
- Grozdena Todorova and Borislav Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001), no.ย 2, 464โ489. MR 1846744, DOI 10.1006/jdeq.2000.3933
- Grozdena Todorova and Borislav Yordanov, Weighted $L^2$-estimates of dissipative wave equations with variable coefficients, J. Differential Equations 246 (2009), no.ย 12, 4497โ4518. MR 2523291, DOI 10.1016/j.jde.2009.03.020
- Yuta Wakasugi, Small data global existence for the semilinear wave equation with space-time dependent damping, J. Math. Anal. Appl. 393 (2012), no.ย 1, 66โ79. MR 2921649, DOI 10.1016/j.jmaa.2012.03.035
- Jens Wirth, Wave equations with time-dependent dissipation. I. Non-effective dissipation, J. Differential Equations 222 (2006), no.ย 2, 487โ514. MR 2208294, DOI 10.1016/j.jde.2005.07.019
- Jens Wirth, Wave equations with time-dependent dissipation. II. Effective dissipation, J. Differential Equations 232 (2007), no.ย 1, 74โ103. MR 2281190, DOI 10.1016/j.jde.2006.06.004
- Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris Sรฉr. I Math. 333 (2001), no.ย 2, 109โ114 (English, with English and French summaries). MR 1847355, DOI 10.1016/S0764-4442(01)01999-1
- Yong Zhou, A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in $\Bbb R^N$, Appl. Math. Lett. 18 (2005), no.ย 3, 281โ286. MR 2121037, DOI 10.1016/j.aml.2003.07.018
Bibliographic Information
- Masahiro Ikeda
- Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan
- Address at time of publication: Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
- MR Author ID: 940764
- Email: mikeda@math.kyoto-u.ac.jp
- Yuta Wakasugi
- Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan
- MR Author ID: 954066
- Email: y-wakasugi@cr.math.sci.osaka-u.ac.jp
- Received by editor(s): February 2, 2013
- Received by editor(s) in revised form: February 28, 2013
- Published electronically: August 19, 2014
- Communicated by: Joachim Krieger
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 163-171
- MSC (2010): Primary 35L71
- DOI: https://doi.org/10.1090/S0002-9939-2014-12201-5
- MathSciNet review: 3272741