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September  2020, 19(9): 4433-4454. doi: 10.3934/cpaa.2020202

A second order fractional differential equation under effects of a super damping

Ruy Coimbra Charão 1,, , Juan Torres Espinoza 1, and  Ryo Ikehata 2,

1. 

Department of Mathematics, Federal University of Santa Catarina, 88040-270, 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

Received  November 2019 Revised  April 2020 Published  June 2020

Fund Project: The first author (R.C.Charão) is partially supported by Print/CAPES process 88881.310536/ 2018-00 and the third author (R. Ikehata) was supported in part by Grant-in-Aid for Scientific Research (C) 15K04958 of JSPS

In this work we study asymptotic properties of global solutions for an initial value problem of a second order fractional differential equation with structural damping. The evolution equation considered includes plate equation problems. We show asymptotic profiles depending on the exponents of the Laplace operators involved in the equation and optimality of the decay rates for the associated energy and the $ L^{2} $ norm of solutions.

Keywords: Second order evolution equation, fractional Laplacian, asymptotic expansion, super damping, optimal decay rates.
Mathematics Subject Classification: Primary:35L15, 35L05;Secondary:35B40, 35B65.
Citation: Ruy Coimbra Charão, Juan Torres Espinoza, Ryo Ikehata. A second order fractional differential equation under effects of a super damping. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4433-4454. doi: 10.3934/cpaa.2020202
References:
[1]

C. I. Christov, G. A. Maugin and M. G. Velarde, Well-posed Boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev. E, 54 (1996), 3621-3638. 

[2]

R. Coimbra 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.

[3]

R. Coimbra Charão, C. R. da Luz and R. Ikehata, New decay rates for a problem of plate dynamics with fractional damping, J. Hyperbolic Differ. Equ., 10 (2013), 563-575.  doi: 10.1142/S0219891613500203.

[4]

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

[5]

A. Esfahani, L. G. Farah and H. Wang, Global existence and blow-up for the generalized sixth-order Boussinesq equation, Nonlinear Anal., 75 (2012), 4325-4338.  doi: 10.1016/j.na.2012.03.019.

[6]

P. G. Geredeli and I. Lasiecka, Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Kármán plates with geometrically localized dissipation and critical nonlinearity, Nonlinear Anal., 91 (2013), 72-92.  doi: 10.1016/j.na.2013.06.008.

[7]

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.

[8]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246.  doi: 10.1007/BF01390079.

[9]

J. L. Horbach, R. Ikehata and R. C. Charão, Optimal Decay Rates and Asymptotic Profile for the Plate Equation with Structural Damping, J. Math. Anal. Appl., 440 (2016), 529-560.  doi: 10.1016/j.jmaa.2016.03.046.

[10]

T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Meth. Appl. Sci., 16 (2006), 1839-1859.  doi: 10.1142/S021820250600173X.

[11]

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

[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.

[13]

R. Ikehata and S. Iyota, Asymptotic profile of solutions for some wave equations with very strong structural damping, Math. Meth. Appl. Sci., 41 (2018), 5074-5090.  doi: 10.1002/mma.4954.

[14]

R. Ikehata and M. Natsume, Energy decay estimates for wave equations with a fractional damping, Differ. Int. Equ., 25 (2012), 939-956. 

[15]

R. Ikehata and M. Soga, Asymptotic profile for a strongly damped plate equation with lower order perturbation, Commun. Pure Appl. Anal., 14 (2015), 1759-1780.  doi: 10.3934/cpaa.2015.14.1759.

[16]

R. Ikehata, G. 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.

[17]

C. R. da Luz and R. Coimbra Charão, Asymptotic properties for a semilinear plate equation in unbounded domains, J. Hyperbolic Differ. Equ., 6 (2009), 269-294.  doi: 10.1142/S0219891609001824.

[18]

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

[19] G. A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford Mathematical Monographs Series, Oxford University Press, Oxford, 1999. 
[20]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS, Kyoto Univ., 12 (1976), 169-189.  doi: 10.2977/prims/1195190962.

[21]

J. E. Muños Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625-1639.  doi: 10.3934/dcds.2003.9.1625.

[22]

B. Muckenhoupt, Weighted norm inequalities for the Fourier transform, Trans. Amer. Math. Soc., 276 (1983), 729-742.  doi: 10.1090/S0002-9947-1983-0688974-X.

[23]

D. H. Peregrine, Long waves on a beach, J. Fluid Mech., 27 (1967), 815-827. 

[24]

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

[25]

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.

[26]

J. Sander and K. Hutter, On the development of the theory of solitary wave, Acta Mechanica, 86 (1991), 11-152.  doi: 10.1007/BF01175953.

[27]

F. Serre, Contribution a l'etude des ecoulements permanents et variables dans les canaux, La Houille Blanche, 8 (1953), 374-388. 

[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]

Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation, J. Hyperbolic Differ. Equ., 7 (2010), 471-501.  doi: 10.1142/S0219891610002207.

[30]

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

[31]

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), 207-222.  doi: 10.1007/BF03167068.

[32]

S. Wang and G. Chen, The Cauchy problem for the generalized IMBq equation in $W^{s,p}(\mathbb{R})$, J. Math. Anal. Appl., 266 (2002), 38-54.  doi: 10.1006/jmaa.2001.7670.

[33]

S. Wang and G. Chen, Small amplitude solutions of the generalized IMBq equation, J. Math. Anal. Appl., 274 (2002), 846-866.  doi: 10.1016/S0022-247X(02)00401-8.

[34]

S. Wang and H. Xu, On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term, J. Differ. Equ., 252 (2012), 4243-4258.  doi: 10.1016/j.jde.2011.12.016.

[35]

S. Wang and H. Xu, On the asymptotic behavior of solution for the generalized IBq equation with Stokes damped term, Z. Angew. Math. Phys., 64 (2013), 719-731.  doi: 10.1007/s00033-012-0257-1.

[36]

S. Wang and H. Xue, Global Solution for a Generalized Boussinesq Equation, Appl. Math. Comput., 204 (2008), 130-136.  doi: 10.1016/j.amc.2008.06.059.

show all references

References:
[1]

C. I. Christov, G. A. Maugin and M. G. Velarde, Well-posed Boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev. E, 54 (1996), 3621-3638. 

[2]

R. Coimbra 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.

[3]

R. Coimbra Charão, C. R. da Luz and R. Ikehata, New decay rates for a problem of plate dynamics with fractional damping, J. Hyperbolic Differ. Equ., 10 (2013), 563-575.  doi: 10.1142/S0219891613500203.

[4]

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

[5]

A. Esfahani, L. G. Farah and H. Wang, Global existence and blow-up for the generalized sixth-order Boussinesq equation, Nonlinear Anal., 75 (2012), 4325-4338.  doi: 10.1016/j.na.2012.03.019.

[6]

P. G. Geredeli and I. Lasiecka, Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Kármán plates with geometrically localized dissipation and critical nonlinearity, Nonlinear Anal., 91 (2013), 72-92.  doi: 10.1016/j.na.2013.06.008.

[7]

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.

[8]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246.  doi: 10.1007/BF01390079.

[9]

J. L. Horbach, R. Ikehata and R. C. Charão, Optimal Decay Rates and Asymptotic Profile for the Plate Equation with Structural Damping, J. Math. Anal. Appl., 440 (2016), 529-560.  doi: 10.1016/j.jmaa.2016.03.046.

[10]

T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Meth. Appl. Sci., 16 (2006), 1839-1859.  doi: 10.1142/S021820250600173X.

[11]

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

[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.

[13]

R. Ikehata and S. Iyota, Asymptotic profile of solutions for some wave equations with very strong structural damping, Math. Meth. Appl. Sci., 41 (2018), 5074-5090.  doi: 10.1002/mma.4954.

[14]

R. Ikehata and M. Natsume, Energy decay estimates for wave equations with a fractional damping, Differ. Int. Equ., 25 (2012), 939-956. 

[15]

R. Ikehata and M. Soga, Asymptotic profile for a strongly damped plate equation with lower order perturbation, Commun. Pure Appl. Anal., 14 (2015), 1759-1780.  doi: 10.3934/cpaa.2015.14.1759.

[16]

R. Ikehata, G. 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.

[17]

C. R. da Luz and R. Coimbra Charão, Asymptotic properties for a semilinear plate equation in unbounded domains, J. Hyperbolic Differ. Equ., 6 (2009), 269-294.  doi: 10.1142/S0219891609001824.

[18]

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

[19] G. A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford Mathematical Monographs Series, Oxford University Press, Oxford, 1999. 
[20]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS, Kyoto Univ., 12 (1976), 169-189.  doi: 10.2977/prims/1195190962.

[21]

J. E. Muños Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625-1639.  doi: 10.3934/dcds.2003.9.1625.

[22]

B. Muckenhoupt, Weighted norm inequalities for the Fourier transform, Trans. Amer. Math. Soc., 276 (1983), 729-742.  doi: 10.1090/S0002-9947-1983-0688974-X.

[23]

D. H. Peregrine, Long waves on a beach, J. Fluid Mech., 27 (1967), 815-827. 

[24]

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

[25]

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.

[26]

J. Sander and K. Hutter, On the development of the theory of solitary wave, Acta Mechanica, 86 (1991), 11-152.  doi: 10.1007/BF01175953.

[27]

F. Serre, Contribution a l'etude des ecoulements permanents et variables dans les canaux, La Houille Blanche, 8 (1953), 374-388. 

[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]

Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation, J. Hyperbolic Differ. Equ., 7 (2010), 471-501.  doi: 10.1142/S0219891610002207.

[30]

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

[31]

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), 207-222.  doi: 10.1007/BF03167068.

[32]

S. Wang and G. Chen, The Cauchy problem for the generalized IMBq equation in $W^{s,p}(\mathbb{R})$, J. Math. Anal. Appl., 266 (2002), 38-54.  doi: 10.1006/jmaa.2001.7670.

[33]

S. Wang and G. Chen, Small amplitude solutions of the generalized IMBq equation, J. Math. Anal. Appl., 274 (2002), 846-866.  doi: 10.1016/S0022-247X(02)00401-8.

[34]

S. Wang and H. Xu, On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term, J. Differ. Equ., 252 (2012), 4243-4258.  doi: 10.1016/j.jde.2011.12.016.

[35]

S. Wang and H. Xu, On the asymptotic behavior of solution for the generalized IBq equation with Stokes damped term, Z. Angew. Math. Phys., 64 (2013), 719-731.  doi: 10.1007/s00033-012-0257-1.

[36]

S. Wang and H. Xue, Global Solution for a Generalized Boussinesq Equation, Appl. Math. Comput., 204 (2008), 130-136.  doi: 10.1016/j.amc.2008.06.059.

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