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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 $u_1 \leq u_2$

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

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  • $\begingroup$ oh thanks!! i got it.. $\endgroup$ – Rosy Dec 11 '17 at 5:43

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