= Source-driven porous-medium similarity front
{title2=$u(x,t)=[1-x^2/(4t)]_+$}
For the nonlinear <diffusion equation> $u_t=(uu_x)_x+u/(2t)$ on $x>0,t>0$, the fixed boundary value $u(0,t)=1$ and zero slope at $x=0$ admit the displayed compact-support <similarity solution>, with front $x_F=2\sqrt t$. Substitution $u=f(x/\sqrt t)$ reduces the equation to $(ff'+\eta f/2)'=0$. The boundary data make the integration constant zero. While $f>0$, division gives $f'=-\eta/2$, hence $f=1-\eta^2/4$. At its first zero the diffusive flux $uu_x$ vanishes, so extension by zero creates no singular flux term. The solution is smooth inside its positive region and is a <weak solution> at the moving front.
Back to article page
