Consider a Hamilton-Jacobi equation: $$u_{t} + f(u_{x}) = 0 \quad (x,t) \in \mathbb{R}\times [0,+\infty)$$ with two possible initial conditions $u(x,0) = g_{i}(x)$ for $x \in \mathbb{R}$ and $i=1,2$. The question is pretty much straightforward: if $g_{1} \le g_{2}$, do we have $u_{1} \le u_{2}$?
When $f$ is convex, I know the answer is yes and the proof is an easy consequence of the Hopf-Lax formula. However, I would like to know conditions for this to hold when $f$ is not convex. I am particularly interested in the case $f(x) = -\frac{1}{2}x^{2}$.
I did a little research before coming here, but all references I know give special attention to Hamilton-Jacobi equations in which $f$ is convex and satisfy $\lim_{p\to\infty}f(p)/p \to +\infty$. The only source I found which differs from this setting is this paper, which states that the Hopf-Lax formula holds when either $f$ or the initial condition is convex, an information I found nowhere else. In this case, if both $g_{1}$ and $g_{2}$ are convex and satisfy $g_{1} \le g_{2}$, its seems that the previous argument using the Hopf-Lax formula applies here and one would have $u_{1} \le u_{2}$ indeed.
