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Positive solutions for Choquard equation in exterior domains
Fractional oscillon equations; solvability and connection with classical oscillon equations
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 |
$ u_{tt}-\mu(t)\Delta u+\omega(t)u_t = f(u),\ x\in\Omega,\ t\in{\mathbb{R}}, $ |
$ \partial \Omega $ |
$ \Omega $ |
$ {\mathbb{R}}^N $ |
$ N\geq 3 $ |
$ \omega $ |
$ \mu $ |
$ f $ |
$ \omega $ |
$ \mu $ |
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.
|

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