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_+ $

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 $.

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. ChenB. 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. DengW. 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. DengW. 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. UedaT. 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. ChenB. 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. DengW. 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. DengW. 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. UedaT. 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]

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

[2]

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

[3]

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

[4]

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

[5]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[6]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[7]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

[8]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453

[9]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017

[10]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448

[11]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021

[12]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019

[13]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028

[14]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[15]

Alessandro Gondolo, Fernando Guevara Vasquez. Characterization and synthesis of Rayleigh damped elastodynamic networks. Networks & Heterogeneous Media, 2014, 9 (2) : 299-314. doi: 10.3934/nhm.2014.9.299

[16]

Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781

[17]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709

[18]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817

[19]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[20]

Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018

2019 Impact Factor: 1.053

Metrics

  • PDF downloads (53)
  • HTML views (107)
  • Cited by (0)

Other articles
by authors

[Back to Top]