A weak solution for a point mass camphor motion. (Q2200751)

The authors consider a model system of equations consisting of diffusion equation \[\partial_t u=D\Delta u-\alpha u+\sum_{j=1}^Jf(x,x_c^j(t);\epsilon),\qquad (t,x)\in \Omega_T\] coupled with a system of nonlinear ordinary differential equations with respect to \(x_c^j(t)\), which describes the selfpropelled motion of point mass objects driven by camphor. Here \(x_c^j(t)\) denote the positions of center of objects at time \(t\) and \(\Omega\subset\mathbb{R}^2\) is a bounded domain with \(C^2\) boundary. Since the objects have masses, their motion becomes very complicated when some of the objects hit the boundary of a water surface or collide each other. To avoid such complexity the authors consider point mass objects obtaining an existence of a weak solution of this model system by giving an a priori estimate for the solution. The key to this estimate is the choice of a special test function and the Tichonov's fixed point theorem.