Comparison of solutions of Hamilton Jacobi Equationfor different initial value

Question

Let $u_i$ for $i=1,2$ be the solutions of Hamilton Jacobi equation

$u_t+f(u_x)=0 ; (x,t) \in R\times (0,\infty)$

$u(x,0)=g_i(x)$ ; $x \in R$ $i=1,2$

(which are obtained by Hopf-Lax formula)

Where $g_i \in L^ \infty $ for $i=1,2$

if $g_1 \leq g_2$ how to show $u_1 \leq u_2$

Answer 1

Hint: write explicitly the two solutions with the Hopf-Lax formula, and compare the arguments of the two "min".