American Institute of Mathematical Sciences

December  2018, 13(4): 549-565. doi: 10.3934/nhm.2018025

Long time behavior for the visco-elastic damped wave equation in $\mathbb{R}^n_+$ and the boundary effect

 Department of Applied Mathematics, Donghua University, Shanghai, China

* Corresponding author: Linglong Du

Received  January 2018 Published  September 2018

Fund Project: The author is supported by Fundamental Research Funds for the Central Universities (No. 2232016D3-32), Natural Science Foundation of Shanghai (No. 18ZR1401300) and partly by National Natural Science Foundation of China (No. 11671075)

In this paper, we investigate the existence and long time behavior of the solution for the nonlinear visco-elastic damped wave equation in $\mathbb{R}^n_+$, provided that the initial data is sufficiently small. It is shown that for the long time, one can use the convected heat kernel to describe the hyperbolic wave transport structure and damped diffusive mechanism. The Green's function for the linear initial boundary value problem can be described in terms of the fundamental solution (for the full space problem) and reflected fundamental solution coupled with the boundary operator. Using the Duhamel's principle, we get the $L^p$ decaying rate for the nonlinear solution $\partial_{{\bf x}}^{\alpha}u$ for $|\alpha|\le 1$.

Citation: Linglong Du. Long time behavior for the visco-elastic damped wave equation in $\mathbb{R}^n_+$ and the boundary effect. Networks & Heterogeneous Media, 2018, 13 (4) : 549-565. doi: 10.3934/nhm.2018025
References:
 [1] F. X. Chen, B. l. Guo and P. Wang, Long time behavior of strongly damped nonlinear wave equations, J. Differ. Equ., 147 (1998), 231-241. doi: 10.1006/jdeq.1998.3447. Google Scholar [2] R. Chill and A. Haraux, An optimal estimate for the difference of solutions of two abstract evolution equations, J. Differ. Equ., 193 (2003), 385-395. doi: 10.1016/S0022-0396(03)00057-3. Google Scholar [3] S. J. Deng, W. K. Wang and S. H. Yu, Green's functions of wave equations in $R^n_+ × R_+$, Arch. Ration. Mech. Anal., 216 (2015), 881-903. doi: 10.1007/s00205-014-0821-2. Google Scholar [4] S. J. Deng, W. K. Wang and H. L. Zhao, Existence theory and Lp estimates for the solution of nonlinear viscous wave equation, Nonlinear Anal.: Real World Appl., 11 (2010), 4404-4414. doi: 10.1016/j.nonrwa.2010.05.024. Google Scholar [5] S. J. Deng and W. K. Wang, Half space problem for Euler equations with damping in 3-D, J. Differ. Equ., 263 (2017), 7372-7411. doi: 10.1016/j.jde.2017.08.013. 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Zumbrun, Pointwise decay estimates for multi-dimensional Navier-Stokes diffusion waves, Z. angew. Math. Phys., 48 (1997), 1-18. doi: 10.1007/s000330050049. Google Scholar [12] R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differ. Equ., 257 (2014), 2159-2177. doi: 10.1016/j.jde.2014.05.031. Google Scholar [13] R. Ikehata, Some remarks on the asymptotic profiles of solutions for strongly damped wave equations on the 1-D half space, J. Math. Anal. Appl., 421 (2015), 905-916. doi: 10.1016/j.jmaa.2014.07.055. Google Scholar [14] R. Ikehata and Y. Inoue, Global existence of weak solutions for two-dimensional semilinear wave equations with strong damping in an exterior domain, Nonlinear Anal., 69 (2008), 1396-1401. doi: 10.1016/j.na.2006.10.038. Google Scholar [15] R. Ikehata and A. Sawada, Asymptotic profile of solutions for wave equations with frictional and viscoelastic damping terms, Asymptot. Anal., 98 (2016), 59-77. doi: 10.3233/ASY-161361. 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Nishihara, The Lp-Lq estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differ. Equ., 191 (2003), 445-469. doi: 10.1016/S0022-0396(03)00026-3. Google Scholar [22] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738. Google Scholar [23] T. Narazaki, Lp-Lq estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan, 56 (2004), 586-626. doi: 10.2969/jmsj/1191418647. Google Scholar [24] G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X. Google Scholar [25] I. Segal, Quantization and dispersion for nonlinear relativistic equations, in Mathematical Theory of Elementary Particles, MIT Press, Cambridge, MA, (1966), 79–108. Google Scholar [26] Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Meth. Appl. Sci., 23 (2000), 203-226. doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M. Google Scholar [27] S. X. Tang, J. Qi and J. Zhang, Formation tracking control for multi-agent systems: a waveequation based approach, preprint.Google Scholar [28] 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. Google Scholar [29] H. T. Wang, Some Studies in Initial-Boundary Value Problem, Ph.D thesis, National University of Singapore, 2014.Google Scholar [30] G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math., 32 (1980), 631-643. doi: 10.4153/CJM-1980-049-5. Google Scholar [31] R. Z. Xu and Y. C. Liu, Asymptotic behavior of solutions for initial-boundary value problems for strongly damped nonlinear wave equations, Nonlinear Anal., 69 (2008), 2492-2495. doi: 10.1016/j.na.2007.08.027. Google Scholar [32] Z. J. Yang, Initial boundary value problem for a class of nonlinear strongly damped wave equations, Math. Meth. Appl. Sci., 26 (2003), 1047-1066. doi: 10.1002/mma.412. Google Scholar [33] S. F. Zhou, Dimension of the global attractor for strongly damped nonlinear wave equation, J. Math. Anal. Appl., 233 (1999), 102-115. doi: 10.1006/jmaa.1999.6269. Google Scholar

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References:
 [1] F. X. Chen, B. l. Guo and P. Wang, Long time behavior of strongly damped nonlinear wave equations, J. Differ. Equ., 147 (1998), 231-241. doi: 10.1006/jdeq.1998.3447. Google Scholar [2] R. Chill and A. Haraux, An optimal estimate for the difference of solutions of two abstract evolution equations, J. Differ. Equ., 193 (2003), 385-395. doi: 10.1016/S0022-0396(03)00057-3. Google Scholar [3] S. J. Deng, W. K. Wang and S. H. Yu, Green's functions of wave equations in $R^n_+ × R_+$, Arch. Ration. Mech. Anal., 216 (2015), 881-903. doi: 10.1007/s00205-014-0821-2. Google Scholar [4] S. J. Deng, W. K. Wang and H. L. Zhao, Existence theory and Lp estimates for the solution of nonlinear viscous wave equation, Nonlinear Anal.: Real World Appl., 11 (2010), 4404-4414. doi: 10.1016/j.nonrwa.2010.05.024. Google Scholar [5] S. J. Deng and W. K. Wang, Half space problem for Euler equations with damping in 3-D, J. Differ. Equ., 263 (2017), 7372-7411. doi: 10.1016/j.jde.2017.08.013. Google Scholar [6] S. J. Deng, Initial-boundary value problem for p-system with damping in half space, Nonlinear Anal., 143 (2016), 193-210. doi: 10.1016/j.na.2016.05.009. Google Scholar [7] S. J. Deng and S. H. Yu, Green's function and pointwise convergence for compressible Navier-Stokes equations, Quart. Appl. Math., 75 (2017), 433-503. doi: 10.1090/qam/1461. Google Scholar [8] L. L. Du and H. T. Wang, Long time wave behavior of the Navier-Stokes equations in half space, Discrete Contin. Dyn. Syst., 38 (2018), 1349-1363. doi: 10.3934/dcds.2018055. Google Scholar [9] L. L. Du and C. X. Ren, Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $R^n_+$, preprint.Google Scholar [10] L. L. Du, Initial boundary value problem of Euler equations with damping in $\mathbb{R}^n_+$, Nonlinear Anal., 176 (2018), 157-177. Google Scholar [11] D. Hoff and K. Zumbrun, Pointwise decay estimates for multi-dimensional Navier-Stokes diffusion waves, Z. angew. Math. Phys., 48 (1997), 1-18. doi: 10.1007/s000330050049. Google Scholar [12] R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differ. Equ., 257 (2014), 2159-2177. doi: 10.1016/j.jde.2014.05.031. Google Scholar [13] R. Ikehata, Some remarks on the asymptotic profiles of solutions for strongly damped wave equations on the 1-D half space, J. Math. Anal. Appl., 421 (2015), 905-916. doi: 10.1016/j.jmaa.2014.07.055. Google Scholar [14] R. Ikehata and Y. Inoue, Global existence of weak solutions for two-dimensional semilinear wave equations with strong damping in an exterior domain, Nonlinear Anal., 69 (2008), 1396-1401. doi: 10.1016/j.na.2006.10.038. Google Scholar [15] R. Ikehata and A. Sawada, Asymptotic profile of solutions for wave equations with frictional and viscoelastic damping terms, Asymptot. Anal., 98 (2016), 59-77. doi: 10.3233/ASY-161361. Google Scholar [16] R. Ikehata and H. Takeda, Critical exponent for nonlinear wave equations with frictional and viscoelastic damping terms, Nonlinear Anal., 148 (2017), 228-253. doi: 10.1016/j.na.2016.10.008. Google Scholar [17] Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the Compressible Navier-Stokes Equations on the half space, Arch. Ration. Mech. Anal., 177 (2005), 231-330. doi: 10.1007/s00205-005-0365-6. Google Scholar [18] T. P. Liu and S. H. Yu, Green's function of Boltzmann equation, 3-D waves, Bullet. Inst. of Math. Academia Sinica, 1 (2006), 1-78. Google Scholar [19] T. P. Liu and S. H. Yu, On boundary relation for some dissipative systems, Bullet. Inst. of Math. Academia Sinica, 6 (2011), 245-267. Google Scholar [20] T. P. Liu and S. H. Yu, Boundary wave propagator for compressible Navier-Stokes equations, Found. Comput. Math., 14 (2014), 1287-1335. doi: 10.1007/s10208-013-9180-x. Google Scholar [21] P. Marcatia and K. Nishihara, The Lp-Lq estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differ. Equ., 191 (2003), 445-469. doi: 10.1016/S0022-0396(03)00026-3. Google Scholar [22] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738. Google Scholar [23] T. Narazaki, Lp-Lq estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan, 56 (2004), 586-626. doi: 10.2969/jmsj/1191418647. Google Scholar [24] G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X. Google Scholar [25] I. Segal, Quantization and dispersion for nonlinear relativistic equations, in Mathematical Theory of Elementary Particles, MIT Press, Cambridge, MA, (1966), 79–108. Google Scholar [26] Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Meth. Appl. Sci., 23 (2000), 203-226. doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M. Google Scholar [27] S. X. Tang, J. Qi and J. Zhang, Formation tracking control for multi-agent systems: a waveequation based approach, preprint.Google Scholar [28] 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. Google Scholar [29] H. T. Wang, Some Studies in Initial-Boundary Value Problem, Ph.D thesis, National University of Singapore, 2014.Google Scholar [30] G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math., 32 (1980), 631-643. doi: 10.4153/CJM-1980-049-5. Google Scholar [31] R. Z. Xu and Y. C. Liu, Asymptotic behavior of solutions for initial-boundary value problems for strongly damped nonlinear wave equations, Nonlinear Anal., 69 (2008), 2492-2495. doi: 10.1016/j.na.2007.08.027. Google Scholar [32] Z. J. Yang, Initial boundary value problem for a class of nonlinear strongly damped wave equations, Math. Meth. Appl. Sci., 26 (2003), 1047-1066. doi: 10.1002/mma.412. Google Scholar [33] S. F. Zhou, Dimension of the global attractor for strongly damped nonlinear wave equation, J. Math. Anal. Appl., 233 (1999), 102-115. doi: 10.1006/jmaa.1999.6269. Google Scholar
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