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Fractional oscillon equations; solvability and connection with classical oscillon equations

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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
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  • 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].

    Mathematics Subject Classification: Primary: 37B55, 35B40, 35B41; Secondary: 34A08, 35L71.

    Citation:

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

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    [2] F. D. M. BezerraA. N. CarvalhoJ. 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.
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    [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.
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    [10] F. Di PlinioG. 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.
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