doi: 10.3934/dcds.2021189
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A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

1. 

Department of Mathematics, Graduate Program in Pure and Applied Mathematics, Federal University of Santa Catarina, 88040-900, Florianopolis, Brazil

2. 

Department of Mathematics, Division of Educational Sciences, Graduate School of Humanities and Social Sciences, Hiroshima University, Higashi-Hiroshima 739-8524, Japan

*Corresponding author: Ruy Coimbra Charão

Received  May 2021 Revised  October 2021 Early access December 2021

Fund Project: The work of the first author (R. C. CHARÃO) was partially supported by PRINT/CAPES - Process 88881.310536/2018-00, the work of the second author (A. PISKE) was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and the work of the third author (R. IKEHATA) was supported in part by Grant-in-Aid for Scientific Research (C)20K03682 of JSPS

We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in $ {{\bf R}}^{n} $, and study the asymptotic profile and optimal decay rates of solutions as $ t \to \infty $ in $ L^{2} $-sense. The operator $ L $ considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charão-Ikehata [7]. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the property of the solutions into three parts from the viewpoint of regularity of the initial data, that is, diffusion-like, wave-like, and both of them.

Citation: Ruy Coimbra Charão, Alessandra Piske, Ryo Ikehata. A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021189
References:
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J. Barrera and H. Volkmer, Asymptotic expansion of the L2-norm of a solution of the strongly damped wave equation in space dimension 1 and 2, Asymptotic Anal., 121 (2021), 367-399.  doi: 10.3233/ASY-201606.  Google Scholar

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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

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R. C. CharãoM. D'Abbicco and R. Ikehata, Asymptotic profile for a wave equations with parameter dependent logarithmic damping, Math. Methods Appl. Sci., 44 (2021), 14003-14024.  doi: 10.1002/mma.7671.  Google Scholar

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R. C. CharãoJ. T. Espinoza and R. Ikehata, A second order fractional differential equation under effects of a super damping, Comm. Pure Appl. Analysis, 19 (2020), 4433-4454.  doi: 10.3934/cpaa.2020202.  Google Scholar

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R. C. Charão and J. L. Horbach, Existence and decay rates for a semilinear dissipative fractional second order evolution equation, Ciência e Natura, 42 (2020), 1-44.  doi: 10.5902/2179460X40996.  Google Scholar

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W. Chen, Cauchy problem for thermoelastic plate equations with different damping mechanisms, Commun. Math. Sci., 18 (2020), 429-457.  doi: 10.4310/CMS.2020.v18.n2.a7.  Google Scholar

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W. Chen, Dissipative structure and diffusion phenomena for doubly dissipative elastic waves in two space dimensions, J. Math. Anal. Appl., 486 (2020), 123922.  doi: 10.1016/j.jmaa.2020.123922.  Google Scholar

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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

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C. R. da LuzR. Ikehata and R. C. Charão, Asymptotic behavior for abstract evolution differential equations of second order, J. Diff. Eqns., 259 (2015), 5017-5039.  doi: 10.1016/j.jde.2015.06.012.  Google Scholar

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M. D'Abbicco and M. R. Ebert, Diffusion phenomena for the wave equation with structural damping in the Lp-Lq framework, J. Diff. Eqns., 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 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'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

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M. D'AbbiccoG. Girardi and J. Liang, L1-L1 estimates for the strongly damped plate equation, J. Math. Anal. Appl., 478 (2019), 476-498.  doi: 10.1016/j.jmaa.2019.05.039.  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

[17]

T. A. Dao and M. Reissig, L1 estimates for oscillating integrals and their applications to semi-linear models with σ-evolution like structural damping, Discrete Contin. Dyn. Syst., 39 (2019), 5431-5463.  doi: 10.3934/dcds.2019222.  Google Scholar

[18]

T. A. Dao and M. Reissig, An application of L1 estimates for oscillating integrals to parabolic-like semi-linear structurally damped σ-models, J. Math. Anal. Appl., 476 (2019), 426-463.  doi: 10.1016/j.jmaa.2019.03.048.  Google Scholar

[19]

P. M. N. Dharmawardane, Global solutions and decay property of regularity-loss type for quasi-linear hyperbolic systems with dissipation, J. Hyperbolic Differ. Equ., 10 (2013), 37-76.  doi: 10.1142/S0219891613500021.  Google Scholar

[20]

P. M. N. DharmawardaneT. Nakamura and S. Kawashima, Decay estimates of solutions for quasi-linear hyperbolic systems of viscoelasticity, SIAM J. Math. Anal., 44 (2012), 1976-2001.  doi: 10.1137/11083900X.  Google Scholar

[21]

T. FukushimaR. Ikehata and H. Michihisa, Asymptotic profiles for damped plate equations with rotational inertia terms, J. Hyperbolic Differ. Equ., 17 (2020), 569-589.  doi: 10.1142/S0219891620500162.  Google Scholar

[22]

T. FukushimaR. Ikehata and H. Michihisa, Thresholds for low regularity solutions to wave equations with structural damping, J. Math. Anal. Appl., 494 (2021), 124669.  doi: 10.1016/j.jmaa.2020.124669.  Google Scholar

[23]

M. GhisiM. Gobbino and A. Haraux, Local and global smoothing effects for somr linear hyperbolic equations with a strong dissipation, Trans. Amer. Math. Soc., 368 (2016), 2039-2079.  doi: 10.1090/tran/6520.  Google Scholar

[24]

K. IdeK. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci., 18 (2008), 647-667.  doi: 10.1142/S0218202508002802.  Google Scholar

[25]

R. Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem, Math. Meth. Appl. Sci., 27 (2004), 865-889.  doi: 10.1002/mma.476.  Google Scholar

[26]

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

[27]

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

[28]

R. Ikehata and M. Onodera, Remark on large time behavior of the L2-norm of solutions to strongly damped wave equations, Diff. Int. Eqns., 30 (2017), 505-520.   Google Scholar

[29]

R. Ikehata and A. Sawada, Asymptotic profile of solutions for wave equations with frictional and viscoelastic damping terms, Asymptotic Anal., 98 (2016), 59-77.  doi: 10.3233/ASY-161361.  Google Scholar

[30]

R. Ikehata and H. Takeda, Asymptotic profiles of solutions for structural damped wave equations, J. Dynamics and Diff. Eqns., 31 (2019), 537-571.  doi: 10.1007/s10884-019-09731-8.  Google Scholar

[31]

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

[32]

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

[33]

J. R. Luyo Sánchez, O Sistema Dinâmico de Von Kármán em Domínios não Limitados é Globalmente Bem Posto no Sentido de Hadamard: Análise do seu Limite Singular (Portuguese), Ph.D thesis, Institute of Mathematics, Federal University of Rio de Janeiro, Brazil, 2003. Google Scholar

[34]

H. Michihisa, Optimal leading term of solutions to wave equations with strong damping terms, Hokkaido Math. J., 50 (2021), 165-186.  doi: 10.14492/hokmj/2018-920.  Google Scholar

[35]

T. Narazaki and M. Reissig, L1 estimates for oscillating integrals related to structural damped wave models, Studies in phase space analysis with applications to PDEs, Studies in Phase Space Analysis with Applications to PDEs, 215–258, Progr. Nonlinear Differential Equations Appl., 84, Birkhäuser/Springer, New York, 2013. doi: 10.1007/978-1-4614-6348-1_11.  Google Scholar

[36]

D. T. PhamM. K. Mezadek and M. Reissig, Global existence for semi-linear structurally damped σ-evolution models, J. Math. Anal. Appl., 431 (2015), 569-596.  doi: 10.1016/j.jmaa.2015.06.001.  Google Scholar

[37]

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

[38]

M. Reissig, Structurally damped elastic waves in 2D, Math. Methods Appl. Sci., 39 (2016), 4618-4628.  doi: 10.1002/mma.3888.  Google Scholar

[39]

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

[40]

X. Su and S. Wang, Optimal decay rates and small global solutions to the dissipative Boussinesq equation, Math. Meth. Appl. Sci., 43 (2020), 174-198.  doi: 10.1002/mma.5843.  Google Scholar

[41]

H. Takeda and S. Yoshikawa, On the initial value problem of the semilinear beam equation with weak damping Ⅱ, Asymptotic profiles, J. Diff. Eqns., 253 (2012), 3061-3080.  doi: 10.1016/j.jde.2012.07.014.  Google Scholar

[42]

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

[43]

G. N. Watson, Asymptotic expansions of hypergeometric functions, Trans. Cambridge Philos. Soc., 22 (1918), 277-308.   Google Scholar

show all references

References:
[1]

J. Barrera and H. Volkmer, Asymptotic expansion of the L2-norm of a solution of the strongly damped wave equation, J. Diff. Eqns., 267 (2019), 902-937.  doi: 10.1016/j.jde.2019.01.028.  Google Scholar

[2]

J. Barrera and H. Volkmer, Asymptotic expansion of the L2-norm of a solution of the strongly damped wave equation in space dimension 1 and 2, Asymptotic Anal., 121 (2021), 367-399.  doi: 10.3233/ASY-201606.  Google Scholar

[3]

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

[4]

R. C. CharãoM. D'Abbicco and R. Ikehata, Asymptotic profile for a wave equations with parameter dependent logarithmic damping, Math. Methods Appl. Sci., 44 (2021), 14003-14024.  doi: 10.1002/mma.7671.  Google Scholar

[5]

R. C. CharãoJ. T. Espinoza and R. Ikehata, A second order fractional differential equation under effects of a super damping, Comm. Pure Appl. Analysis, 19 (2020), 4433-4454.  doi: 10.3934/cpaa.2020202.  Google Scholar

[6]

R. C. Charão and J. L. Horbach, Existence and decay rates for a semilinear dissipative fractional second order evolution equation, Ciência e Natura, 42 (2020), 1-44.  doi: 10.5902/2179460X40996.  Google Scholar

[7]

R. C. Charão and R. Ikehata, Asymptotic profile and optimal decay of solutions of some wave equations with logarithmic damping, Z. Angew. Math. Phys., 71 (2020), Paper No. 148, 26 pp. doi: 10.1007/s00033-020-01373-x.  Google Scholar

[8]

W. Chen, Cauchy problem for thermoelastic plate equations with different damping mechanisms, Commun. Math. Sci., 18 (2020), 429-457.  doi: 10.4310/CMS.2020.v18.n2.a7.  Google Scholar

[9]

W. Chen, Dissipative structure and diffusion phenomena for doubly dissipative elastic waves in two space dimensions, J. Math. Anal. Appl., 486 (2020), 123922.  doi: 10.1016/j.jmaa.2020.123922.  Google Scholar

[10]

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

[11]

C. R. da LuzR. Ikehata and R. C. Charão, Asymptotic behavior for abstract evolution differential equations of second order, J. Diff. Eqns., 259 (2015), 5017-5039.  doi: 10.1016/j.jde.2015.06.012.  Google Scholar

[12]

M. D'Abbicco and M. R. Ebert, Diffusion phenomena for the wave equation with structural damping in the Lp-Lq framework, J. Diff. Eqns., 256 (2014), 2307-2336.  doi: 10.1016/j.jde.2014.01.002.  Google Scholar

[13]

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

[14]

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

[15]

M. D'AbbiccoG. Girardi and J. Liang, L1-L1 estimates for the strongly damped plate equation, J. Math. Anal. Appl., 478 (2019), 476-498.  doi: 10.1016/j.jmaa.2019.05.039.  Google Scholar

[16]

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

[17]

T. A. Dao and M. Reissig, L1 estimates for oscillating integrals and their applications to semi-linear models with σ-evolution like structural damping, Discrete Contin. Dyn. Syst., 39 (2019), 5431-5463.  doi: 10.3934/dcds.2019222.  Google Scholar

[18]

T. A. Dao and M. Reissig, An application of L1 estimates for oscillating integrals to parabolic-like semi-linear structurally damped σ-models, J. Math. Anal. Appl., 476 (2019), 426-463.  doi: 10.1016/j.jmaa.2019.03.048.  Google Scholar

[19]

P. M. N. Dharmawardane, Global solutions and decay property of regularity-loss type for quasi-linear hyperbolic systems with dissipation, J. Hyperbolic Differ. Equ., 10 (2013), 37-76.  doi: 10.1142/S0219891613500021.  Google Scholar

[20]

P. M. N. DharmawardaneT. Nakamura and S. Kawashima, Decay estimates of solutions for quasi-linear hyperbolic systems of viscoelasticity, SIAM J. Math. Anal., 44 (2012), 1976-2001.  doi: 10.1137/11083900X.  Google Scholar

[21]

T. FukushimaR. Ikehata and H. Michihisa, Asymptotic profiles for damped plate equations with rotational inertia terms, J. Hyperbolic Differ. Equ., 17 (2020), 569-589.  doi: 10.1142/S0219891620500162.  Google Scholar

[22]

T. FukushimaR. Ikehata and H. Michihisa, Thresholds for low regularity solutions to wave equations with structural damping, J. Math. Anal. Appl., 494 (2021), 124669.  doi: 10.1016/j.jmaa.2020.124669.  Google Scholar

[23]

M. GhisiM. Gobbino and A. Haraux, Local and global smoothing effects for somr linear hyperbolic equations with a strong dissipation, Trans. Amer. Math. Soc., 368 (2016), 2039-2079.  doi: 10.1090/tran/6520.  Google Scholar

[24]

K. IdeK. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci., 18 (2008), 647-667.  doi: 10.1142/S0218202508002802.  Google Scholar

[25]

R. Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem, Math. Meth. Appl. Sci., 27 (2004), 865-889.  doi: 10.1002/mma.476.  Google Scholar

[26]

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

[27]

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

[28]

R. Ikehata and M. Onodera, Remark on large time behavior of the L2-norm of solutions to strongly damped wave equations, Diff. Int. Eqns., 30 (2017), 505-520.   Google Scholar

[29]

R. Ikehata and A. Sawada, Asymptotic profile of solutions for wave equations with frictional and viscoelastic damping terms, Asymptotic Anal., 98 (2016), 59-77.  doi: 10.3233/ASY-161361.  Google Scholar

[30]

R. Ikehata and H. Takeda, Asymptotic profiles of solutions for structural damped wave equations, J. Dynamics and Diff. Eqns., 31 (2019), 537-571.  doi: 10.1007/s10884-019-09731-8.  Google Scholar

[31]

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

[32]

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

[33]

J. R. Luyo Sánchez, O Sistema Dinâmico de Von Kármán em Domínios não Limitados é Globalmente Bem Posto no Sentido de Hadamard: Análise do seu Limite Singular (Portuguese), Ph.D thesis, Institute of Mathematics, Federal University of Rio de Janeiro, Brazil, 2003. Google Scholar

[34]

H. Michihisa, Optimal leading term of solutions to wave equations with strong damping terms, Hokkaido Math. J., 50 (2021), 165-186.  doi: 10.14492/hokmj/2018-920.  Google Scholar

[35]

T. Narazaki and M. Reissig, L1 estimates for oscillating integrals related to structural damped wave models, Studies in phase space analysis with applications to PDEs, Studies in Phase Space Analysis with Applications to PDEs, 215–258, Progr. Nonlinear Differential Equations Appl., 84, Birkhäuser/Springer, New York, 2013. doi: 10.1007/978-1-4614-6348-1_11.  Google Scholar

[36]

D. T. PhamM. K. Mezadek and M. Reissig, Global existence for semi-linear structurally damped σ-evolution models, J. Math. Anal. Appl., 431 (2015), 569-596.  doi: 10.1016/j.jmaa.2015.06.001.  Google Scholar

[37]

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

[38]

M. Reissig, Structurally damped elastic waves in 2D, Math. Methods Appl. Sci., 39 (2016), 4618-4628.  doi: 10.1002/mma.3888.  Google Scholar

[39]

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

[40]

X. Su and S. Wang, Optimal decay rates and small global solutions to the dissipative Boussinesq equation, Math. Meth. Appl. Sci., 43 (2020), 174-198.  doi: 10.1002/mma.5843.  Google Scholar

[41]

H. Takeda and S. Yoshikawa, On the initial value problem of the semilinear beam equation with weak damping Ⅱ, Asymptotic profiles, J. Diff. Eqns., 253 (2012), 3061-3080.  doi: 10.1016/j.jde.2012.07.014.  Google Scholar

[42]

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

[43]

G. N. Watson, Asymptotic expansions of hypergeometric functions, Trans. Cambridge Philos. Soc., 22 (1918), 277-308.   Google Scholar

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