In Cannarsi-Sinestrari's book 'Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control' there is a proof, via the Method of Characteristics, of global-in-time existence of classical solutions to the Cauchy problem
$$u_t +H(Du)=0 \quad \text{in}\quad \mathbb{R}^n\times(0,\infty),\\ u = g \quad \text{on}\quad \mathbb{R}^n\times\{t=0\}$$ under the assumption that both H and g are $C^2$ and convex.
As far as I can see the proof is flawed: It uses a previous theorem where $Dg$ is assumed bounded, without re-stating that assumption (and $g$ convex with $Dg$ bounded is a bit restrictive). In the proof of that theorem, the assumption is used to prove properness of the map one wish to prove is a diffeomorphism (via the Hadamard-Caccioppoli Theorem; note, in Lions's book, 'Generalized Solutions of Hamilton-Jacobi Equations', which contains a proof of the same theorem above, the necessity of properness is ignored a couple of times).

Question: Does anyway know a reference to a correct proof only under the assumption that $H$ and $g$ are $C^2$ and convex? (If one adds the assumption (??) that the map $x + DH(Dg(x))$ is proper (which holds if $Dg$ is bounded), then the Cannarsi-Sinestrari proof is correct).

Of course, the example $H(p)=p^2, g(x)=x^2$ shows that boundedness of $Dg$ is not needed, and I expect the result is true.

NB Let me know if I am asking this in the wrong forum....