On self-similar solutions of the equation of motion of a glacier in one-dimensional approximation (Q1915688)

The dynamics of the border of a one-dimensional glacier is described by the equation \[ {\partial u\over \partial t}= {\partial Q\over \partial x}+ f, \] where \(t> 0\), \(0< x< x_0(t)\), \(Q= \text{sgn}(\partial u/\partial x) u^2(u|\partial u/\partial x|)^n\), \(u\geq 0\) with initial condition \(u|_{t= 0}= u_0(x)\), and the boundary conditions \(Q|_{x= 0}= q_0(t)\), \(Q|_{x= x_0}= q_1(t)\). Here \(t\) is time, \(x\) is the longitudinal horizontal coordinate, \(x_0\) is the coordinate of the moving boundary of the glacier, i.e., \(u(x_0(t), t)\equiv 0\), \(f\) is the mass budget of the ice (we assume below that \(f= 0\)), \(u_0(x)\) is the initial shape of the surface, and \(q_0(t)\) and \(q_1(t)\) are sinks for the ice mass on the boundary of the glacier \(x= 0\) and \(x= x_0\). We consider solutions that are monotone decreasing with respect to \(x\), nonnegative and with compact support. We obtain necessary conditions on the parameter \(\lambda\) for the existence of self-similar solutions of the form \[ u(x, t)= t^{1/(1+ \lambda(n+ 1))} \varphi(\eta),\quad \eta= xt^{- (2+ \lambda)/(1+ \lambda(n+ 1))}. \]