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March  2021, 16(1): 49-67. doi: 10.3934/nhm.2020033

Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $

Linglong Du 1,, and  Min Yang 2,

1. 

Department of Mathematics and Institute for Nonlinear Science, Donghua University, Shanghai, China
2. 

Department of Mathematics, Donghua University, Shanghai, China

* Corresponding author: Linglong Du

Received  May 2020 Revised  August 2020 Published  December 2020

Fund Project: The first author is supported by the Fundamental Research Funds for the Central Universities, National Natural Science Foundation of China (No. 11671075, No.12001097) and Natural Science Foundation of Shanghai (No. 18ZR1401300)

In this paper, we investigate the long time behavior of the solution for the nonlinear wave equation with frictional and visco-elastic damping terms in $ \mathbb{R}^n_+ $. It is shown that for the long time, the frictional damped effect is dominated. The Green's functions for the linear initial boundary value problem can be described in terms of the fundamental solutions for the full space problem and reflected fundamental solutions coupled with the boundary operator. Using the Duhamel's principle, we get the pointwise long time behavior of the solution $ \partial_{{\bf{x}}}^{\alpha}u $ for $ |\alpha|\le 1 $.

Keywords: Initial boundary value problem, mixed damped, nonlinear wave equation, Green's function method, pointwise estimate.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35A08.
Citation: Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2021, 16 (1) : 49-67. doi: 10.3934/nhm.2020033
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 $R^n_+$ $\times$ $R_+$, Arch. Ration. Mech. Anal., 216 (2015), 881-903.  doi: 10.1007/s00205-014-0821-2.  Google Scholar

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S. J. Deng, W. K. Wang and H. L. Zhao, Existence theory and $L^p$ 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

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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

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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

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L. L. Du and H. T. Wang, Long time wave behavior of the Navier-Stokes equations in half space, Discrete Contin. Dyn. Syst. A, 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 $R^n_+$, Discrete Contin. Dyn. Syst. B, 22 (2019), 3265-3280.  doi: 10.3934/dcdsb.2018319.  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.  doi: 10.1016/j.na.2018.06.014.  Google Scholar

[11]

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

[12]

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

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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., 68 (2008), 154-169.  doi: 10.1016/j.na.2006.10.038.  Google Scholar

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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

[15]

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

[16]

R. Ikehata and H. Takeda, Large time behavior of global solutions to nonlinear wave equations with frictional and viscoelastic damping terms, Osaka J. Math., 56 (2019), 807–830. Available from: https://projecteuclid.org/euclid.ojm/1571623223.  Google Scholar

[17]

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

[18]

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

[19]

P. Marcatia and K. Nishihara, The $L^p-L^q$ 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

[20]

T. Narazaki, $L^p-L^q$ 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

[21]

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

[22]

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

[23]

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

[24]

H. T. Wang, Some Studies in Initial-boundary Value Problem, Ph.D thesis, National University of Singapore, 2014. Google Scholar

[25]

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

[26]

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

[27]

T. Yamazaki, Diffusion phenomenon for abstract wave equations with decaying dissipation, Adv. Stud. Pure Math., 47-1 (2007), 363-381.  doi: 10.2969/aspm/04710363.  Google Scholar

[28]

Z. J. Yang, Initial boundary value problem for a class of non-linear strongly damped wave equations, Math. Meth. Appl. Sci., 26 (2003), 1047-1066.  doi: 10.1002/mma.412.  Google Scholar

[29]

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

show all references

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 $R^n_+$ $\times$ $R_+$, 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 $L^p$ 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., 11 (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. A, 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 $R^n_+$, Discrete Contin. Dyn. Syst. B, 22 (2019), 3265-3280.  doi: 10.3934/dcdsb.2018319.  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.  doi: 10.1016/j.na.2018.06.014.  Google Scholar

[11]

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

[12]

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

[13]

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., 68 (2008), 154-169.  doi: 10.1016/j.na.2006.10.038.  Google Scholar

[14]

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

[15]

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

[16]

R. Ikehata and H. Takeda, Large time behavior of global solutions to nonlinear wave equations with frictional and viscoelastic damping terms, Osaka J. Math., 56 (2019), 807–830. Available from: https://projecteuclid.org/euclid.ojm/1571623223.  Google Scholar

[17]

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

[18]

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

[19]

P. Marcatia and K. Nishihara, The $L^p-L^q$ 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

[20]

T. Narazaki, $L^p-L^q$ 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

[21]

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

[22]

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

[23]

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

[24]

H. T. Wang, Some Studies in Initial-boundary Value Problem, Ph.D thesis, National University of Singapore, 2014. Google Scholar

[25]

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

[26]

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

[27]

T. Yamazaki, Diffusion phenomenon for abstract wave equations with decaying dissipation, Adv. Stud. Pure Math., 47-1 (2007), 363-381.  doi: 10.2969/aspm/04710363.  Google Scholar

[28]

Z. J. Yang, Initial boundary value problem for a class of non-linear strongly damped wave equations, Math. Meth. Appl. Sci., 26 (2003), 1047-1066.  doi: 10.1002/mma.412.  Google Scholar

[29]

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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