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Authors: ISLEM BAAZIZ, BENYATTOU BENABDERRAHMANE, SALAH DRABLA.
Received 17 December 2021; Accepted 10 April 2022.
pp. 383-397
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In a bounded domain, we consider a viscoelastic equation utt−Δu+∫t0g(t−τ)Δu(τ)dτ=|u|γu𝑢𝑡𝑡−Δ𝑢+∫0𝑡𝑔(𝑡−𝜏)Δ𝑢(𝜏)𝑑𝜏=|𝑢|𝛾𝑢 with a nonlinear feedback localized on a part of the boundary, where γ>0𝛾>0 and the relaxation function g𝑔 satisfied g′(t)≤ξ(t)gp(t), 1≤p<32,𝑔’(𝑡)≤𝜉(𝑡)𝑔𝑝(𝑡), 1≤𝑝<32, and certain initial data. We establish an explicit and general decay rate result, using some properties of the convex functions. Our new results substantially improve several earlier related results in the literature.
Mathematics Subject Classification (2010): 35L05, 35L70, 35L15, 93D20, 74D05.
Keywords: General decay, nonlinear source, viscoelastic, wave equation, relaxation function
