Abstract
This paper describes a polynomial decay rate of the solution of the Kirchhoff type in viscoelasticity with logarithmic nonlinearity, where an asymptotically-stable result of the global solution is obtained taking into account that the kernel is not necessarily decreasing.
Highlights
Consider the quasilinear viscoelastic problem:|ut |ρ utt + M (t) ∆2 u (t) + ∆2 utt (t) −Z t h (t − s) ∆2 u (s) ds = u |u|γ−2 ln |u|k, in Ω × (0, ∞),∂u ( x, t) = 0 in ∂Ω × (0, ∞), ∂v (2)u ( x, 0) = u0, ut ( x, 0) = u1 for x ∈ Ω (3)M (t) := ξ 0 + ξ 1 k∆u (t)k L2 (Ω), β ≥ 1, (4) u ( x, t) = with:
Our decay rate obtained in the third section is less general than that obtained in [7,8], where a common decay rate outcome was established in order to let the functions of the relaxation satisfy: g0 (t) ≤ − H ( g(t)), t ≥ 0, H (0) = 0 (5)
We follow up with the same steps of the previous result in [7,10] for a new class of Kirchhoff hyperbolic equations on bounded domain polynomial decay of the Kirchhoff type in viscoelasticity combined with the right-hand side defined as a logarithmic nonlinearity, and the kernel is not necessarily decreasing taking into account that the similar conditions in the last ones in ([7,10]) are considered
Summary
The mathematicians had an exponential decay outcome as time tended to infinity h0 (t) + γh(t) ≥ 0 for all t ≥ 0 provided that (h0 (t) + γh(t)) eαt ∈ L1 (0, ∞) for some α > 0 Their proof depended on another technique, the so-called “Lyapunov functional”. In this present work, we follow up with the same steps of the previous result in [7,10] for a new class of Kirchhoff hyperbolic equations on bounded domain polynomial decay of the Kirchhoff type in viscoelasticity combined with the right-hand side defined as a logarithmic nonlinearity, and the kernel is not necessarily decreasing taking into account that the similar conditions in the last ones in ([7,10]) are considered.
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