I starting to learn Partial differential equation and I got the following question:
u=u(x,y) is a solution for cauchy problem:
$u_t+u_x-u=0$,
$u|_{t=0}= \begin{cases} 0,&x\leq 0\\\\ x/\epsilon,&0\leq x\leq \epsilon\\\\ 1, &\epsilon \leq x \end{cases} $
calcultate: $\lim_{\epsilon\to 0}\int_{-\infty}^\infty u_xxdx$
what is the part of $u_t$ in the equation? it should not be a function of only x and y?
I don't really know how to treat it.