# American Institute of Mathematical Sciences

July  2016, 15(4): 1265-1283. doi: 10.3934/cpaa.2016.15.1265

## The lifespan of solutions to semilinear damped wave equations in one space dimension

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

Received  September 2015 Revised  January 2016 Published  April 2016

In the present paper, we consider the initial value problem for semilinear damped wave equations in one space dimension. Wakasugi [7] have obtained an upper bound of the lifespan for the problem only in the subcritical case. On the other hand, D'Abbicco $\&$ Lucente $\&$ Reissig [3] showed a blow-up result in the critical case. The aim of this paper is to give an estimate of the upper bound of the lifespan in the critical case, and show the optimality of the upper bound. Also, we derive an estimate of the lower bound of the lifespan in the subcritical case which shows the optimality of the upper bound in [7]. Moreover, we show that the critical exponent changes when the initial data are odd functions.
Citation: Kyouhei Wakasa. The lifespan of solutions to semilinear damped wave equations in one space dimension. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1265-1283. doi: 10.3934/cpaa.2016.15.1265
##### References:
 [1] R. Agemi, Y. Kurokawa and H. Takamura, Critical curve for p-q systems of nonlinear wave equations in three space dimensions,, \emph{J. Differential Equations}, 167 (2000), 87.  doi: 10.1006/jdeq.2000.3766.  Google Scholar [2] M. D'Abbicco, The threshold of effective damping for semilinear wave equations,, \emph{Mathematical Methods in Applied Sciences}, 38 (2015), 1032.  doi: 10.1002/mma.3126.  Google Scholar [3] M. D'Abbicco, S. Lucente and M. Reissig, A shift in the Strauss exponent for semilinear wave equations with a not effective damping,, \emph{Journal of Differential Equations}, 259 (2015), 5040.  doi: 10.1016/j.jde.2015.06.018.  Google Scholar [4] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions,, \emph{Manuscripta Math.}, 28 (1979), 235.  doi: 10.1007/BF01647974.  Google Scholar [5] H. Kubo, A. Osaka and M. Yazici, Global existence and blow-up for wave equations with weighted nonlinear terms in one space dimension,, \emph{Interdisciplinary Information Sciences}, 19 (2013), 143.  doi: 10.4036/iis.2013.143.  Google Scholar [6] K. Wakasa, The lifespan of solutions to wave equations with weighted nonlinear terms in one space dimension,, \emph{Hokkaido Mathematical Journal}, ().   Google Scholar [7] Y. Wakasugi, On the Diffusive Structure for the Damped Wave Equation with Variable Coefficients,, Doctoral thesis, (2014).   Google Scholar

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##### References:
 [1] R. Agemi, Y. Kurokawa and H. Takamura, Critical curve for p-q systems of nonlinear wave equations in three space dimensions,, \emph{J. Differential Equations}, 167 (2000), 87.  doi: 10.1006/jdeq.2000.3766.  Google Scholar [2] M. D'Abbicco, The threshold of effective damping for semilinear wave equations,, \emph{Mathematical Methods in Applied Sciences}, 38 (2015), 1032.  doi: 10.1002/mma.3126.  Google Scholar [3] M. D'Abbicco, S. Lucente and M. Reissig, A shift in the Strauss exponent for semilinear wave equations with a not effective damping,, \emph{Journal of Differential Equations}, 259 (2015), 5040.  doi: 10.1016/j.jde.2015.06.018.  Google Scholar [4] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions,, \emph{Manuscripta Math.}, 28 (1979), 235.  doi: 10.1007/BF01647974.  Google Scholar [5] H. Kubo, A. Osaka and M. Yazici, Global existence and blow-up for wave equations with weighted nonlinear terms in one space dimension,, \emph{Interdisciplinary Information Sciences}, 19 (2013), 143.  doi: 10.4036/iis.2013.143.  Google Scholar [6] K. Wakasa, The lifespan of solutions to wave equations with weighted nonlinear terms in one space dimension,, \emph{Hokkaido Mathematical Journal}, ().   Google Scholar [7] Y. Wakasugi, On the Diffusive Structure for the Damped Wave Equation with Variable Coefficients,, Doctoral thesis, (2014).   Google Scholar
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