Consider the partial differential equation with boundary conditions \begin{equation} \frac{\partial u}{\partial t} = \varepsilon \bigg( (u+1)\frac{\partial u}{\partial x}\bigg), \qquad \frac{\partial u}{\partial x}=0 \quad \text{at} \quad x=0,1 \tag{1} \end{equation} for $t\in[0, \infty)$, and where $\varepsilon\rightarrow 0$.
Let's suppose I want to find an asymptotic solution of (1). To leading order $\frac{\partial u}{\partial t}=0$, so that $u$ just takes on the form of the initial conditions, which can't be right. Alternatively, one could try scaling $t=\tau/\epsilon$ where $\tau =\mathcal{O}(1)$. This would remove $\varepsilon$ from (1) but wouldn't leave an equation that's easier to solve than the original.
My questions: (1) How would you approach finding an asymptotic solution to this problem? (2) if the correct approach is to introduce the coordinate $\tau =\mathcal{O}(1)$, then why? Surely this coordinate just implies that you need to scale time to be large, which isn't intuitive to me. (3) If anyone knows of a solution to equation (1) then please feel free to provide it. Thanks
Define $u_\varepsilon$ by $u(x,t) = u_\varepsilon(x,\varepsilon t)$. Then $\partial_t u = \varepsilon \partial_t u_\varepsilon$ and $\partial_x u = \partial_x u_\varepsilon$. Therefore $$\partial_t u_\varepsilon = (u_\varepsilon + 1)\partial_x u_\varepsilon.$$ The function $u_\varepsilon$ clearly independent of $\varepsilon$, so let's write $w = u_\varepsilon$. Then $$\partial_t w = (w + 1)\partial_x w,$$ and $$u(x,t) = w(x,\varepsilon t) \approx w(x,0) + w_t(x,0)\varepsilon t = u(x,0) + (u(x,0)+1)u_x(x,0) \varepsilon t.$$ This may be the kind of expansion you're looking for?