Exponential decay for waves with indefinite memory dissipation. (Q1686828)

The authors consider the equation \[ u_{tt}-u_{xx}+\partial_x[a(x)+\int^t_0 g(t-s)u_x(x,s) ds]=0 \] in \((0,L]\times(0,\infty)\) under Dirichlet boundary conditions and with initial data \(u(0)\in H_0^1\) and \(u_t(0)\in L^2(0,L).\) Assuming \(\int_0^La(x)dx>0\) and that \(a'\), \(a''\) are uniformly bounded and that the kernel \(g\) is small and its two first derivatives decay exponentially, and that the bound on the derivatives of \(a\) is small compared to \(g\) the authors prove that the norms of \(u(t)\) and \(u_t(t)\) in respectively \(H_0^1\) and \(L^2\) decay exponentially. The proof is by classical resolvent theory for Volterra equations, by fixed point theory and by a result of \textit{G. Menz} [J. Differ. Equations 242, No. 1, 171--191 (2007; Zbl 1143.35007)] on exponential stability of certain semigroups.