March  2020, 19(3): 1581-1608. doi: 10.3934/cpaa.2020079

Study of semi-linear $ \sigma $-evolution equations with frictional and visco-elastic damping

1. 

School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, No.1 Dai Co Viet road, Hanoi, Vietnam

2. 

Faculty for Mathematics and Computer Science, TU Bergakademie Freiberg, Prüferstr. 9, 09596, Freiberg, Germany

3. 

Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan

* Corresponding author

Received  July 2019 Revised  September 2019 Published  November 2019

Fund Project: The first author is supported by Vietnamese Government's Scholarship (Grant number: 2015/911)

In this article, we study semi-linear $ \sigma $-evolution equations with double damping including frictional and visco-elastic damping for any $ \sigma\ge 1 $. We are interested in investigating not only higher order asymptotic expansions of solutions but also diffusion phenomenon in the $ L^p-L^q $ framework, with $ 1\le p\le q\le \infty $, to the corresponding linear equations. By assuming additional $ L^{m} $ regularity on the initial data, with $ m\in [1, 2) $, we prove the global (in time) existence of small data energy solutions and indicate the large time behavior of global obtained solutions as well to semi-linear equations. Moreover, we also determine the so-called critical exponent when $ \sigma $ is integers.

Citation: Tuan Anh Dao, Hironori Michihisa. Study of semi-linear $ \sigma $-evolution equations with frictional and visco-elastic damping. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1581-1608. doi: 10.3934/cpaa.2020079
References:
[1]

M. D'Abbicco, $L^1-L^1$ estimates for a doubly dissipative semilinear wave equation, Nonlinear Differ. Equ. Appl., 24 (2017), 1-23.  doi: 10.1007/s00030-016-0428-4.  Google Scholar

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M. D'Abbicco and M. R. Ebert, Diffusion phenomena for the wave equation with structural damping in the $L^p-L^q$ framework, J. Differ. Equ., 256 (2014), 2307-2336.  doi: 10.1016/j.jde.2014.01.002.  Google Scholar

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M. D'Abbicco and M. R. Ebert, A classifiation of structural dissipations for evolution operators, Math. Methods Appl. Sci., 39 (2016), 2558-2582.  doi: 10.1002/mma.3713.  Google Scholar

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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

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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

[6]

T. A. Dao and M. Reissig, An application of $L^1$ estimates for oscillating integrals to parabolic like semi-linear structurally damped $\sigma$-evolution models, J. Math. Anal. Appl., 476 (2019), 426-463.  doi: 10.1016/j.jmaa.2019.03.048.  Google Scholar

[7]

T. A. Dao and M. Reissig, $L^1$ estimates for oscillating integrals and their applications to semi-linear models with $\sigma$-evolution like structural damping, Discrete Contin. Dyn. Syst. A, 39 (2019), 5431-5463.  doi: 10.3934/dcds.2019222.  Google Scholar

[8]

P. T. Duong and M. Reissig, The external damping Cauchy problems with general powers of the Laplacian, in, New Trends in Analysis and Interdisciplinary Applications: Trends in Mathematics, Birkhäuser, Cham (2017), pp. 537–543. doi: 10.1007/978-3-319-48812-7_68.  Google Scholar

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M. R. Ebert and M. Reissig, Methods for Partial Differential Equations, Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models, Birkhäuser, 2018. doi: 10.1007/978-3-319-66456-9.  Google Scholar

[10]

V. A. Galaktionov, E. L. Mitidieri and S. I. Pohozaev, Blow-up for higher-order prabolic, hyperbolic, dispersion and Schrödinger equations, in, Monogr. Res. Notes Math., Chapman and Hall/CRC, 2014.  Google Scholar

[11]

H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, Harmonic Analysis and Nonlinear Partial Differential Equations, Res. Inst. Math. Sci. (RIMS), RIMS Kokyuroku Bessatsu, B26, Kyoto, (2011), 159–175.  Google Scholar

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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

[13]

R. Ikehata and H. Michihisa, Moment conditions and lower bounds in expanding solutions of wave equations with double damping terms, Asymptot. Anal., 114 (2019), 19–36.  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. IkehataG. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces, J. Differ. Equ., 254 (2013), 3352-3368.  doi: 10.1016/j.jde.2013.01.023.  Google Scholar

[17]

H. Michihisa, Optimal leading term of solutions to wave equations with strong damping terms, Hokkaido Math. J., to appear. Google Scholar

[18]

H. Michihisa, Expanding methods for evolution operators of strongly damped wave equations, submitted, (2018). Google Scholar

[19]

T. Narazaki, $L^p-L^q$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626.  doi: 10.2969/jmsj/1191418647.  Google Scholar

[20]

H. Takeda, Higher-order expansion of solutions for a damped wave equation, Asymptot. Anal., 94 (2015), 1-31.  doi: 10.3233/ASY-151295.  Google Scholar

[21]

Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1.  Google Scholar

show all references

References:
[1]

M. D'Abbicco, $L^1-L^1$ estimates for a doubly dissipative semilinear wave equation, Nonlinear Differ. Equ. Appl., 24 (2017), 1-23.  doi: 10.1007/s00030-016-0428-4.  Google Scholar

[2]

M. D'Abbicco and M. R. Ebert, Diffusion phenomena for the wave equation with structural damping in the $L^p-L^q$ framework, J. Differ. Equ., 256 (2014), 2307-2336.  doi: 10.1016/j.jde.2014.01.002.  Google Scholar

[3]

M. D'Abbicco and M. R. Ebert, A classifiation of structural dissipations for evolution operators, Math. Methods Appl. Sci., 39 (2016), 2558-2582.  doi: 10.1002/mma.3713.  Google Scholar

[4]

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

[5]

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

[6]

T. A. Dao and M. Reissig, An application of $L^1$ estimates for oscillating integrals to parabolic like semi-linear structurally damped $\sigma$-evolution models, J. Math. Anal. Appl., 476 (2019), 426-463.  doi: 10.1016/j.jmaa.2019.03.048.  Google Scholar

[7]

T. A. Dao and M. Reissig, $L^1$ estimates for oscillating integrals and their applications to semi-linear models with $\sigma$-evolution like structural damping, Discrete Contin. Dyn. Syst. A, 39 (2019), 5431-5463.  doi: 10.3934/dcds.2019222.  Google Scholar

[8]

P. T. Duong and M. Reissig, The external damping Cauchy problems with general powers of the Laplacian, in, New Trends in Analysis and Interdisciplinary Applications: Trends in Mathematics, Birkhäuser, Cham (2017), pp. 537–543. doi: 10.1007/978-3-319-48812-7_68.  Google Scholar

[9]

M. R. Ebert and M. Reissig, Methods for Partial Differential Equations, Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models, Birkhäuser, 2018. doi: 10.1007/978-3-319-66456-9.  Google Scholar

[10]

V. A. Galaktionov, E. L. Mitidieri and S. I. Pohozaev, Blow-up for higher-order prabolic, hyperbolic, dispersion and Schrödinger equations, in, Monogr. Res. Notes Math., Chapman and Hall/CRC, 2014.  Google Scholar

[11]

H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, Harmonic Analysis and Nonlinear Partial Differential Equations, Res. Inst. Math. Sci. (RIMS), RIMS Kokyuroku Bessatsu, B26, Kyoto, (2011), 159–175.  Google Scholar

[12]

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

[13]

R. Ikehata and H. Michihisa, Moment conditions and lower bounds in expanding solutions of wave equations with double damping terms, Asymptot. Anal., 114 (2019), 19–36.  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. IkehataG. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces, J. Differ. Equ., 254 (2013), 3352-3368.  doi: 10.1016/j.jde.2013.01.023.  Google Scholar

[17]

H. Michihisa, Optimal leading term of solutions to wave equations with strong damping terms, Hokkaido Math. J., to appear. Google Scholar

[18]

H. Michihisa, Expanding methods for evolution operators of strongly damped wave equations, submitted, (2018). Google Scholar

[19]

T. Narazaki, $L^p-L^q$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626.  doi: 10.2969/jmsj/1191418647.  Google Scholar

[20]

H. Takeda, Higher-order expansion of solutions for a damped wave equation, Asymptot. Anal., 94 (2015), 1-31.  doi: 10.3233/ASY-151295.  Google Scholar

[21]

Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1.  Google Scholar

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