Consider the following PDE's system (KS-system)

$$u_t=\Delta u -\nabla \cdot (u\nabla c)$$ $$c_t=\Delta c +n-c$$ $$\frac{\partial n}{\partial \nu}=\frac{\partial c}{\partial \nu}=0 \ \ noflux-condition$$ $$M1>u(x,0)=u_0>0$$ $$M2>c(x,0)=c_0>0$$

where $u=u(x,t)$, $c=c(x,t)$, $(x,t)\in \Omega\times[0,T]$, $\Omega \subset \mathbb{R}^2$ is bounded domain with smooth boundary, this is a classic keller Segel model where $u$ denotes the cell density and $c$ represents the concentration of an attractive cue and the initial data are positive and bounded by $M1$ and $M2$ respectively.

It is a great interest that $u$, $c$ solutions are positive functions, in several articles it is common for the authors to assure that positivity is thanks to the maximum principle or comparison , but for me it is not clear and I would like to understand how the argument works.

I will try to show you that I know, I am studying the principle of maximum in parabolic equations, my reference is this book [1], hence for me it is clear that if we assume $u,c$ classic solution of KS-system with $u>0$ in the second equation then we obtain that $c> 0$ by the principle of the maximum. Now let's rewrite the first equation as

$$u_t=\Delta u -\nabla u \cdot \nabla c - \Delta c u$$

And now this is my concern because if we want to apply theorem 7 (page 182) of [1] we need to remove the restriction on $h$ for us on $\Delta c$

I thank you in advance for your time and help, as always excuse my English


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