Unique continuation property for the Rosenau equation

Ponente(s): Ricardo Córdoba Gómez, Anyi Daniela Corredor

In this work we establish a unique continuation result for the the high order nonlinear partial differential equation, \begin{equation}\label{BLe} u_{tt}+a u_{xxxx}+b u_{xxxxtt}-\gamma u_{xx}= \big(f(u)\big)_{xx}, \end{equation} that describes the dynamics of dense discrete systems with high order effects, where $a > 0,$ $b > 0,$ and $\gamma > 0$ are constants, $f(u) =-\beta |u|^p u$ with $\beta > 0$ and $p > 0$. More precisely, when $ f(u)=-\beta u^{2k+1},$ $k\in\mathbb N,$ using a Carleman type estimate, we show that if $u = u(x, t)$ is a solution of the model \eqref{BLe} with $$ u \in L^{2}\left(-T,T; H^{6}_{loc}(\mathbb R)\right), \quad u_{t}\in L^{2}\left(-T,T; H^{2}_{loc}(\mathbb R)\right), $$ and $u$ vanishes on an open subset $\Omega$ of $\mathbb R \times [-T,T],$ then $u\equiv 0$ in the horizontal component of $\Omega.$ We recall that the horizontal component $\Omega_1$ of an open subset $\Omega\subseteq \mathbb R\times\mathbb R$ is defined as the union of all segments $t = constant$ in $\mathbb R\times\mathbb R$ which contain a point of $\Omega,$ this is, $$ \Omega_1= \bigl\{(x,t)\in \mathbb R \times [-T,T] \ : \ \exists x_1\in \mathbb R, \ (x_1,t)\in \Omega \bigr\}. $$