Theory of unique existence of the solution for one model problem of mechanics

Abstract

The work is aimed at proving the unique existence of the problem $u_t^i + (H_{u^i})x = \nu u_{xx}^i$, $\nu > 0$ on the segment $[0,l]$ with initial and boundary conditions of the following type: $u_t^i(0,t) = u^i(l,t) = 0$, $u^i(x,0) = \varphi^i(x)$.