On the asymptotic stabilization of a generalized hyperelastic-rod wave equation

We discuss the problem of asymptotic stabilization of the hyper-elastic-rod wave equation on the real line egin{equation*} partial_t u-partial_{txx}^3 u+3u partial_x u= gammaleft(2partial_x u, partial_{xx}^2 u+u, partial_{xxx}^3 u ight),quad t > 0,>>xin mathbb{R}. end{equation*} We consider the equation with an additional forcing term of the form $ f:H^1(mathbb{R}) o H^{-1}(mathbb{R}),, f[u]=-lambda(u-partial_{xx}^2 u),$ % for some $lambda>0$. We resume the results of cite{AC} on the existence of a semigroup of global weak dissipative solutions of the corresponding closed-loop system defined for every initial data $u_0in H^1(mathbb{R})$. Any such solution decays exponentially to 0 as $t oinfty$.