The point-wise estimates of solutions for semi-linear dissipative wave equation

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January 2013, 12(1): 237-252. doi: 10.3934/cpaa.2013.12.237

The point-wise estimates of solutions for semi-linear dissipative wave equation

1.	Department of Mathematics, North China Electric Power University, Beijing 102208

Received May 2011 Revised September 2011 Published September 2012

Abstract
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In this paper we focus on the global-in-time existence and the point-wise estimates of solutions to the initial value problem for the semi-linear dissipative wave equation in multi-dimensions. By using the method of Green function combined with the energy estimates, we obtain the point-wise decay estimates of solutions to the problem.

Keywords: Green function., point-wise estimates, Semi-linear dissipative wave equation.

Mathematics Subject Classification: 35E15, 35L05, 35L1.

Citation: Yongqin Liu. The point-wise estimates of solutions for semi-linear dissipative wave equation. Communications on Pure & Applied Analysis, 2013, 12 (1) : 237-252. doi: 10.3934/cpaa.2013.12.237

References:

[1]	V. Belleri and V. Pata, Attractors for semi-linear strongly damped wave equations on $R^3$,, Discrete Contin. Dyn. Syst., 7 (2001), 719. doi: 10.3934/dcds.2001.7.719. Google Scholar
[2]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Math., 19,, Amer. Math. Soc., (1998). Google Scholar
[3]	D. Hoff and K. Zumbrun, Point-wise decay estimates for multidimensional Navier-Stokes diffusion waves,, Z. angew Math. Phys., 48 (1997), 597. doi: 10.1007/s000330050049. Google Scholar
[4]	T. Hosono and T. Ogawa, Large time behavior and $L^p-L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations,, J. Differential Equations, 203 (2004), 82. doi: 10.1016/j.jde.2004.03.034. Google Scholar
[5]	R. Ikehata, K. Nishihara and H. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption,, J. Differential Equations, 226 (2006), 1. doi: 10.1016/j.jde.2006.01.002. Google Scholar
[6]	R. Ikehata and M. Ohta, Critical exponent for semi-linear dissipative wave equation in $\mathbbR^n$,, J. Math. Anal. Appl., 269 (2002), 87. doi: 10.1016/S0022-247X(02)00021-5. Google Scholar
[7]	N. I. Karachalios and N. M. Stavrakakis, Estimates on the dimension of a global attractor for a semi-linear dissipative wave equation on $R^n$,, Discrete Contin. Dyn. Syst., 8 (2002), 939. doi: 10.3934/dcds.2002.8.939. Google Scholar
[8]	S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semi-linear wave equation with a dissipative term,, J. Math. Soc. Japan, 47 (1995), 617. doi: 10.2969/jmsj/04740617. Google Scholar
[9]	T.-T. Li and Y. Zhou, Breakdown of solutions to $\Box u+u_t=\|u\|^{1+\alpha}$,, Discrete Contin. Dyn. Syst., 1 (1995), 503. doi: 10.3934/dcds.1995.1.503. Google Scholar
[10]	J. Lin, K. Nishihara and J. Zhai, $L^2$-estimates of solutions for damped wave equations with space-time dependent damping term,, J. Differential Equations, 248 (2010), 403. doi: 10.1016/j.jde.2009.09.022. Google Scholar
[11]	Y. Liu and W. Wang, The point-wise estimates of solutions for dissipative wave equation in multi-dimensions,, Discrete Contin. Dyn. Syst., 20 (2008), 1013. doi: 10.3934/dcds.2008.20.1013. Google Scholar
[12]	M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semi-linear dissipative wave equations,, Math. Z., 214 (1993), 325. doi: .10.1007/BF02572407. Google Scholar
[13]	K. Nishihara, Global asymptotics for the damped wave equation with absorption in higher dimensional space,, J. Math. Soc. Japan, 58 (2006), 805. doi: 10.2969/jmsj/1156342039. Google Scholar
[14]	K. Nishihara and J. Zhai, Asymptotic behaviors of solutions for time dependent damped wave equations,, J. Math. Anal. Appl., 360 (2009), 412. doi: 10.1016/j.jmaa.2009.06.065. Google Scholar
[15]	K. Nishihara and H. Zhao, Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption,, J. Math. Anal. Appl., 313 (2006), 598. doi: 10.1016/j.jmaa.2005.08.059. Google Scholar
[16]	K. Ono, Asymptotic behavior of solutions for semi-linear telegraph equations,, J. Math. Tokushima Univ., 31 (1997), 11. Google Scholar
[17]	K. Ono, Global existence and asymptotic behavior of small solutions for semi-linear dissipative wave equations,, Discrete Contin. Dyn. Syst., 9 (2003), 651. doi: 10.3934/dcds.2003.9.651. Google Scholar
[18]	G. Todorova and B. Yordnov, Critical exponent for a nonlinear wave equation with damping,, J. Differential Equations, 174 (2001), 464. doi: 10.1006/jdeq.2000.3933. Google Scholar
[19]	W. Wang and T. Yang, The point-wise estimates of solutions for Euler equations with damping in multi-dimensions,, J. Differential Equations, 173 (2001), 410. doi: 10.1006/jdeq.2000.3937. Google Scholar

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References:

[1]	V. Belleri and V. Pata, Attractors for semi-linear strongly damped wave equations on $R^3$,, Discrete Contin. Dyn. Syst., 7 (2001), 719. doi: 10.3934/dcds.2001.7.719. Google Scholar
[2]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Math., 19,, Amer. Math. Soc., (1998). Google Scholar
[3]	D. Hoff and K. Zumbrun, Point-wise decay estimates for multidimensional Navier-Stokes diffusion waves,, Z. angew Math. Phys., 48 (1997), 597. doi: 10.1007/s000330050049. Google Scholar
[4]	T. Hosono and T. Ogawa, Large time behavior and $L^p-L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations,, J. Differential Equations, 203 (2004), 82. doi: 10.1016/j.jde.2004.03.034. Google Scholar
[5]	R. Ikehata, K. Nishihara and H. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption,, J. Differential Equations, 226 (2006), 1. doi: 10.1016/j.jde.2006.01.002. Google Scholar
[6]	R. Ikehata and M. Ohta, Critical exponent for semi-linear dissipative wave equation in $\mathbbR^n$,, J. Math. Anal. Appl., 269 (2002), 87. doi: 10.1016/S0022-247X(02)00021-5. Google Scholar
[7]	N. I. Karachalios and N. M. Stavrakakis, Estimates on the dimension of a global attractor for a semi-linear dissipative wave equation on $R^n$,, Discrete Contin. Dyn. Syst., 8 (2002), 939. doi: 10.3934/dcds.2002.8.939. Google Scholar
[8]	S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semi-linear wave equation with a dissipative term,, J. Math. Soc. Japan, 47 (1995), 617. doi: 10.2969/jmsj/04740617. Google Scholar
[9]	T.-T. Li and Y. Zhou, Breakdown of solutions to $\Box u+u_t=\|u\|^{1+\alpha}$,, Discrete Contin. Dyn. Syst., 1 (1995), 503. doi: 10.3934/dcds.1995.1.503. Google Scholar
[10]	J. Lin, K. Nishihara and J. Zhai, $L^2$-estimates of solutions for damped wave equations with space-time dependent damping term,, J. Differential Equations, 248 (2010), 403. doi: 10.1016/j.jde.2009.09.022. Google Scholar
[11]	Y. Liu and W. Wang, The point-wise estimates of solutions for dissipative wave equation in multi-dimensions,, Discrete Contin. Dyn. Syst., 20 (2008), 1013. doi: 10.3934/dcds.2008.20.1013. Google Scholar
[12]	M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semi-linear dissipative wave equations,, Math. Z., 214 (1993), 325. doi: .10.1007/BF02572407. Google Scholar
[13]	K. Nishihara, Global asymptotics for the damped wave equation with absorption in higher dimensional space,, J. Math. Soc. Japan, 58 (2006), 805. doi: 10.2969/jmsj/1156342039. Google Scholar
[14]	K. Nishihara and J. Zhai, Asymptotic behaviors of solutions for time dependent damped wave equations,, J. Math. Anal. Appl., 360 (2009), 412. doi: 10.1016/j.jmaa.2009.06.065. Google Scholar
[15]	K. Nishihara and H. Zhao, Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption,, J. Math. Anal. Appl., 313 (2006), 598. doi: 10.1016/j.jmaa.2005.08.059. Google Scholar
[16]	K. Ono, Asymptotic behavior of solutions for semi-linear telegraph equations,, J. Math. Tokushima Univ., 31 (1997), 11. Google Scholar
[17]	K. Ono, Global existence and asymptotic behavior of small solutions for semi-linear dissipative wave equations,, Discrete Contin. Dyn. Syst., 9 (2003), 651. doi: 10.3934/dcds.2003.9.651. Google Scholar
[18]	G. Todorova and B. Yordnov, Critical exponent for a nonlinear wave equation with damping,, J. Differential Equations, 174 (2001), 464. doi: 10.1006/jdeq.2000.3933. Google Scholar
[19]	W. Wang and T. Yang, The point-wise estimates of solutions for Euler equations with damping in multi-dimensions,, J. Differential Equations, 173 (2001), 410. doi: 10.1006/jdeq.2000.3937. Google Scholar

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