# American Institute of Mathematical Sciences

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

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.
Citation: Yongqin Liu. The point-wise estimates of solutions for semi-linear dissipative wave equation. Communications on Pure and 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), 719-735. doi: 10.3934/dcds.2001.7.719. [2] L. C. Evans, "Partial Differential Equations," Graduate Studies in Math., 19, Amer. Math. Soc., Providence, RI, 1998. [3] D. Hoff and K. Zumbrun, Point-wise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. angew Math. Phys., 48 (1997), 597-614. doi: 10.1007/s000330050049. [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), 82-118. doi: 10.1016/j.jde.2004.03.034. [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), 1-29. doi: 10.1016/j.jde.2006.01.002. [6] R. Ikehata and M. Ohta, Critical exponent for semi-linear dissipative wave equation in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 269 (2002), 87-97. doi: 10.1016/S0022-247X(02)00021-5. [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), 939-951. doi: 10.3934/dcds.2002.8.939. [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), 617-653. doi: 10.2969/jmsj/04740617. [9] T.-T. Li and Y. Zhou, Breakdown of solutions to $\Box u+u_t=|u|^{1+\alpha}$, Discrete Contin. Dyn. Syst., 1 (1995), 503-520. doi: 10.3934/dcds.1995.1.503. [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), 403-422. doi: 10.1016/j.jde.2009.09.022. [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), 1013-1028. doi: 10.3934/dcds.2008.20.1013. [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), 325-342. doi: .10.1007/BF02572407. [13] K. Nishihara, Global asymptotics for the damped wave equation with absorption in higher dimensional space, J. Math. Soc. Japan, 58 (2006), 805-836. doi: 10.2969/jmsj/1156342039. [14] K. Nishihara and J. Zhai, Asymptotic behaviors of solutions for time dependent damped wave equations, J. Math. Anal. Appl., 360 (2009), 412-421. doi: 10.1016/j.jmaa.2009.06.065. [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), 598-610. doi: 10.1016/j.jmaa.2005.08.059. [16] K. Ono, Asymptotic behavior of solutions for semi-linear telegraph equations, J. Math. Tokushima Univ., 31 (1997), 11-22. [17] K. Ono, Global existence and asymptotic behavior of small solutions for semi-linear dissipative wave equations, Discrete Contin. Dyn. Syst., 9 (2003), 651-662. doi: 10.3934/dcds.2003.9.651. [18] G. Todorova and B. Yordnov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489. doi: 10.1006/jdeq.2000.3933. [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), 410-450. doi: 10.1006/jdeq.2000.3937.

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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), 719-735. doi: 10.3934/dcds.2001.7.719. [2] L. C. Evans, "Partial Differential Equations," Graduate Studies in Math., 19, Amer. Math. Soc., Providence, RI, 1998. [3] D. Hoff and K. Zumbrun, Point-wise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. angew Math. Phys., 48 (1997), 597-614. doi: 10.1007/s000330050049. [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), 82-118. doi: 10.1016/j.jde.2004.03.034. [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), 1-29. doi: 10.1016/j.jde.2006.01.002. [6] R. Ikehata and M. Ohta, Critical exponent for semi-linear dissipative wave equation in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 269 (2002), 87-97. doi: 10.1016/S0022-247X(02)00021-5. [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), 939-951. doi: 10.3934/dcds.2002.8.939. [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), 617-653. doi: 10.2969/jmsj/04740617. [9] T.-T. Li and Y. Zhou, Breakdown of solutions to $\Box u+u_t=|u|^{1+\alpha}$, Discrete Contin. Dyn. Syst., 1 (1995), 503-520. doi: 10.3934/dcds.1995.1.503. [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), 403-422. doi: 10.1016/j.jde.2009.09.022. [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), 1013-1028. doi: 10.3934/dcds.2008.20.1013. [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), 325-342. doi: .10.1007/BF02572407. [13] K. Nishihara, Global asymptotics for the damped wave equation with absorption in higher dimensional space, J. Math. Soc. Japan, 58 (2006), 805-836. doi: 10.2969/jmsj/1156342039. [14] K. Nishihara and J. Zhai, Asymptotic behaviors of solutions for time dependent damped wave equations, J. Math. Anal. Appl., 360 (2009), 412-421. doi: 10.1016/j.jmaa.2009.06.065. [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), 598-610. doi: 10.1016/j.jmaa.2005.08.059. [16] K. Ono, Asymptotic behavior of solutions for semi-linear telegraph equations, J. Math. Tokushima Univ., 31 (1997), 11-22. [17] K. Ono, Global existence and asymptotic behavior of small solutions for semi-linear dissipative wave equations, Discrete Contin. Dyn. Syst., 9 (2003), 651-662. doi: 10.3934/dcds.2003.9.651. [18] G. Todorova and B. Yordnov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489. doi: 10.1006/jdeq.2000.3933. [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), 410-450. doi: 10.1006/jdeq.2000.3937.
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