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Optimal decay rates for the viscoelastic wave equation
Author(s) -
Mustafa Muhammad I.
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4604
Subject(s) - mathematics , viscoelasticity , function (biology) , range (aeronautics) , infinity , exponential function , relaxation (psychology) , polynomial , convex function , mathematical analysis , generality , exponential growth , regular polygon , exponential decay , wave equation , energy (signal processing) , statistics , thermodynamics , geometry , physics , materials science , psychology , social psychology , evolutionary biology , nuclear physics , composite material , psychotherapist , biology
In this paper, we consider a viscoelastic equation with minimal conditions on the L 1 ( 0 , ∞ ) relaxation function g , namely, g ′ ( t ) ≤ − ξ ( t ) H ( g ( t ) ) , where H is an increasing and convex function near the origin and ξ 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.