I have a PDE in the form $$u_t=F(u,u_x)$$ where the unknown is $u(x,t)$ on say $\mathbb R\times[0,\infty)$. $F$ is very nonlinear so I was told to assume self-similarity in the form $$u(x,t)=t\tilde u(x)$$ to reduce it to an ODE. I have convinced myself based on the dimensions of the variables involved that this is the appropriate scaling.
My confusion is how then to deal with the boundary conditions at $t=0$. I'm interested in having some particular Dirichlet conditions there, but the scaling assumption seems to force $u(x,0)=0$ for all $x$. I guess this would be because the equation is only approximately self-similar, and then it somehow ``forgets'' about the boundary condition over time? Is there any way around this?