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

Ryo Ikehata; Shingo Kitazaki

doi:10.3934/eect.2019040

Article Contents

2019, Volume 8, Issue 4: 825-846. Doi: 10.3934/eect.2019040

This issue Previous Article A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem Next Article Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

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

Ryo Ikehata^, and
Shingo Kitazaki

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

^* Corresponding author: Ryo Ikehata
^* Corresponding author: Ryo Ikehata

Received: September 30, 2018

Revised: January 31, 2019

Early access: June 2019

Published: June 2019

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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].

Keywords:

Mathematics Subject Classification: Primary: 35L15, 35L05; Secondary: 35B40.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592. doi: 10.1002/mma.2913.
[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.
[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.
[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.
[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.
[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.
[14]	R. Ikehata and M. Natsume, Energy decay estimates for wave equations with a fractional damping, Diff. Int. Eqns, 25 (2012), 939-956.
[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.
[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.
[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.
[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).
[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.
[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.
[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.
[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.
[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.
[24]	H. Michihisa, Expanding methods for evolution operators of strongly damped wave equations, preprint.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.