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Polynomial decay to a class of abstract coupled system with past history. (English) Zbl 1274.35024

The authors study an initial value problem for a class of coupled systems of the form \[ u_{tt}+ A_1 u-\int _0^{\infty }g(s)A_2 u(t-s) ds+\beta v=0, \]
\[ v_{tt} +Bv+\beta u=0, \] where \(A_i, B\) are self-adjoint operators on a Hilbert space satisfying \(A_2=f(A_1),\;B=h(A_1)\) with \(f(s)=o(s^{\alpha }),\;h(s)=o(s^{\gamma })\) as \(s\to \infty \). The function \(g\) is assumed to be a nonnegative integrable function satisfying \(g'(s)\leq a g(s)\), \(s\in \mathbb {R}^{+}\). Using the relative history of \(u\), the authors rewrite the system in the abstract form \[ \frac {\text{d}}{\text{d}t}U(t)=\mathcal {A} U(t), \] and show that the semigroup associated with the system is not exponentially stable when \(\gamma >0\) and \(\alpha <1.\) It is shown that solutions decay polynomially to zero in an appropriate norm, with rates that can be improved by taking more regular initial data. Some examples conclude the paper.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
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