In Cannarsa-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 wishes 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 anyone 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 Cannarsa-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....