Uniform energy decay rates of hyperbolic equations with nonlinear boundary and interior dissipation

We consider the problem of uniform stabilization of nonlinear hyperbolic equations, epitomized by the following three canonical dynamics: (1) the wave equation in the natural state space L2(Ω) × H(Ω), under nonlinear (and non-local) boundary dissipation in the Dirichlet B.C., as well as nonlinear internal damping; (2) a corresponding Kirchhoff equation in the natural state space [H(Ω) ∩H1 0 (Ω)]×H1 0 (Ω), under nonlinear boundary dissipation in the ‘moment’ B.C. as well as nonlinear internal damping; (3) the system of dynamic elasticity corresponding to (1). All three dynamics possess a strong, hard-to-show ‘boundary → boundary’ regularity property, which was proved, also by invoking a micro-local argument, in Lasiecka and Triggiani (2004, 2008). This is by no means a general property of hyperbolic or hyperbolic-like dynamics (Lasiecka and Triggiani, 2003, 2008). The present paper, as a continuation of Lasiecka and Triggiani (2008), seeks to take advantage of this strong regularity property in the case of those PDE dynamics where it holds true. Thus, under the above boundary → boundary regularity, as well as exact controllability of the corresponding linear model, uniform stabilization of nonlinear models is obtained under minimal nonlinear assumptions, provided that a corresponding unique continuation property holds true. The treatment of the present paper is cast in the abstract setting (Lasiecka, 1989, 2001; Lasiecka and Triggiani, 2000, Ch. 7, 2003, 2008), which is proper for these hyperbolic dynamics and recovers the results of Lasiecka and Triggiani (2003, 2008) in the absence of the nonlinear interior damping, in particular in the linear case.