The Initial Trace of a Solution of the Porous Medium Equation

Let <!-- MATH $u = u(x,t)$ --> be a continuous weak solution of the porous medium equation in <!-- MATH ${{\mathbf{R}}^d} \times (0,T)$ --> for some 0$">. We show that corresponding to there is a unique nonnegative Borel measure on <!-- MATH ${{\mathbf{R}}^d}$ --> which is the initial trace of . Moreover, we show that the initial trace 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 as <!-- MATH $|x| \to \infty$ --> .