Dissipative boundary conditions for nonlinear 1-D hyperbolic systems: sharp conditions through an approach via time-delay systems

2014/03/07 by Coron, Jean-Michel, Nguyen, Hoai-Minh
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1403.1747

Abstract

We analyse dissipative boundary conditions for nonlinear hyperbolic systems in one space dimension. We show that a previous known sufficient condition for exponential stability with respect to the C¹-norm is optimal. In particular a known weaker sufficient condition for exponential stability with respect to the H²-norm is not sufficient for the exponential stability with respect to the C¹-norm. Hence, due to the nonlinearity, even in the case of classical solutions, the exponential stability depends strongly on the norm considered. We also give a new sufficient condition for the exponential stability with respect to the W^2,p-norm. The methods used are inspired from the theory of the linear time-delay systems and incorporate the characteristic method.

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