I'm studying a specific reaction-diffusion equation, and to prove well-posedness I need to find a convex set such that the vector field points inwards at the border.
The vector field is: $ F(u_1,u_2,u_3)=(\alpha(u_{3}^{\gamma}-u_{1}^{\alpha} u_{2}^{\beta}),\beta(u_{3}^{\gamma}-u_{1}^{\alpha} u_{2}^{\beta}), \gamma(-u_{3}^{\gamma}+u_{1}^{\alpha} u_{2}^{\beta}))$, with $u_1,u_2,u_3,\alpha,\beta, \gamma>0$
What are the techniques or approaches to find convex sets for this type of vector field? Any source is appreciated.
Thanks.
You basically need to find a set such that the vector field $F$ is the inward pointing normal vector. To achieve this let $G$ be the function $G(x,y,z)=\langle -F(x,y,z), (x,y,z) \rangle$. I don't want to do the algebra but it should work out that $DG_w$, the differential of $G$ at $w$, has full rank and each $w$ such that $G(w)=0$ (you need to show this). If that is the case then you can use the submersion theorem to state that the set $G^{-1}(0)$ is a manifold and, moreover, the inward $F$ is by definition the inward pointing vector to that manifold. If they didn't ask to prove all that then you can probably get away by just saying the set $G^{-1}(0)$ is the one you want. Convexity can be shown just by saying that if $G(w)=0=G(z)$ then linear combinations of w and z are also zero after applying G to it.