A singular initial value problem and self-similar solutions of a nonlinear dissipative wave equation
- Liang, Jianfeng
DOIAbstract
AbstractWe present a systematic study of local solutions of the ODE of the form x″=1tf(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
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