The conservative model of a dissipative dynamical system

Let $R_\sigma$ be the response operator of a dissipative dynamical system (DS) governed by the equation $u_{tt}+\sigma u_t-u_{xx}=0$, $x>0$, where $\sigma=\sigma(x)\ge0$. Let $R_q$ be the response operator of a conservative DS governed by the equation $u_{tt}-u_{xx}+q(x)u=0$, $x>0$, where $q=q(x)$ is real. We demonstrate that for any dissipative DS there exists a unique conservative DS (the “model”) such that $R_\sigma=R_q$ is valid. Bibl. 10 titles.