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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. CharãoC. 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'AbbiccoM. 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. GhisiM. 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. IkehataG. 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. UmedaS. 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. CharãoC. 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'AbbiccoM. 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. GhisiM. 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. IkehataG. 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. UmedaS. 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

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