  As a profound example of spontaneous motion, we analyze the motion of a
camphor particle on a water surface. The motion is modeled as an
initial-boundary value problem for a coupled nonlinear system of a diffusion
equation and an ordinary differential equation in a two-dimensional domain.
Since it seems that the well-posedness of this initial boundary value problem
is missing, we provided its proof. Then, by constructing an approximate
solution to this initial boundary value problem, we gave a mathematically
rigorous interpretation of a camphor motion. That is we showed that the motion
of camphor locally in time has a self-avoiding orbit. We also gave the
numerical performance of the approximate solution.
