Energy decay for solutions of the wave equation with general memory boundary conditions

Stability for semilinear wave equation of variable coefficients with acoustic boundary conditions and general boundary memory feedback

“…). We would like to highlight [10], where a general borelian measure is involved, the authors recovered and extended some of the results from the literature.…”

“…Inspired by Cornilleau et al [10], we define the energy of the solution (1.1) by the following formula:…”

“…In the context of singularities, using Proposition 3 of [10], the following Rellich inequality is useful.…”

“…where µ 0 is some positive constant and µ is a Borel measure on IR + . When ρ = 0 and A(x) = I, the problem of proving uniform decay rates for wave equations with boundary dissipations but without acoustic boundary conditions has attracted a lot of attention in recent years, see [1,2,7,10,21,28,29,30]. The problem (1.1) covers the case of a problem with memory type as studied in the references [1,2,7,28], when the measure µ is given by µ(s) = k(s)ds, where ds stands for Lebesgue measure and k is a nonnegative kernel.…”

“…[10]) Let µ be a Borel positive measure on IR + and µ 0 > 0. The following properties are equivalent:(1) ∃β > 0 such that +∞ 0 e βs dµ(s) < µ 0 .…”