A nonlocal singular parabolic problem modelling gravitational interaction of particles
This interesting paper is devoted to the existence and nonexistence of solutions (both stationary and evolution) for the parabolic-elliptic system containing the equations \(u_t= \nabla\cdot(\nabla u+ u\nabla\varphi)\) and \(\Delta(\varphi- V)= u\). Here \(u(x,t)\geq 0\) is the density of a cloud of particles, and the potential \(\varphi(x,t)\) depends on a given external potential \(V(x)\) and (in a nonlocal way) on the distribution of the particles. Thus the particles create the gravitational potential \(\varphi\) and move directed by the gradient of \(-\varphi\). This system describes a cloud of self-interacting particles surrounding a fixed mass. Also similar systems resembling the above stated can be interpreted as models for gravitational interaction of particles, including the evolution version of the Chandrasekhar equation for the gravitational equilibrium of polytropic stars. Existence and uniqueness as well as nonexistence of solutions (steady states and nonstationary case) for the radially symmetric problem with Hölder-continuous initial data is considered. Another interesting result is the existence of local-in-time and global-in-time solutions under sharp conditions. The existence of finite-time blow-up of large solutions in star-shaped domains is shown as well.
finite-time blow-up of large solutions in star-shaped domains
scientific article; zbMATH DE number 1449013