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Consider the diffusive Hamilton-Jacobi equation $$u_t-\Delta u=|\nablau|^p+h(x)\ \ \text{ in } \Omega\times(0,T)$$ with Dirichlet conditions, whicharises in stochastic control problems as well as in KPZ type models. We studythe question of the gradient blowup rate for classical solutions with $p>2$.  We first consider the case of time-increasing solutions. For such solutions,the precise rate was obtained by Guo and Hu (2008) in one space dimension, butthe higher dimensional case has remained an open question (except for radiallysymmetric solutions in a ball). Here, we partially answer this question byestablishing the optimal estimate $$C_1(T-t)^{-1/(p-2)}\leq \|\nablau(t)\|_{\infty} \leq C_2(T-t)^{-1/(p-2)} \tag{1}$$ for time-increasing gradientblowup solutions in any convex, smooth bounded domain $\Omega$ with $2<p<3$. Wealso cover the case of (nonradial) solutions in a ball for $p=3$. Moreover weobtain the almost sharp rate in general (nonconvex) domains for $2<p\le 3$. Theproofs rely on suitable auxiliary functionals, combined with the following, newBernstein-type gradient estimate with sharp constant: $$|\nabla u|\led_\Omega^{-1/(p-1)}\bigl(d_p+C d_\Omega^\alpha\bigr)  \ \ \text{ in } \Omega\times(0,T),\qquad d_p=(p-1)^{-1/(p-1)},  \tag{2}$$ where $d_\Omega$ is the function distance to the boundary. Thisclose connection between the temporal and spatial estimates (1) and (2) seemsto be a completely new observation.  Next, for any $p>2$, we show that more singular rates may occur for solutionswhich are $\textit{not}$ time-increasing. Namely, for a suitable class ofsolutions in one space-dimension, we prove the lower estimate$\|u_x(t)\|_\infty \geq C(T-t)^{-2/(p-2)}$.

Gradient blow-up rates and sharp gradient estimates for diffusive Hamilton-Jacobi equations