The object of this paper is the regularity of solutions to the Hamilton-Jacobi equation \[ u_t(x,t)-\text{Tr}(a(x,t)D^2u(x,t))+H(x,t,Du(x,t))=0 \quad\text{in }\mathbb R^N\times(0,T), \]
where \(H\) and \(a\) satisfy the following hypotheses:
i) there are real numbers \(q>2\), \(\delta>1\) and \(\eta_{\pm}\geq 0\) such that
\[ \frac{1}{\delta}|z|^q-\eta_-\leq H(x,t,z)\leq \delta |z|^q+\eta_+, \quad\forall (x,t,z)\in \mathbb R^N\times(0,T)\times \mathbb R^N; \]
ii) \(a=\sigma\sigma^*\) for some locally Lipschitz-continuous map \(\sigma:(x,t) \rightarrow \sigma(x,t)\), with values in the \(\mathbb N\times D\) real matrices \((D\geq 1)\) such that \(\|\sigma(x,t)|\leq\delta\) for all \((x,t)\in\mathbb R^N\times(0,T)\). Note that no initial condition is needed for analysis, nor convexity of \(H\) in \(Du\). For a given viscosity solution, the authors are interested by uniform continuity estimates in \((x,t)\) that do not depend on the smoothness of coefficients. The main result of the paper is the Hölder continuity of solutions to first-order equations
\[ u_t+H(x,t,Du)=0\quad\text{in }\mathbb R^N\times(0,T), \]
and to second-order problems.
The results apply to degenerate parabolic equations and require superlinear growth at infinity, in the gradient variables, of the Hamiltonian. Proofs are based on comparison arguments and representation formulas for viscosity solutions.