The effects connected with coincidence of velocities in the two-velocities dynamical system

M. I. Belishev ; A. V. Zurov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 29, Tome 264 (2000), pp. 44-65

Résumé

The paper deals with the system \begin{align*} \rho u_{tt}-u_{xx}+Vu=0,\quad x>0,\quad t>0;\\ |_{t=0}=u_t|_{t=0}=0;\\ |_{x=0} = f, \end{align*} where $\rho=\rho(x)$ and $V=V(x)$ are $2\times2$-matrix functions; $\rho=\operatorname{diag}\{\rho_1,\rho_2\},\rho_{\alpha}>0$; $f$ is a boundary control; $u=u(x,t)$ is the solution. The singularities of the fundamental solution corresponding to the controls $\binom{\delta}0$ and $\binom0{\delta}$ ($\delta=\delta(t)$ is the Dirac $\delta$-function) are under investigation. In the case of $\rho_1(x)\ne\rho_2(x)$ the singularities of the fundamental solution are described in terms of the standard scale $\delta,\int\delta, \iint\delta,\ldots$. In the presence of points $x=x_*:\rho_1(x_*)=\rho_2(x_*)$ an interesting effect occurs: the singularities of intermediate (fractional) orders appear.