= Barenblatt solution
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A Barenblatt solution is a mass-preserving source-type <similarity solution> of the <porous medium equation>. For the one-dimensional normalization
$$
u_t=\beta(uu_x)_x,\qquad \beta>0,
$$
the nonnegative profile is
$$
u(x,t)=\left[A t^{-1/3}-\frac{x^2}{6\beta t}\right]_+,\qquad A>0,\quad t>0.
$$
Its support is $|x|\leq R(t)$ with $R(t)=\sqrt{6\beta A}\,t^{1/3}$, and its peak is $At^{-1/3}$. Substituting the quadratic profile verifies the <partial differential equation> wherever it is positive. At the edge, $uu_x=0$, so extending by zero gives a <weak solution> without a singular interface flux.
The total <mass> determines $A$:
$$
M=\int_{-R(t)}^{R(t)}u(x,t)\,dx=\frac43A\sqrt{6\beta A}.
$$
It is independent of time. As $t\downarrow0$, 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: $\beta(uu_x)_x=(\beta/2)(u^2)_{xx}$.
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