Optimal decay for second‐order abstract viscoelastic equation in Hilbert spaces with infinite memory and time delay

2020 ◽  
Vol 44 (2) ◽  
pp. 2071-2095
Author(s):  
Houria Chellaoua ◽  
Yamna Boukhatem
Keyword(s):  
Time Delay ◽  
Hilbert Spaces ◽  
Second Order ◽  
Infinite Memory ◽  
Optimal Decay ◽  
Viscoelastic Equation

Optimal decay of an abstract nonlinear viscoelastic equation in Hilbert spaces with delay term in the nonlinear internal damping

2020 ◽  
pp. 1-30
Author(s):  
Houria Chellaoua ◽  
Yamna Boukhatem ◽  
Baowei Feng
Keyword(s):  
Hilbert Spaces ◽  
Nonlinear Damping ◽  
Delayed Feedback ◽  
Internal Damping ◽  
General Decay Rate ◽  
Delay Term ◽  
Nonlinear Viscoelastic ◽  
Optimal Decay ◽  
Viscoelastic Equation ◽  
Nonlinear Internal Damping

In this paper, we consider a second-order abstract viscoelastic equation in Hilbert spaces with delay term in the nonlinear internal damping and a nonlinear source term. Under some suitable assumptions on the weight of the delayed feedback, the weight of the non-delayed feedback and the behavior of the relaxation function, we establish two explicit and general decay rate results of the energy by introducing a suitable Lyaponov functional and some properties of the convex functions. Moreover, we give some applications and examples. This work generalizes the previous results without time delay term to those with delay in the nonlinear damping.

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