American Institute of Mathematical Sciences

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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

Yongqin Liu 1,

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.
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), 719-735. doi: 10.3934/dcds.2001.7.719.  Google Scholar

[2]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Math., 19, Amer. Math. Soc., Providence, RI, 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), 597-614. 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), 82-118. 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), 1-29. 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), 87-97. 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), 939-951. 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), 617-653. 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), 503-520. 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), 403-422. 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), 1013-1028. 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), 325-342. 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), 805-836. 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), 412-421. 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), 598-610. 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), 11-22.  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), 651-662. 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), 464-489. 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), 410-450. doi: 10.1006/jdeq.2000.3937.  Google Scholar

show all references

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.  Google Scholar

[2]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Math., 19, Amer. Math. Soc., Providence, RI, 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), 597-614. 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), 82-118. 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), 1-29. 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), 87-97. 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), 939-951. 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), 617-653. 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), 503-520. 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), 403-422. 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), 1013-1028. 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), 325-342. 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), 805-836. 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), 412-421. 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), 598-610. 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), 11-22.  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), 651-662. 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), 464-489. 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), 410-450. doi: 10.1006/jdeq.2000.3937.  Google Scholar

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