LpL^p-asymptotic stability analysis of a 1D wave equation with a boundary nonmonotone damping

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation with a nonlinear non-monotone damping acting at a boundary. The study is performed in an $L^p$-functional framework, $p\in [1,\infty]$. Some well-posedness results are provided together with exponential decay to zero of trajectories, with an estimation of the decay rate. The well-posedness results rely mainly on some results collected in [7]. Asymptotic behavior results are obtained by the use of a suitable Lyapunov functional if $p$ is finite and on a trajectory-based analysis if $p=\infty$

The convergence problem for dissipative autonomous systems: classical methods and recent advances

The initial motivation of this text was to provide an up to date translation of the monograph [45] written in french by the first author, taking account of more recent developments of infinite dimensional dynamics based on the {\L}ojasiewicz gradient inequality. In order to keep the present work within modest size bounds and to make it available to the readers without too much delay, we decided to make a first volume entirely dedicated to the so-called convergence problem for autonomous systems of dissipative type. We hope that this volume will help the interested reader to make the connection between the rather simple background developed in the french monograph and the rather technical specialized literature on the convergence problem which grew up rather fast in the recent years