## 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

### Citations:

Zbl 1263.37002; Zbl 1223.37100
Full Text:

### References:

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