Hello everyone. I'm about to use a comparison principle that I belive is true, but I can't find any precise reference to be sure of it. Here is what I would like to say :
I have a parabolic equation $u_t - Lu = 0$ in $(t,x,y) \in \mathbb R \times \mathbb R \times \omega$ where $\omega$ is a bounded domain.

Note : L has a $0$-order term with unknown sign, but I would like to compare with $0$ so this is not a problem.

I have a function $U$ smooth enough (that is $\mathcal C^1$ in time, $\mathcal C^2$ in space in the whole closed cylinder $\mathbb R\times \overline\omega$) that verifies :

$U_t - LU \geqslant 0$

$\partial_\nu U = 0$ on $\partial\omega$ (so $\geqslant 0$...)

$U(0,x,y) \geqslant 0$

I would like to conclude $U \geqslant 0$.

But every maximum principle I know "works with Dirichlet conditions" : $U$ must also be non-negative at any time on the boundary of $\mathbb R\times \omega$.
In *Smoller, Shock Waves and Reaction-Diffusion Equations* p.93 I found a principle comparison that is almost exactly what I want (far more general in fact, because it tolerates nonlinearities, and works with mixed conditions), but with $L$ as follows :

$Lu = \sum_{i,j = 1}^n (a_{ij}(x,t)u_{x_i})_{x_j})$

what is somehow restrictive : in my case, the equation is an advection-reaction-diffusion one, of the type $u_t - \Delta u + q(x,y)\cdot\nabla u -k(t,x,y)u = 0$ that is not like the one above.

So my question is : does anyone knows about such a comparison principle ? Does the one in Smoller's book extend to more general operators ?

Thank you very much.