How can we show that a smooth solution of the problem $\begin{cases} u_t +uu_x = 0 \\ u(x, 0) = \cos(\pi x) \end{cases}$ satisfies the equation $u = \cos \pi(x − ut)$ and that $u$ ceases to exist (as a single-valued continuous function) when $t = 1/\pi$? The only thing I can think of is maybe graphically doing it, but I don't see how.
Can someonpe please edit Robert's answer below for the situtation at hand? I made a mistake before typing it. Thanks