The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space

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January 2014, 13(1): 307-330. doi: 10.3934/cpaa.2014.13.307

The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space

Jiao Chen ^1, and Weike Wang ^2,

1.	Mathematics department, Shanghai Jiao Tong University, Shanghai 200240, China
2.	Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, 200240, Shanghai

Received January 2013 Revised May 2013 Published July 2013

Abstract
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In this paper, we study the time-asymptotic behavior of the solution for the Cauchy problem of the damped wave equation with a nonlinear convection term in the multi-dimensional space. When the initial data is a small perturbation around a constant state $u^*$, we obtain the point-wise decay estimates of the solution under the so-called dissipative condition $|b| < 1$, where $b$ depends on $u^*$ and the nonlinear term.

Keywords: Energy estimates, Green's function methods, Damped wave equation, Point-wise estimates., Nonlinear convection.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Jiao Chen, Weike Wang. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Communications on Pure and Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307

References:

[1]	L. L. Fan, H. X. Liu and H. Yin, Dacay estimates of planar stationary waves for damped wave equations with nonlinear convection in mutil-dimensional half space, Acta Math Sci, 31(B) (2011), 1389-1410. doi: 10.1016/S0252-9602(11)60326-3.
[2]	T. Hosono and T. Ogawa, Large time behavior and $L^p-L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differ. Equations, 203 (2004), 82-118. doi: 10.1016/j.jde.2004.03.034.
[3]	L. C. Evans, "Partial Differential Equations," Graduate in Math., 19, Amer. Math. Soc., Providence, RI, 1998.
[4]	T. Li and Y. M. Chen, "Global Classical Solutions for Nonlinear Evolution Equations," Pitman Monogr. Surv. Pure Appl. Math., vol. 45, Longman Scientific and Technical, Harlow, 1992.
[5]	T. P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math, 50 (1997), 1113-1182. doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.3.CO;2-8.
[6]	T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Comm. Math. Phys., 169 (1998), 145-173. doi: 10.1007/s002200050418.
[7]	T. P. Liu and Y. Zeng, Large time behavior of solutions general quasilinear hyperbolic-parabolic systems of conservation laws, A. M. S. memoirs, 599 (1997).
[8]	Y. Q. Liu, The point-wise estimates of solutions for semi-linear dissipative wave equation, Comm. Pure Appl. Anal., 12 (2013), 237-252. doi: 10.3934/cpaa.2013.12.237.
[9]	Y. Q. Liu and W. K. Wang, The pointwise 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.
[10]	M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semiliear dissipative wave equation, Math. Z., 214 (1993), 325-241. doi: 10.1007/BF02572407.
[11]	K. Nishihara, Global asymptotics for the damped wace equation with absotption in higher dimensional space, J. Math. Soc., Japan, 58 (2006), 805-836.
[12]	K. Nishihara and H. J. Zhao, Dacay properties of solutions to the Cauchy problem for the damped wace equation with absorption, J. Math. Anal. Appl., 313 (2006), 698-610. doi: 10.1016/j.jmaa.2005.08.059.
[13]	K. Ono, Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations, Discrete Contin. Dyn. Syst., 9 (2003), 651-662. doi: 10.3934/dcds.2003.9.651.
[14]	R. Ikehata, A remark on a critical exponent for the semilinear dissipative wave equation in the one dimensional half space, Differential Integral Equations, 16 (2003), 727-736.
[15]	R. Ikehata, K. Nishihara and H. J. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differ. Equation, 226 (2006), 1-29. doi: 10.1016/j.jde.2006.01.002.
[16]	S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (1995), 617-653. doi: 10.2969/jmsj/04740617.
[17]	Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term, Adv. Math. Sci. Appl., 18, (2008), 329-343.
[18]	Y. Ueda, T. Nakamura and S. Kawashima, Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space, Kinet. Relat. Models, 1 (2008), 49-64. doi: 10.3934/krm.2008.1.49.
[19]	W. K. Wang and W. J. Wang, The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions, J. Math. Anal. Appl., 366 (2010), 226-241. doi: 10.1016/j.jmaa.2009.12.013.
[20]	W. K. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differ. Equations, 173 (2001), 410-450. doi: 10.1006/jdeq.2000.3937.

show all references

References:

[1]	L. L. Fan, H. X. Liu and H. Yin, Dacay estimates of planar stationary waves for damped wave equations with nonlinear convection in mutil-dimensional half space, Acta Math Sci, 31(B) (2011), 1389-1410. doi: 10.1016/S0252-9602(11)60326-3.
[2]	T. Hosono and T. Ogawa, Large time behavior and $L^p-L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differ. Equations, 203 (2004), 82-118. doi: 10.1016/j.jde.2004.03.034.
[3]	L. C. Evans, "Partial Differential Equations," Graduate in Math., 19, Amer. Math. Soc., Providence, RI, 1998.
[4]	T. Li and Y. M. Chen, "Global Classical Solutions for Nonlinear Evolution Equations," Pitman Monogr. Surv. Pure Appl. Math., vol. 45, Longman Scientific and Technical, Harlow, 1992.
[5]	T. P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math, 50 (1997), 1113-1182. doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.3.CO;2-8.
[6]	T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Comm. Math. Phys., 169 (1998), 145-173. doi: 10.1007/s002200050418.
[7]	T. P. Liu and Y. Zeng, Large time behavior of solutions general quasilinear hyperbolic-parabolic systems of conservation laws, A. M. S. memoirs, 599 (1997).
[8]	Y. Q. Liu, The point-wise estimates of solutions for semi-linear dissipative wave equation, Comm. Pure Appl. Anal., 12 (2013), 237-252. doi: 10.3934/cpaa.2013.12.237.
[9]	Y. Q. Liu and W. K. Wang, The pointwise 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.
[10]	M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semiliear dissipative wave equation, Math. Z., 214 (1993), 325-241. doi: 10.1007/BF02572407.
[11]	K. Nishihara, Global asymptotics for the damped wace equation with absotption in higher dimensional space, J. Math. Soc., Japan, 58 (2006), 805-836.
[12]	K. Nishihara and H. J. Zhao, Dacay properties of solutions to the Cauchy problem for the damped wace equation with absorption, J. Math. Anal. Appl., 313 (2006), 698-610. doi: 10.1016/j.jmaa.2005.08.059.
[13]	K. Ono, Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations, Discrete Contin. Dyn. Syst., 9 (2003), 651-662. doi: 10.3934/dcds.2003.9.651.
[14]	R. Ikehata, A remark on a critical exponent for the semilinear dissipative wave equation in the one dimensional half space, Differential Integral Equations, 16 (2003), 727-736.
[15]	R. Ikehata, K. Nishihara and H. J. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differ. Equation, 226 (2006), 1-29. doi: 10.1016/j.jde.2006.01.002.
[16]	S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (1995), 617-653. doi: 10.2969/jmsj/04740617.
[17]	Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term, Adv. Math. Sci. Appl., 18, (2008), 329-343.
[18]	Y. Ueda, T. Nakamura and S. Kawashima, Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space, Kinet. Relat. Models, 1 (2008), 49-64. doi: 10.3934/krm.2008.1.49.
[19]	W. K. Wang and W. J. Wang, The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions, J. Math. Anal. Appl., 366 (2010), 226-241. doi: 10.1016/j.jmaa.2009.12.013.
[20]	W. K. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differ. Equations, 173 (2001), 410-450. doi: 10.1006/jdeq.2000.3937.

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