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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

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.

Citation: Ruy Coimbra Charão, Juan Torres Espinoza, Ryo Ikehata. A second order fractional differential equation under effects of a super damping. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4433-4454. doi: 10.3934/cpaa.2020202
References:
[1]

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

[2]

R. Coimbra 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

[3]

R. Coimbra CharãoC. 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.  Google Scholar

[4]

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

[5]

A. EsfahaniL. 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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

J. L. HorbachR. 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.  Google Scholar

[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.  Google Scholar

[11]

K. IdeK. 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.  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 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.  Google Scholar

[14]

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

[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.  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]

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.  Google Scholar

[18]

C. R. da LuzR. 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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

D. T. PhamM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

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.  Google Scholar

[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.  Google Scholar

[31]

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), 207-222.  doi: 10.1007/BF03167068.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

R. Coimbra 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

[3]

R. Coimbra CharãoC. 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.  Google Scholar

[4]

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

[5]

A. EsfahaniL. 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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

J. L. HorbachR. 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.  Google Scholar

[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.  Google Scholar

[11]

K. IdeK. 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.  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 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.  Google Scholar

[14]

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

[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.  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]

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.  Google Scholar

[18]

C. R. da LuzR. 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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

D. T. PhamM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

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.  Google Scholar

[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.  Google Scholar

[31]

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), 207-222.  doi: 10.1007/BF03167068.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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