Convergence rate to the singular solution for the Gelfand equation and its stability

Abstract

We study the asymptotic behaviors of the solution to the Gelfand equation. The Gelfand equation appears in the kinetic theory of gravitational steady state and the theory of nonlinear diffusion. We present a convergence rate of the solutions of the Gelfand equation to the unique singular solution as r r goes to infinity and prove asymptotic stability of the solution by considering the initial value problem for the Gelfand equation. To obtain the convergence rate and the point-wise stability estimate, we construct a uniform lower bound function and use the solution for the linearized Gelfand equation.