Solution of equations describing thin layer flow of heavy viscous liquid on a curvilinear surface (Q1389336)

The authors consider the system of differential equations \[\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = A(x) + \frac{\partial^2 u}{\partial y^2},\quad \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0\], \[\frac{\partial u}{\partial y} = 0, \quad v = \frac{\partial h}{\partial t} + u \frac{\partial h}{\partial x}\text{ for }y=h(t,x), \quad v=0\text{ for }y = 0\] which describes an non-stationary flow of a thin layer of heavy viscous liquid on a fixed impenetrable surface. The problem is reduced to the Cauchy problem for equations with a smaller number of variables.