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June  2021, 20(6): 2257-2277. doi: 10.3934/cpaa.2021067

Fractional oscillon equations; solvability and connection with classical oscillon equations

Flank D. M. Bezerra 1, , Rodiak N. Figueroa-López 2, and  Marcelo J. D. Nascimento 2,,

1. 

Universidade Federal da Paraíba, Departamento de Matemática, 58051-900 João Pessoa PB, Brazil
2. 

Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos SP, Brazil

* Corresponding author

Received  March 2020 Revised  March 2021 Published  June 2021 Early access  April 2021

Fund Project: The second author is supported by CAPES/Finance Code 001/2019, Brazil. The third author is partially supported by FAPESP grant # 2017/06582-2, Brazil

In this paper we are concerned with the asymptotic behavior of nonautonomous fractional approximations of oscillon equation
$ u_{tt}-\mu(t)\Delta u+\omega(t)u_t = f(u),\ x\in\Omega,\ t\in{\mathbb{R}}, $
subject to Dirichlet boundary condition on
$ \partial \Omega $
, where
$ \Omega $
 is a bounded smooth domain in
$ {\mathbb{R}}^N $
,
$ N\geq 3 $
, the function
$ \omega $
 is a time-dependent damping,
$ \mu $
 is a time-dependent squared speed of propagation, and
$ f $
 is a nonlinear functional. Under structural assumptions on
$ \omega $
 and
$ \mu $
 we establish the existence of time-dependent attractor for the fractional models in the sense of Carvalho, Langa, Robinson [6], and Di Plinio, Duane, Temam [10].
Keywords: oscillon equation, fractional powers, fractional equations, semilinear hyperbolic equations, pullback attractor.
Mathematics Subject Classification: Primary: 37B55, 35B40, 35B41; Secondary: 34A08, 35L71.
Citation: Flank D. M. Bezerra, Rodiak N. Figueroa-López, Marcelo J. D. Nascimento. Fractional oscillon equations; solvability and connection with classical oscillon equations. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2257-2277. doi: 10.3934/cpaa.2021067
References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

F. D. M. Bezerra, A. N. Carvalho, J. W. Cholewa and M. J. D. Nascimento, Parabolic approximation of damped wave equations via fractional powers: fast growing nonlinearities and continuity of the dynamics, J. Math. Anal. Appl., 450 (2017), 377-405.  doi: 10.1016/j.jmaa.2017.01.024.

[3]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283.  doi: 10.1016/j.na.2010.11.032.

[4]

A. N. Carvalho and J. W. Cholewa, Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Aust. Math. Soc., 66 (2002), 443-463. doi: 10.1017/S0004972700040296.

[5]

A. N. Carvalho and J. W. Cholewa, Strongly damped wave equations in $W^{1,p}_0(\Omega) \times L^{p}(\Omega)$, Discrete Contin. Dyn. Syst., 2007 (2007), 230-239. 

[6]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences 182, Springer-Verlag, New York, 2012. doi: 10.1007/978-1-4614-4581-4.

[7]

A. N. Carvalho and M. J. D. Nascimento, Singularly non-autonomous semilinear parabolic problems with critical exponents and applications, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 449-471.  doi: 10.3934/dcdss.2009.2.449.

[8]

S. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems., Pacific J. Math., 136 (1989), 15-55. 

[9]

M. Conti, V. Pata and R. Temam, Attractors for processes on time-dependent spaces. Applications to wave equations, J. Differ. Equ., 255 (2013), 1254-1277.  doi: 10.1016/j.jde.2013.05.013.

[10]

F. Di Plinio, G. S. Duane and R. Temam, Time dependent attractor for the oscillon equation, Discrete Contin. Dyn. Syst., 29 (2011), 141-167.  doi: 10.3934/dcds.2011.29.141.

[11]

F. Di Plinio, G. S. Duane and R. Temam, The 3-dimensional oscillon equation, Boll. Unione Mat. Ital., 5 (2012), 19-53. 

[12]

J. K. Hale, Asymptotic Behavior of Dissipative System, American Mathematical Society, 1989. doi: 10.1090/surv/025.

[13]

T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96. 

[14]

M. J. D. Nascimento and F. D. M. Bezerra, Non-autonomous approximations governed by the fractional powers of damped wave operators, Electron. J. Differ. Equ., 2019 (2019), 1-19. 

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[16]

P. E. Sobolevskiĭ, Equations of parabolic type in a Banach space, Amer. Math. Soc. Trans., 49 (1966), 1-62. 

show all references

References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

F. D. M. Bezerra, A. N. Carvalho, J. W. Cholewa and M. J. D. Nascimento, Parabolic approximation of damped wave equations via fractional powers: fast growing nonlinearities and continuity of the dynamics, J. Math. Anal. Appl., 450 (2017), 377-405.  doi: 10.1016/j.jmaa.2017.01.024.

[3]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283.  doi: 10.1016/j.na.2010.11.032.

[4]

A. N. Carvalho and J. W. Cholewa, Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Aust. Math. Soc., 66 (2002), 443-463. doi: 10.1017/S0004972700040296.

[5]

A. N. Carvalho and J. W. Cholewa, Strongly damped wave equations in $W^{1,p}_0(\Omega) \times L^{p}(\Omega)$, Discrete Contin. Dyn. Syst., 2007 (2007), 230-239. 

[6]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences 182, Springer-Verlag, New York, 2012. doi: 10.1007/978-1-4614-4581-4.

[7]

A. N. Carvalho and M. J. D. Nascimento, Singularly non-autonomous semilinear parabolic problems with critical exponents and applications, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 449-471.  doi: 10.3934/dcdss.2009.2.449.

[8]

S. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems., Pacific J. Math., 136 (1989), 15-55. 

[9]

M. Conti, V. Pata and R. Temam, Attractors for processes on time-dependent spaces. Applications to wave equations, J. Differ. Equ., 255 (2013), 1254-1277.  doi: 10.1016/j.jde.2013.05.013.

[10]

F. Di Plinio, G. S. Duane and R. Temam, Time dependent attractor for the oscillon equation, Discrete Contin. Dyn. Syst., 29 (2011), 141-167.  doi: 10.3934/dcds.2011.29.141.

[11]

F. Di Plinio, G. S. Duane and R. Temam, The 3-dimensional oscillon equation, Boll. Unione Mat. Ital., 5 (2012), 19-53. 

[12]

J. K. Hale, Asymptotic Behavior of Dissipative System, American Mathematical Society, 1989. doi: 10.1090/surv/025.

[13]

T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96. 

[14]

M. J. D. Nascimento and F. D. M. Bezerra, Non-autonomous approximations governed by the fractional powers of damped wave operators, Electron. J. Differ. Equ., 2019 (2019), 1-19. 

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[16]

P. E. Sobolevskiĭ, Equations of parabolic type in a Banach space, Amer. Math. Soc. Trans., 49 (1966), 1-62. 

Figure 1.  Partial description of the fractional power spaces scale for $ \varLambda(t) , t\in{\mathbb{R}} $.
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