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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.

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may I ask if you solved this problem and indeed what field do you work in? –  lost1 Dec 3 '13 at 1:00

2 Answers 2

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.

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Thank you. I did not really found what I wanted, but by reading one of his proofs, I think I understood a little more what's going on. If I understand correctly, it states that the max cannot be reached at a point where there is Neumann condition, since at this point, the normal derivative should be > 0, so the maximum on the frontier is the maximum on the other part of the frontier. What I have a hard time dealing with now, is the 0-order term. I believe everything falls down if it is arbitrary, (except if the sup is 0 - which is kind of what I want to show !), doesn't it ? –  Laurent Mar 20 '12 at 17:21

In Gary M. Lieberman, Second Order Parabolic Partial Differential Equations p.13 I found a comparison principle that gives exactly what I want for a large class of boundary operators, but needs a parabolic operator with non-positive $0$-order term. I'm starting to think that maybe there is an error in the paper I'm reading !

On the other hand, Lieberman says just after, that the theorem still holds with the $0$-order term only bounded from above, but with stronger hypotheses on the parabolic boundary and the boundary operator. Still, mine is not really pretty since it is of type $a\cdot\nabla u + bu$ with vector field $a$ being either $-\vec n$ or $0$ and scalar function being either $-1$ or $0$, depending on the part of the parabolic boundary.

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