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.


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