If $f:\mathbb{R}^n\rightarrow\mathbb{R}_+$ is a **nonnegative real analytic** function and $g:\mathbb{R}^n\rightarrow\mathbb{R}$ is a **strongly convex smooth** function with a **surjective** gradient $\nabla g:\mathbb{R}^n\rightarrow\mathbb{R}^n$. Is it possible that the solution $x:\mathbb{R}_+\rightarrow\mathbb{R}^n$ to
$$\dot{x}=-\nabla f(x), \quad x(0)=x_0$$
is bounded for all $x_0\in\mathbb{R}^n$, i.e., $\|x(t)\|\leq c(x_0)$ for all $t\in\mathbb{R}_+$ for some constant $c(x_0)>0$ (assume $c(x_0)$ is smooth in $x_0$), but the solution to the Hessian Riemannian gradient flow
$$\dot{x}=-\left(\nabla^2g(x)\right)^{-1}\nabla f(x), \quad x(0)=x_0,$$
where $\nabla^2g$ denotes the Hessian of $g$, is unbounded for some $x_0$.

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