We discuss the problem of asymptotic stabilization of the hyper\-elastic-rod wave equation on the real line \begin{equation*} \partial_t u-\partial_{txx}^3 u+3u \partial_x u= \gamma\left(2\partial_x u\, \partial_{xx}^2 u+u\, \partial_{xxx}^3 u\right),\quad t > 0,\>\>x\in \mathbb{R}. \end{equation*} We consider the equation with an additional forcing term of the form $ f:H^1(\mathbb{R})\to 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 $u_0\in H^1(\mathbb{R})$. Any such solution decays exponentially to 0 as $t\to\infty$.
On the asymptotic stabilization of a generalized hyperelastic-rod wave equation
COCLITE, Giuseppe Maria
2014-01-01
Abstract
We discuss the problem of asymptotic stabilization of the hyper\-elastic-rod wave equation on the real line \begin{equation*} \partial_t u-\partial_{txx}^3 u+3u \partial_x u= \gamma\left(2\partial_x u\, \partial_{xx}^2 u+u\, \partial_{xxx}^3 u\right),\quad t > 0,\>\>x\in \mathbb{R}. \end{equation*} We consider the equation with an additional forcing term of the form $ f:H^1(\mathbb{R})\to 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 $u_0\in H^1(\mathbb{R})$. Any such solution decays exponentially to 0 as $t\to\infty$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
