Optimal decay rates for the abstract viscoelastic equation

  • Muhammad I. MustafaEmail author


In this paper, we consider an abstract viscoelastic equation with minimal conditions on the \(L^{1}(0,\infty )\) relaxation function g namely \( g^{\prime }(t)\le -\xi (t)H(g(t))\), where H is an increasing and convex function near the origin and \(\xi \) is a nonincreasing function. With only these very general assumptions on the behavior of g at infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when \(H(s)=s^{p}\) and p covers the full admissible range [1, 2). We get the best decay rates expected under this level of generality and our new results substantially improve several earlier related results in the literature.


General decay Viscoelastic damping Relaxation function Convexity 

Mathematics Subject Classification

35B40 74D99 93D15 93D20 


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This work was supported by MASEP Research Group in the Research Institute of Sciences and Engineering at University of Sharjah.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SharjahSharjahUnited Arab Emirates

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