Let v E C1(R) and consider the equation

dt(u) + dx(uv) = 0, x E R, t>0,

u(x,0) = u0(x), x E R,

We assume that v satisfies v(x) > 0, ví (x) > 0 for all x E R.

Prove that sup|u(x,t)| < sup|u0(x)| for all t > 0