On one nonlocal problem with free boundary (Q2759333)

The author considers the equation \( \frac{\partial^2 u}{\partial x^2} + \frac{n-1}{x} \frac{\partial u}{\partial x} - g(u) \frac{\partial u} {\partial t}=0\), \(x,t\in{\mathbb R}_+^1\), \(n=1,2,3.\) The function \( g(u)\), \(u>0\) is positive, continuous, decreasing and has integrable singularity at the point \(u=0.\) By means of the functional \( f(u) = \int_0^u g(u) du \) and the invertible functional \( u = f^{-1}(u) \) the original equation is reduced to a quasilinear parabolic equation. The reduced problem describes the wave which extends on disturbed background \( v = 0 \) with unknown front of wave \(x=s(t).\) The reduced problem is investigated by group analysis methods. There are found automodel solutions to the reduced problem and automodel solution to the original problem.