A singular initial value problem and self-similar solutions of a nonlinear dissipative wave equation (Q2378218)

Local existence, uniqueness and regularity of solutions of the initial value problem to the equation \(x''=\frac{1}{t}f(t,x,x')\) near \(t=0\) are proved. The results are generalized to higher order ordinary differential equations. As an application, the singular behavior of self-similar radial solutions of the equation \(w_{tt}-\Delta w+\left| w_t\right| ^pw_t=0\) (\(p>0\)) near an incoming light cone is investigated.