# American Institute of Mathematical Sciences

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

 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

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.
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 & Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307
##### References:
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##### 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. doi: 10.1016/S0252-9602(11)60326-3. Google Scholar [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. doi: 10.1016/j.jde.2004.03.034. Google Scholar [3] L. C. Evans, "Partial Differential Equations,", Graduate in Math., (1998). Google Scholar [4] T. Li and Y. M. Chen, "Global Classical Solutions for Nonlinear Evolution Equations,", Pitman Monogr. Surv. Pure Appl. Math., (1992). Google Scholar [5] T. P. Liu, Pointwise convergence to shock waves for viscous conservation laws,, Comm. Pure Appl. Math, 50 (1997), 1113. doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.3.CO;2-8. Google Scholar [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. doi: 10.1007/s002200050418. Google Scholar [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). Google Scholar [8] Y. Q. Liu, The point-wise estimates of solutions for semi-linear dissipative wave equation,, Comm. Pure Appl. Anal., 12 (2013), 237. doi: 10.3934/cpaa.2013.12.237. Google Scholar [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. doi: 10.3934/dcds.2008.20.1013. Google Scholar [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. doi: 10.1007/BF02572407. Google Scholar [11] K. Nishihara, Global asymptotics for the damped wace equation with absotption in higher dimensional space,, J. Math. Soc., 58 (2006), 805. Google Scholar [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. doi: 10.1016/j.jmaa.2005.08.059. Google Scholar [13] K. Ono, Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations,, Discrete Contin. Dyn. Syst., 9 (2003), 651. doi: 10.3934/dcds.2003.9.651. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jde.2006.01.002. Google Scholar [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. doi: 10.2969/jmsj/04740617. Google Scholar [17] Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term,, Adv. Math. Sci. Appl., 18 (2008), 329. Google Scholar [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. doi: 10.3934/krm.2008.1.49. Google Scholar [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. doi: 10.1016/j.jmaa.2009.12.013. Google Scholar [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. doi: 10.1006/jdeq.2000.3937. Google Scholar
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