Let $u$ solve:
$$ u_t + u u_x = 0 $$ Prove that if the initial condition $u_0(x)$ is compactly supported, then $u$ is compactly supported at any future times.
I'm not sure how to prove this. I was going to use the fact that $u$ is present in the pde and perhaps leverage that fact to prove compact support for future times. Would I have to do this using the method of characteristics?
Suppose that $u_0(x)$ vanishes outside the interval $[x_L,x_R] $. The exact implicit solution to $u_t+uu_x=0$ is $u=u_0(x-ut)$ implying that $u=0$ if $(x-ut)$ is not in $[x_L,x_R]$. Sice $u$ is finite, this gives finite upper and lower bounds for any given $t$