July  2019, 24(7): 3265-3280. doi: 10.3934/dcdsb.2018319

Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $

Department of Applied Mathematics, Donghua University, Shanghai, China

* Corresponding author: Caixuan Ren

Received  November 2017 Published  January 2019

Fund Project: Du 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). Ren is supported by NSFC(No. 11601075) and the Fundamental Research Funds for the Central Universities (No.16D110910).

In this paper, the asymptotic wave behavior of the solution for the nonlinear damped wave equation in $ \mathbb{R}^n_+ $ is investigated. We describe the double mechanism of the hyperbolic effect and the parabolic effect using the explicit functions. With the absorbing and radiative boundary condition, we show that the Green's function for the half space linear problem can be described in terms of the fundamental solution for the Cauchy problem and the reflected fundamental solution coupled with a boundary operator. Using the Duhamel's principle, we see that due to the fast decay property of the Green's function and the high nonlinearity, the pointwise decaying rate for the nonlinear solution and extra time decaying rate for its first order derivative are obtained.

Citation: Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319
References:
[1]

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

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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, Characteristic half space problem for the Broadwell model, Netw. Heterog. Media, 9 (2014), 97-110.  doi: 10.3934/nhm.2014.9.97.  Google Scholar

[5]

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

[6]

L. FanH. Liu and H. Yin, Decay estimates of planar stationary wavs for damped wave equations with nonlinear convection in multi-dimensional half space, Acta Mathematica Scientia, 31 (2011), 1389-1410.  doi: 10.1016/S0252-9602(11)60326-3.  Google Scholar

[7]

R. Ikehata and M. Ohta, Critical exponents for semilinear dissipative wave equations in $R^N$, J. Math. Anal. Appl., 269 (2002), 87-97.  doi: 10.1016/S0022-247X(02)00021-5.  Google Scholar

[8]

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

[9]

C. Y. LanH. E. Lin and S. H. Yu, The Green's function for the Broadwell model with a transonic boundary, J. Hyperbolic Differ. Equ., 5 (2008), 279-294.  doi: 10.1142/S0219891608001489.  Google Scholar

[10]

T. P. Liu and S. H. Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Commun. Pure Appl. Math., 60 (2007), 295-356.  doi: 10.1002/cpa.20172.  Google Scholar

[11]

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

[12]

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

[13]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave quations, Publ. Res. Inst. Math. Sci., 12 (1976), 169-189.  doi: 10.2977/prims/1195190962.  Google Scholar

[14]

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

[15]

K. Nishihara, $L^p-L^q$ estimates for solutions to the damped wave equations in 3-dimensional space and their applications, Math. Z., 244 (2003), 631-649.  doi: 10.1007/s00209-003-0516-0.  Google Scholar

[16]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differ. Equ., 174 (2001), 464-489.  doi: 10.1006/jdeq.2000.3933.  Google Scholar

[17]

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

[18]

Y. UedaT. Nakamura and S. Kawashima, Stability of degenerate stationary waves for viscous gases, Arch. Ration. Mech. Anal., 198 (2010), 735-762.  doi: 10.1007/s00205-010-0369-8.  Google Scholar

[19]

Y. UedaT. Nakamura and S. Kawashima, Energy method in the partial Fourier space and application to stability problems in the half space, J. Differ. Equ., 250 (2011), 1169-1199.  doi: 10.1016/j.jde.2010.10.003.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

L. L. Du, Characteristic half space problem for the Broadwell model, Netw. Heterog. Media, 9 (2014), 97-110.  doi: 10.3934/nhm.2014.9.97.  Google Scholar

[5]

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

[6]

L. FanH. Liu and H. Yin, Decay estimates of planar stationary wavs for damped wave equations with nonlinear convection in multi-dimensional half space, Acta Mathematica Scientia, 31 (2011), 1389-1410.  doi: 10.1016/S0252-9602(11)60326-3.  Google Scholar

[7]

R. Ikehata and M. Ohta, Critical exponents for semilinear dissipative wave equations in $R^N$, J. Math. Anal. Appl., 269 (2002), 87-97.  doi: 10.1016/S0022-247X(02)00021-5.  Google Scholar

[8]

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

[9]

C. Y. LanH. E. Lin and S. H. Yu, The Green's function for the Broadwell model with a transonic boundary, J. Hyperbolic Differ. Equ., 5 (2008), 279-294.  doi: 10.1142/S0219891608001489.  Google Scholar

[10]

T. P. Liu and S. H. Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Commun. Pure Appl. Math., 60 (2007), 295-356.  doi: 10.1002/cpa.20172.  Google Scholar

[11]

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

[12]

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

[13]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave quations, Publ. Res. Inst. Math. Sci., 12 (1976), 169-189.  doi: 10.2977/prims/1195190962.  Google Scholar

[14]

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

[15]

K. Nishihara, $L^p-L^q$ estimates for solutions to the damped wave equations in 3-dimensional space and their applications, Math. Z., 244 (2003), 631-649.  doi: 10.1007/s00209-003-0516-0.  Google Scholar

[16]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differ. Equ., 174 (2001), 464-489.  doi: 10.1006/jdeq.2000.3933.  Google Scholar

[17]

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

[18]

Y. UedaT. Nakamura and S. Kawashima, Stability of degenerate stationary waves for viscous gases, Arch. Ration. Mech. Anal., 198 (2010), 735-762.  doi: 10.1007/s00205-010-0369-8.  Google Scholar

[19]

Y. UedaT. Nakamura and S. Kawashima, Energy method in the partial Fourier space and application to stability problems in the half space, J. Differ. Equ., 250 (2011), 1169-1199.  doi: 10.1016/j.jde.2010.10.003.  Google Scholar

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