Elastic membrane equation with dynamic boundary conditions and infinite memory

Ahlem Merah,Fatiha Mesloub

doi:10.5269/bspm.47621

Ahlem Merah, Fatiha Mesloub

Open Access

https://doi.org/10.5269/bspm.47621

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Abstract

In this paper, we study the elastic membrane equation with dynamic boundary conditions, source term and a nonlinear weak damping localized on a part of the boundary and past history. Under some appropriate assumptions on the relaxation function the general decay for the energy have been established using the perturbed Lyapunov functionals and some properties of convex functions.

Highlights

C3 |s| ≤ |h (s)| ≤ C4 |s|, if |s| > 1 the authors first proved the global existence of solutions, and obtained the energy decays exponentially if p = 1 and decays polynomially if p > 1.The results were generalized by Cavalcanti et al.[9]
They obtained the same results without imposing a growth condition on h and under a weaker assumption on g Messaoudi and Mustafa [26] extended these results and established an explicit and general decay rate result by exploiting some properties of convex functions
Ferhat and Hakem [18] considered a weak viscoelastic wave equation with dynamic boundary conditions and Kelvin Voigt damping and delay term acting on the boundary in a bounded domain, and proved the asymptotic behavior by making use an appropriate Lyapunov functional

Summary

The phase space His defined by

For the relaxation function g, we assume: (A1) g(t) : R+ ֒→ R+ is a nonincreasing C1 function satisfying g(0) > 0 and ξ0 − g(s)ds = l > 0. There exists an increasing strictly convex function G : R+ ֒→ R+ of class C1 (R+) ∩ C2 (R+). (A2) h : R ֒→ R is a nondecreasing C0 function such that there exists a strictly increasing function h0 ∈ C1 (R+) with h0 (0) = 0 and positive constants c1, c2 and ε such that h0 (|s|) ≤ |h(s)| ≤ h−0 1 (|s|) ; if |s| ≤ ε c1 |s| ≤ |h(s)| ≤ c2 |s|

Moreover we suppose that the function

Note that

Elastic Membrane Equation

The same arguments give us