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Longtime dynamics for a nonlinear viscoelastic equation with time-dependent memory kernel. (English) Zbl 1479.35118

Summary: This paper investigates the well-posedness, the existence and the regularity of the time-dependent global attractor for a viscoelastic equation in \(\Omega \subset \mathbb{R}^3\): \[ | \partial_t u |^\rho \partial_{t t} u - \partial_{t t} \Delta u - h_t ( 0 ) \Delta u - \int_0^\infty \partial_s h_t ( s ) \Delta u ( t - s ) \operatorname{d} s + f ( u ) = h\] with time-dependent memory kernel which is used to model aging phenomena of the material. By using the novel theory framework recently developed in literature [M. Conti et al., J. Differ. Equations 264, No. 7, 4235–4259 (2018; Zbl 1391.35282); Am. J. Math. 140, No. 2, 349–389 (2018; Zbl 1480.45010)] and establishing some delicate integration estimates along the trajectory of the solutions in the time-dependent phase space, we show that when \(\rho \in ( 1 , 4 ]\), the growth exponent \(p\) of \(f ( u )\) is up to the critical range \(1 \leq p \leq 5\), and the time-dependent memory kernel satisfies the same conditions as in [loc. cit.], the model is well-posed. Especially, when \(\rho \in ( 1 , 4 )\) and \(1 \leq p < 5\), the related process has an invariant time-dependent global attractor which has optimal regularity.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L72 Second-order quasilinear hyperbolic equations
35R09 Integro-partial differential equations
74D10 Nonlinear constitutive equations for materials with memory
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