Self-similar solutions for a class of non-divergence form equations

Abstract

In this paper, we study the self-similar solutions for a non-divergence form equation of the form $$u(x, t)=(t + 1)^{-\alpha}f((t + 1)^{\beta}|x|^2).$$ We first establish the existence and uniqueness of solutions f with compact supports, which implies that the self-similar solution is shrink. On the basis of this, we also establish the convergent rates of these solutions on the boundary of the supports. On the other hands, we also consider the convergent speeds of solutions, and compare which with Dirac function as t tends to infinity.