Optimal energy decay result for nonlinear abstract viscoelastic dissipative systems. (English) Zbl 1464.35036
Summary: In this paper, we consider the nonlinear abstract equation
\[
u_{tt}+Au-\int_0^tg(t-s)Au(s)\mathrm{d}s+h(u_t)=j(u)
\]
subject to a competing effect of viscoelastic and frictional dampings. With very general assumptions on the behavior of \(g\) at infinity and the behavior of \(h\) near 0, we establish explicit and optimal energy decay result. To the best of our knowledge, this is the first time we have such combination of generality and optimality in one explicit formula for the energy decay rates of this system.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 35L90 | Abstract hyperbolic equations |
| 35R09 | Integro-partial differential equations |
| 74D10 | Nonlinear constitutive equations for materials with memory |
| 93D15 | Stabilization of systems by feedback |
| 93D20 | Asymptotic stability in control theory |
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