Serge Nicaise

TL;DR: This paper considers the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition and proves exponential stability of the solution under suitable assumptions.

TL;DR: In this paper, the authors investigated time harmonic Maxwell equations in heterogeneous media, where the permeability μ and the permittivity e are piecewise constant and the associated boundary value problem can be interpreted as a transmission problem.

TL;DR: In this paper, the authors considered the wave equation in a bounded region with a smooth boundary with distributed delay on the boundary or into the domain, and proved the expo- nential stability of the solution.

Abstract: Exponential stability analysis via Lyapunov method is extended to the one-dimensional heat and wave equations with time-varying delay in the boundary conditions. The delay function is admitted to be time-varying with an a priori given upper bound on its derivative, which is less than $1$. Sufficient and explicit conditions are derived that guarantee the exponential stability. Moreover the decay rate can be explicitly computed if the data are given.

TL;DR: In this paper, the authors studied transmission problems for elliptic operators of order 2m with general boundary and interface conditions, introducing new covering conditions, which allowed them to prove solvability, regularity and asymptotics of solutions in weighted Sobolev spaces.