A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities

¹ IECL, Université de Lorraine and CNRS (UMR 7502), Délégation CNRS at LJLL UMR 7598, 57045 Metz, France.
fatiha.alabau@univ-lorraine.fr
² School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, P.R. China.
wzq@fudan.edu.cn
³ School of Mathematics and Information Sciences, Yantai University, Yantai 264005, P.R. China.
fdylx01@sina.com.cn

Received: 17 March 2015
Accepted: 18 February 2016

Abstract

In this paper, we consider the energy decay of a damped hyperbolic system of wave-wave type which is coupled through the velocities. We are interested in the asymptotic properties of the solutions of this system in the case of indirect nonlinear damping, i.e. when only one equation is directly damped by a nonlinear damping. We prove that the total energy of the whole system decays as fast as the damped single equation. Moreover, we give a one-step general explicit decay formula for arbitrary nonlinearity. Our results shows that the damping properties are fully transferred from the damped equation to the undamped one by the coupling in velocities, different from the case of couplings through displacements as shown in [F. Alabau, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015–1020; F. Alabau, P. Cannarsa and V. Komornik, J. Evol. Equ. 2 (2002) 127–150; F. Alabau, SIAM J. Control Optim. 41 (2002) 511–541; F. Alabau-Boussouira and M. Léautaud, ESAIM: COCV 18 (2012) 548–582] for the linear damping case, and in [F. Alabau-Boussouira, NoDEA 14 (2007) 643–669] for the nonlinear damping case. The proofs of our results are based on multiplier techniques, weighted nonlinear integral inequalities and the optimal-weight convexity method of [F. Alabau-Boussouira, Appl. Math. Optim. 51 (2005) 61–105; F. Alabau-Boussouira, J. Differ. Equ. 248 (2010) 1473–1517].