Consider the problem

(1)

in an open, convex, bounded domain with the boundary condition

on

and intial data .

We assume that .

By maximum principle, we know that if is a classical solution to (1) then for all .

Sometimes we need to divide by . This leads to the a difficulty that can be zero some where. The following lemma gives us a very useful result to avoid zero-value of .

**Lemma. (Moving planes)**

Assume that is a solution to (1). There is a compact subdomain of and a positive constant such that

for all .

Moreover, if is a increasing function, then

for all .

*Proof: It’s too long. May be given in another post.*

**Remark: **By maximum principle, can be zero only on . This implies that

.

Hence, we can divide in the domain .

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## About baotangquoc

Lecturer
School of Applied Mathematics and Informatics
Hanoi University of Science and Technology
No 1, Dai Co Viet Street, Hanoi