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.
Best Answer
Some kind of a maximum principle for the Neumann problem, as well as some comparison result for rather general nonlinear equations and even the third (Robin) boundary condition are proved in Chapter 2 of the book by A. Friedman, "Partial Differential Equations of Parabolic Type", Prentice-Hall, 1964.