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: $\beta (u{u}_x{)}_x=(\beta /2)(u^2{)}_{xx}$.

For the planar porous medium equation $n_t=[D_0/(3n_0^2)]\Delta (n^3)$, the mass-$Q$ point-source similarity solution is $n=n_0{\lambda }^{-2}\sqrt{[1-r^2/(r_0^2{\lambda }^2){]}_{+}}$, with $\lambda =(6D_{0}t/r_0^2{)}^{1/6}$ and $r_0^2=3Q/(2\pi n_0)$. Its front grows as $t^{1/6}$ and its central density falls as $t^{-1/3}$. The mass conservation integral uses the planar area element $2\pi rdr$. Although the density slope diverges at the front, its diffusion flux vanishes there, giving a compactly supported weak solution with initial measure $Q{\delta }^{(2)}$.

For $n,A>0$, the one-dimensional porous medium equation $h_t=A(h^n{h}_x{)}_x$ has a volume-preserving similarity solution $h=[c_n(R^2-x^2)/(At){]}_{+}^{1/n}$, where $c_n=n/[2(n+2)]$ and $R\propto t^{1/(n+2)}$. On a reflecting half-line with conserved area $M$, $R=[M(At/c_n{)}^{1/n}/I_n{]}^{n/(n+2)}$, where $I_n=\frac{1}{2}B(\frac{1}{2},1+1/n)$. For a whole symmetric release use half its total area as $M$. The Beta function fixes normalization, while finite-front zero volume flux fixes the edge condition.

The source-type similarity solution of $h_t=D(h^2{h}_x{)}_x$ with whole-line conserved area $\mathcal{V}$ is $h=H(t)[1-x^2/L(t{)}^2{]}_{+}^{1/2}$, where $L=2(\mathcal{V}/\pi {)}^{1/2}(Dt{)}^{1/4}$ and $H=(\mathcal{V}/\pi {)}^{1/2}(Dt{)}^{-1/4}$. Its volume flux vanishes at the finite front, while the velocity tends to $\dot{L}=L/(4t)$. A reflecting half-line uses twice its prescribed area in these formulas.

For the radially symmetric equationin two dimensions, the mass-$N$ similarity solution has the compactly supported formwhere mass normalization gives $x_0=(16/\pi {)}^{1/4}$. Its support radius grows as $t^{1/4}$ and its occupied area grows as $t^{1/2}$.