We discuss the problem of asymptotic stabilization of the hyper\-elastic-rod wave equation on the real line \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}. We consider the equation with an additional force 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 [F. Ancona and G. M. Coclite, 2015] 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 esponentially to 0 as t\to\infty.
On the asymptotic stabilization of the hyperelastic-rod wave equation
ANCONA, FABIO;
2014
Abstract
We discuss the problem of asymptotic stabilization of the hyper\-elastic-rod wave equation on the real line \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}. We consider the equation with an additional force 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 [F. Ancona and G. M. Coclite, 2015] 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 esponentially to 0 as t\to\infty.Pubblicazioni consigliate
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