I would like to know if there is any kind of Gronwall inequality for a smooth function $u \colon \mathbb{R}^n \to \mathbb{R}$ satisfying $$ |\nabla u | \le K u, $$ where $K$ is a constant.
2026-02-23 06:55:01.1771829701
Gronwall-type inequality in higher dimension
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Well, fix $x\in\mathbb{R}^n$ and consider the one-dimensional function $g(t)=u(tx)$. Then $g'(t)=\nabla u(tx)\cdot x$ and so $$|g'(t)|\le Kg(t)|x|.$$ Now you can apply Gronwall to $g$. You will get $K|x|$ in the exponential.