need to address a branching diffusion equation with a nonlinearity that forces $\phi$ to go to zero over a distance of order 1 at the right of the point at which $\phi(t,x)=1$, and which therefore acts as a tip cutoff. In terms of a smooth equation, we may write, for example, \_t (t,x)=\_x\^2(t,x)+ \[eq:tipcutoffsmooth\] and study the properties of the solutions to this equation in the region $\phi\gg 1$.

In both cases, the equation can be linearized in the respective domain of interest, and one gets \_t (t,x)=\_x\^2(t,x)+(t,x). We shall assume that the nonlinear term is equivalent to a right-moving absorptive boundary at the accuracy at which we want to address the problem. (This assumption was better motivated by Ebert and Van Saarloos in their discussion of what they call the “interior expansion” [@EVS:2000]). In the first case, we study the function $\phi$ to the right of the boundary; in the second case, we study the function to the left.

#### Solution to the linearized equation with an appropriate boundary condition.

Near the boundary, at large times, the function $\phi$ reads (t,x)\~(+)e\^[-]{}, \[eq:phiinterior\] where $\xi=x-\text{[position of the cutoff]}\sim x-2t+\frac32\ln t+\dots$ and this is valid for $1\ll \xi\ll \sqrt{t}$. According to 