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Let $F:\mathbb R^3\to \mathbb R^3$ and $G,H:\mathbb R^3\to\mathbb R$ be some given smooth maps. In order for the PDE $F\cdot \nabla u+Gu+H=0$ to admit a global solution $u:\mathbb R^3\to\mathbb R$, what are the restrictions needed on $F,G,H$?

Typically this type of quasilinear PDE can be solved by method of characteristics. But here we have no boundary condition so we gain some freedom in finding solutions.

The method of characteristic reduces the problem to solving a family of ODEs. To obtain a global solution of $u$, we need all the ODE to have a global solution, and these solutions should patch together smoothly. What condition on $F,G,H$ do we need to achieve this?

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    $\begingroup$ If $F, G, H$ are given and smooth, then all the ODEs will have global solutions. It seems to me that any issue would be only coming from patching. $\endgroup$ Mar 24, 2020 at 4:49

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