American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019
[2]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3063-3092. doi: 10.3934/dcds.2020398
[3]

Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019
[4]

Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021030
[5]

Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021043
[6]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055
[7]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151
[8]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003
[9]

Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021057
[10]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004
[11]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185
[12]

Siqi Chen, Yong-Kui Chang, Yanyan Wei. Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021017
[13]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006
[14]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241
[15]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[16]

Maolin Cheng, Yun Liu, Jianuo Li, Bin Liu. Nonlinear Grey Bernoulli model NGBM (1, 1)'s parameter optimisation method and model application. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021054
[17]

Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012
[18]

Samira Shahsavari, Saeed Ketabchi. The proximal methods for solving absolute value equation. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 449-460. doi: 10.3934/naco.2020037
[19]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072
[20]

Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021094

2019 Impact Factor: 1.053

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences