 Source-driven porous-medium similarity front

For the nonlinear diffusion equation $u_t=(u{u}_x{)}_x+u/(2t)$ 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 $(f{f}^{\prime }+\eta f/2{)}^{\prime }=0$. The boundary data make the integration constant zero. While $f>0$, division gives $f^{\prime }=-\eta /2$, hence $f=1-{\eta }^2/4$. At its first zero the diffusive flux $u{u}_x$ 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.