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| We consider viscoelastic nonlinear plate equation with memory kernel quantified by the following differential inequality:
$ g' + H(g) \leq 0$ where $ H(s)$ is a given continuous, positive, increasing, and convex function such that H(0) = 0. We shall show that the energy of the nonlinear PDE
is driven by the same inequality.
The results presented provide an uniform framework for obtaining optimal decay rates for the energy of nonlinear mechanical systems which contain memory effects.
The study of PDE with a memory will be reduced to solving an appropriate nonlinear ODE systems.
The method is based on the idea introduced in Lasiecka and Tataru in 1993
for determining decay rates of the energy given in terms of the function H(s).
This new method allows to optimize the benefits (both geometric and topological) secured by a combination of dissipative mechanisms generated by both frictional and
memory damping. The obtained results are quantitative and allow for optimal design of materials arising in nonlinear mechanical structures. |
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