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Fractional oscillon equations; solvability and connection with classical oscillon equations. (English) Zbl 1483.35024

Summary: 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 A. N. Carvalho et al. [Attractors for infinite-dimensional non-autonomous dynamical systems. Berlin: Springer (2013; Zbl 1263.37002)], and F. Di Plinio et al. [Discrete Contin. Dyn. Syst. 29, No. 1, 141–167 (2011; Zbl 1223.37100)].

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
34A08 Fractional ordinary differential equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
35R11 Fractional partial differential equations
47D06 One-parameter semigroups and linear evolution equations
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References:

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