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

Abstract

We present a systematic study of local solutions of the ODE of the form x ″ = 1 t f ( t , x , x ′ ) near t = 0 . Such ODEs occur in the study of self-similar radial solutions of some second order PDEs. A general theorem of existence and uniqueness is established. It is shown that there is a dichotomy between the cases γ > 0 and γ < 0 , where γ = ∂ f / ∂ x ′ at t = 0 . As an application, we study the singular behavior of self-similar radial solutions of a nonlinear wave equation with superlinear damping near an incoming light cone. A smoothing effect is observed as the incoming waves are focused at the tip of the cone.