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

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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

Kyouhei Wakasa ^1,

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

Received September 2015 Revised January 2016 Published April 2016

Abstract
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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.

Keywords: semilinear damped wave equation, lifespan, Initial value problem, critical exponent, one space dimension.

Mathematics Subject Classification: Primary: 35L71, 35E15; Secondary: 35A01, 35A09, 35B4.

Citation: Kyouhei Wakasa. The lifespan of solutions to semilinear damped wave equations in one space dimension. Communications on Pure and 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, J. Differential Equations, 167 (2000), 87-133. doi: 10.1006/jdeq.2000.3766.
[2]	M. D'Abbicco, The threshold of effective damping for semilinear wave equations, Mathematical Methods in Applied Sciences, 38 (2015), 1032-1045. doi: 10.1002/mma.3126.
[3]	M. D'Abbicco, S. Lucente and M. Reissig, A shift in the Strauss exponent for semilinear wave equations with a not effective damping, Journal of Differential Equations, 259 (2015), 5040-5073. doi: 10.1016/j.jde.2015.06.018.
[4]	F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. doi: 10.1007/BF01647974.
[5]	H. Kubo, A. Osaka and M. Yazici, Global existence and blow-up for wave equations with weighted nonlinear terms in one space dimension, Interdisciplinary Information Sciences, 19 (2013), 143-148. doi: 10.4036/iis.2013.143.
[6]	K. Wakasa, The lifespan of solutions to wave equations with weighted nonlinear terms in one space dimension, Hokkaido Mathematical Journal, to appear.
[7]	Y. Wakasugi, On the Diffusive Structure for the Damped Wave Equation with Variable Coefficients, Doctoral thesis, Osaka University, 2014.

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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, J. Differential Equations, 167 (2000), 87-133. doi: 10.1006/jdeq.2000.3766.
[2]	M. D'Abbicco, The threshold of effective damping for semilinear wave equations, Mathematical Methods in Applied Sciences, 38 (2015), 1032-1045. doi: 10.1002/mma.3126.
[3]	M. D'Abbicco, S. Lucente and M. Reissig, A shift in the Strauss exponent for semilinear wave equations with a not effective damping, Journal of Differential Equations, 259 (2015), 5040-5073. doi: 10.1016/j.jde.2015.06.018.
[4]	F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. doi: 10.1007/BF01647974.
[5]	H. Kubo, A. Osaka and M. Yazici, Global existence and blow-up for wave equations with weighted nonlinear terms in one space dimension, Interdisciplinary Information Sciences, 19 (2013), 143-148. doi: 10.4036/iis.2013.143.
[6]	K. Wakasa, The lifespan of solutions to wave equations with weighted nonlinear terms in one space dimension, Hokkaido Mathematical Journal, to appear.
[7]	Y. Wakasugi, On the Diffusive Structure for the Damped Wave Equation with Variable Coefficients, Doctoral thesis, Osaka University, 2014.

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