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

摘要

We present a systematic study of local solutions of the ODE of the form x '' = 1/1 integral(t, x, x') near t = 0. Scuh 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 gamma > 0 and gamma < 0, where gamma =partial derivative integral/partial derivative 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. (C) 2008 Elsevier Inc. All rights reserved.