The initial trace of a solution of the porous medium equation

Abstract:

Let $u = u(x,t)$ be a continuous weak solution of the porous medium equation in ${{\mathbf {R}}^d} \times (0,T)$ for some $T > 0$. We show that corresponding to $u$ there is a unique nonnegative Borel measure $\rho$ on ${{\mathbf {R}}^d}$ which is the initial trace of $u$. Moreover, we show that the initial trace $\rho$ must belong to a certain growth class. Roughly speaking, this growth restriction shows that there are no solutions of the porous medium equation whose pressure grows, on average, more rapidly then $|x{|^2}$ as $|x| \to \infty$.