American Institute of Mathematical Sciences

Advanced
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, 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]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381
[2]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[3]

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 - A, 2020  doi: 10.3934/dcds.2020398
[4]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340
[5]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088
[6]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354
[7]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270
[8]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004
[9]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[10]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095
[11]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127
[12]

Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053
[13]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[14]

Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028
[15]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442
[16]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052
[17]

Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380
[18]

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
[19]

Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 597-613. doi: 10.3934/dcdss.2020364
[20]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174

2019 Impact Factor: 1.053

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences