SELF-SIMILAR SOLUTIONS TO A NONLINEAR PARABOLIC-ELLIPTIC SYSTEM

Abstract

We study the forward self-similar solutions to a parabolic-elliptic system $$ u_t = \Delta u - \nabla \cdot (u\nabla v),\quad 0 = \Delta v + u $$ in the whole space $\bf R^2$. First it is proved that self-similar solutions $(u, v)$ must be radially symmetric about the origin. Then the structure of the set of self-similar solutions is investigated. As a consequence, it is shown that there exists a self-similar solution $(u, v)$ if and only if $\|u\|_{L^1({\bf R}^2)} \lt 8\pi$, and that the profile function of $u$ forms a delta function singularity as $\|u\|_{L^1({\bf R}^2)} \to 8\pi$.