# American Institute of Mathematical Sciences

• Previous Article
Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping
• EECT Home
• This Issue
• Next Article
A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem
December  2019, 8(4): 825-846. doi: 10.3934/eect.2019040

## Optimal energy decay rates for some wave equations with double damping terms

 Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan

* Corresponding author: Ryo Ikehata

Received  October 2018 Revised  February 2019 Published  June 2019

Fund Project: The first author is supported by grant-in-Aid for scientific Research (C)15K04958 of JSPS

We consider the Cauchy problem in ${\bf R}^{n}$ for some wave equations with double damping terms, that is, one is the frictional damping $u_{t}(t, x)$ and the other is very strong structural damping expressed as $(-\Delta)^{\theta}u_{t}(t, x)$ with $\theta > 1$. We will derive optimal decay rates of the total energy and the $L^{2}$-norm of solutions as $t \to \infty$. These results can be obtained in the case when the initial data have a sufficient high regularity in order to guarantee that the corresponding high frequency parts of such energy and $L^{2}$-norm of solutions are remainder terms. A strategy to get such results comes from a method recently developed by the first author [11].

Citation: Ryo Ikehata, Shingo Kitazaki. Optimal energy decay rates for some wave equations with double damping terms. Evolution Equations & Control Theory, 2019, 8 (4) : 825-846. doi: 10.3934/eect.2019040
##### References:
 [1] R. C. Charão, C. R. da Luz and R. Ikehata, Sharp decay rates for wave equations with a fractional damping via new method in the Fourier space, J. Math Anal. Appl, 408 (2013), 247-255. doi: 10.1016/j.jmaa.2013.06.016. Google Scholar [2] R. Chill and A. Haraux, An optimal estimate for the difference of solutions of two abstract evolution equations, J. Diff. Eqns, 193 (2003), 385-395. doi: 10.1016/S0022-0396(03)00057-3. Google Scholar [3] M. D'Abbicco, $L^1$-$L^ 1$ estimates for a doubly dissipative semilinear wave equation, NoDEA Nonlinear Differential Equations Appl, 24 (2017), Art. 5, 23 pp. doi: 10.1007/s00030-016-0428-4. Google Scholar [4] M. D'Abbicco and M. R. Ebert, Diffusion phenomena for the wave equation with structural damping in the $L^{p}$-$L^{q}$ framework, J. Diff. Eqns, 256 (2014), 2307-2336. doi: 10.1016/j.jde.2014.01.002. Google Scholar [5] M. D'Abbicco and M. R. Ebert, A classification of structural dissipations for evolution operators, Math. Methods Appl. Sci., 39 (2016), 2558-2582. doi: 10.1002/mma.3713. Google Scholar [6] M. D'Abbicco and M. R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations, Nonlinear Anal., 149 (2017), 1-40. doi: 10.1016/j.na.2016.10.010. Google Scholar [7] M. D'Abbicco, M. R. Ebert and T. Picon, Long time decay estimates in real Hardy spaces for evolution equations with structural dissipation, J. Pseudo-Differ. Oper. Appl., 7 (2016), 261-293. doi: 10.1007/s11868-015-0141-9. Google Scholar [8] M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592. doi: 10.1002/mma.2913. Google Scholar [9] M. Ghisi, M. Gobbino and A. Haraux, Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation, Trans. Amer. Math. Soc., 368 (2016), 2039-2079. doi: 10.1090/tran/6520. Google Scholar [10] R. Ikehata, Decay estimates of solutions for the wave equations with strong damping terms in unbounded domains, Math. Methods Appl. Sci., 24 (2001), 659-670. doi: 10.1002/mma.235. Google Scholar [11] R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Diff. Eqns, 257 (2014), 2159-2177. doi: 10.1016/j.jde.2014.05.031. Google Scholar [12] R. Ikehata and S. Iyota, Asymptotic profile of solutions for some wave equations with very strong structural damping, Math. Methods Appl. Sci., 41 (2018), 5074-5090. doi: 10.1002/mma.4954. Google Scholar [13] R. Ikehata and H. Michihisa, Moment conditions and lower bounds in expanding solutions of wave equations with double damping terms, arXiv: 1807.10020, Asymptotic Analysis doi: 10.3233/ASY-181516. Google Scholar [14] R. Ikehata and M. Natsume, Energy decay estimates for wave equations with a fractional damping, Diff. Int. Eqns, 25 (2012), 939-956. Google Scholar [15] R. Ikehata and M. Onodera, Remarks on large time behavior of the $L^{2}$-norm of solutions to strongly damped wave equations, Diff. Int. Eqns, 30 (2017), 505-520. Google Scholar [16] R. Ikehata and A. Sawada, Asymptotic profiles of solutions for wave equations with frictional and viscoelastic damping terms, Asymptotic Anal., 98 (2016), 59-77. doi: 10.3233/ASY-161361. Google Scholar [17] 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 [18] R. Ikehata and H. Takeda, Large time behavior of global solutions to nonlinear wave equations with frictional and viscoelastic damping terms, Osaka Math. J., (in press).Google Scholar [19] R. Ikehata and H. Takeda, Asymptotic profiles of solutions for structural damped wave equations, arXiv: 1607.01839, Journal of Dynamics and Differential Equations, 31 (2019), 537–571. doi: 10.1007/s10884-019-09731-8. Google Scholar [20] R. Ikehata, G. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces, J. Diff. Eqns, 254 (2013), 3352-3368. doi: 10.1016/j.jde.2013.01.023. Google Scholar [21] G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math, 143 (2000), 175-197. doi: 10.4064/sm-143-2-175-197. Google Scholar [22] X. Lu and M. Reissig, Rates of decay for structural damped models with decreasing in time coefficients, Int. J. Dynamical Systems and Diff. Eqns, 2 (2009), 21-55. doi: 10.1504/IJDSDE.2009.028034. Google Scholar [23] A. Matsumura, On the asymptotic behavior of solutions of semilinear wave equations, Publ. RIMS Kyoto Univ, 12 (1976), 169-189. doi: 10.2977/prims/1195190962. Google Scholar [24] H. Michihisa, Expanding methods for evolution operators of strongly damped wave equations, preprint.Google Scholar [25] K. Nishihara, $L^{p}$-$L^{q}$ estimates to the damped wave equation in $3$-dimensional space and their application, Math. Z., 244 (2003), 631-649. doi: 10.1007/s00209-003-0516-0. Google Scholar [26] 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 [27] R. Racke, Non-homogeneous non-linear damped wave equations in unbounded domains, Math. Meth. Appl. Sci., 13 (1990), 481-491. doi: 10.1002/mma.1670130604. Google Scholar [28] 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 [29] M. Sobajima and Y. Wakasug, Diffusion phenomena for the wave equation with space dependent damping in an exterior domain, J. Diff. Eqns, 261 (2016), 5690-5718. doi: 10.1016/j.jde.2016.08.006. Google Scholar [30] M. Taylor, The diffusion phenomenon for damped wave equations with space-time dependent coefficients, Discrete Continuous Dynamical Systems, 38 (2018), 5921-5941. doi: 10.3934/dcds.2018257. Google Scholar [31] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Diff. Eqns, 174 (2001), 464-489. doi: 10.1006/jdeq.2000.3933. Google Scholar [32] T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457. doi: 10.1007/BF03167068. Google Scholar

show all references

##### References:
 [1] R. C. Charão, C. R. da Luz and R. Ikehata, Sharp decay rates for wave equations with a fractional damping via new method in the Fourier space, J. Math Anal. Appl, 408 (2013), 247-255. doi: 10.1016/j.jmaa.2013.06.016. Google Scholar [2] R. Chill and A. Haraux, An optimal estimate for the difference of solutions of two abstract evolution equations, J. Diff. Eqns, 193 (2003), 385-395. doi: 10.1016/S0022-0396(03)00057-3. Google Scholar [3] M. D'Abbicco, $L^1$-$L^ 1$ estimates for a doubly dissipative semilinear wave equation, NoDEA Nonlinear Differential Equations Appl, 24 (2017), Art. 5, 23 pp. doi: 10.1007/s00030-016-0428-4. Google Scholar [4] M. D'Abbicco and M. R. Ebert, Diffusion phenomena for the wave equation with structural damping in the $L^{p}$-$L^{q}$ framework, J. Diff. Eqns, 256 (2014), 2307-2336. doi: 10.1016/j.jde.2014.01.002. Google Scholar [5] M. D'Abbicco and M. R. Ebert, A classification of structural dissipations for evolution operators, Math. Methods Appl. Sci., 39 (2016), 2558-2582. doi: 10.1002/mma.3713. Google Scholar [6] M. D'Abbicco and M. R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations, Nonlinear Anal., 149 (2017), 1-40. doi: 10.1016/j.na.2016.10.010. Google Scholar [7] M. D'Abbicco, M. R. Ebert and T. Picon, Long time decay estimates in real Hardy spaces for evolution equations with structural dissipation, J. Pseudo-Differ. Oper. Appl., 7 (2016), 261-293. doi: 10.1007/s11868-015-0141-9. Google Scholar [8] M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592. doi: 10.1002/mma.2913. Google Scholar [9] M. Ghisi, M. Gobbino and A. Haraux, Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation, Trans. Amer. Math. Soc., 368 (2016), 2039-2079. doi: 10.1090/tran/6520. Google Scholar [10] R. Ikehata, Decay estimates of solutions for the wave equations with strong damping terms in unbounded domains, Math. Methods Appl. Sci., 24 (2001), 659-670. doi: 10.1002/mma.235. Google Scholar [11] R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Diff. Eqns, 257 (2014), 2159-2177. doi: 10.1016/j.jde.2014.05.031. Google Scholar [12] R. Ikehata and S. Iyota, Asymptotic profile of solutions for some wave equations with very strong structural damping, Math. Methods Appl. Sci., 41 (2018), 5074-5090. doi: 10.1002/mma.4954. Google Scholar [13] R. Ikehata and H. Michihisa, Moment conditions and lower bounds in expanding solutions of wave equations with double damping terms, arXiv: 1807.10020, Asymptotic Analysis doi: 10.3233/ASY-181516. Google Scholar [14] R. Ikehata and M. Natsume, Energy decay estimates for wave equations with a fractional damping, Diff. Int. Eqns, 25 (2012), 939-956. Google Scholar [15] R. Ikehata and M. Onodera, Remarks on large time behavior of the $L^{2}$-norm of solutions to strongly damped wave equations, Diff. Int. Eqns, 30 (2017), 505-520. Google Scholar [16] R. Ikehata and A. Sawada, Asymptotic profiles of solutions for wave equations with frictional and viscoelastic damping terms, Asymptotic Anal., 98 (2016), 59-77. doi: 10.3233/ASY-161361. Google Scholar [17] 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 [18] R. Ikehata and H. Takeda, Large time behavior of global solutions to nonlinear wave equations with frictional and viscoelastic damping terms, Osaka Math. J., (in press).Google Scholar [19] R. Ikehata and H. Takeda, Asymptotic profiles of solutions for structural damped wave equations, arXiv: 1607.01839, Journal of Dynamics and Differential Equations, 31 (2019), 537–571. doi: 10.1007/s10884-019-09731-8. Google Scholar [20] R. Ikehata, G. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces, J. Diff. Eqns, 254 (2013), 3352-3368. doi: 10.1016/j.jde.2013.01.023. Google Scholar [21] G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math, 143 (2000), 175-197. doi: 10.4064/sm-143-2-175-197. Google Scholar [22] X. Lu and M. Reissig, Rates of decay for structural damped models with decreasing in time coefficients, Int. J. Dynamical Systems and Diff. Eqns, 2 (2009), 21-55. doi: 10.1504/IJDSDE.2009.028034. Google Scholar [23] A. Matsumura, On the asymptotic behavior of solutions of semilinear wave equations, Publ. RIMS Kyoto Univ, 12 (1976), 169-189. doi: 10.2977/prims/1195190962. Google Scholar [24] H. Michihisa, Expanding methods for evolution operators of strongly damped wave equations, preprint.Google Scholar [25] K. Nishihara, $L^{p}$-$L^{q}$ estimates to the damped wave equation in $3$-dimensional space and their application, Math. Z., 244 (2003), 631-649. doi: 10.1007/s00209-003-0516-0. Google Scholar [26] 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 [27] R. Racke, Non-homogeneous non-linear damped wave equations in unbounded domains, Math. Meth. Appl. Sci., 13 (1990), 481-491. doi: 10.1002/mma.1670130604. Google Scholar [28] 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 [29] M. Sobajima and Y. Wakasug, Diffusion phenomena for the wave equation with space dependent damping in an exterior domain, J. Diff. Eqns, 261 (2016), 5690-5718. doi: 10.1016/j.jde.2016.08.006. Google Scholar [30] M. Taylor, The diffusion phenomenon for damped wave equations with space-time dependent coefficients, Discrete Continuous Dynamical Systems, 38 (2018), 5921-5941. doi: 10.3934/dcds.2018257. Google Scholar [31] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Diff. Eqns, 174 (2001), 464-489. doi: 10.1006/jdeq.2000.3933. Google Scholar [32] T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457. doi: 10.1007/BF03167068. Google Scholar
 [1] Tae Gab Ha. On viscoelastic wave equation with nonlinear boundary damping and source term. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1543-1576. doi: 10.3934/cpaa.2010.9.1543 [2] Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, Irena Lasiecka, Flávio A. Falcão Nascimento. Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1987-2011. doi: 10.3934/dcdsb.2014.19.1987 [3] Abdelaziz Soufyane, Belkacem Said-Houari. The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evolution Equations & Control Theory, 2014, 3 (4) : 713-738. doi: 10.3934/eect.2014.3.713 [4] Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37 [5] Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361 [6] Wenjun Liu, Biqing Zhu, Gang Li, Danhua Wang. General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term. Evolution Equations & Control Theory, 2017, 6 (2) : 239-260. doi: 10.3934/eect.2017013 [7] Ferhat Mohamed, Hakem Ali. Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 491-506. doi: 10.3934/dcdsb.2017024 [8] Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064 [9] Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations & Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017 [10] Jing Zhang. The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework. Evolution Equations & Control Theory, 2017, 6 (1) : 135-154. doi: 10.3934/eect.2017008 [11] Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251 [12] Hideo Kubo. On the pointwise decay estimate for the wave equation with compactly supported forcing term. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1469-1480. doi: 10.3934/cpaa.2015.14.1469 [13] Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459 [14] Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45 [15] Belkacem Said-Houari, Flávio A. Falcão Nascimento. Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction. Communications on Pure & Applied Analysis, 2013, 12 (1) : 375-403. doi: 10.3934/cpaa.2013.12.375 [16] Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541 [17] Jong Yeoul Park, Sun Hye Park. On uniform decay for the coupled Euler-Bernoulli viscoelastic system with boundary damping. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 425-436. doi: 10.3934/dcds.2005.12.425 [18] Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407 [19] Moez Daoulatli, Irena Lasiecka, Daniel Toundykov. Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 67-94. doi: 10.3934/dcdss.2009.2.67 [20] Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021

2018 Impact Factor: 1.048