 Barenblatt solution

A Barenblatt solution is a mass-preserving source-type similarity solution of the porous medium equation. For the one-dimensional normalizationthe nonnegative profile isIts support is $|x|\le R(t)$ with $R(t)=\sqrt{6\beta A}{t}^{1/3}$, and its peak is $A{t}^{-1/3}$. Substituting the quadratic profile verifies the partial differential equation wherever it is positive. At the edge, $u{u}_x=0$, so extending by zero gives a weak solution without a singular interface flux.

The total mass determines $A$:It is independent of time. As $t\downarrow 0$, the support shrinks and the measures $u(x,t)dx$ converge to $M{\delta }_0$, a point source. This is a nonlinear counterpart of the heat kernel, with a finite moving front instead of positive Gaussian tails. The normalization matters: $\beta (u{u}_x{)}_x=(\beta /2)(u^2{)}_{xx}$.