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May  2022, 42(5): 2215-2255. doi: 10.3934/dcds.2021189

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 Published  May 2022 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 and Continuous Dynamical Systems, 2022, 42 (5) : 2215-2255. doi: 10.3934/dcds.2021189
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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592.  doi: 10.1002/mma.2913.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[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.

[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.

[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.

[43]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592.  doi: 10.1002/mma.2913.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[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.

[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.

[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.

[43]

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

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