Article Fisher–KPP equation with Robin boundary conditions on the re...

A solution of an equation is stable: if the data are changed a little, the corresponding solution changes only a little.

This question is about Theorem 1.1 and its proof. I almost understand why, if i is odd, the solution of the Fisher-KPP equation is stable by its proof with a triangle inequality. We can get limsup|u_1(x,t；phi_{A}(x)-u_2(x,t;phi_{A}(x)|=0. However, I don't understand why if i is even, then the solution is unstable. Why does this problem matter with even or odd?

Article Entire solutions of time periodic Fisher–KPP equation on the half line

Can we also show that the Fisher-KPP equation is stable with the Dirichlet condition by the triangle inequality, by Theorem 1.1 in this article?

If so, then I infer that the Fisher-KPP equation with Neumann condition is also stable for V_i(x), with i is an odd integer, but unstable with i is an even integer because the Dirichlet condition+Neumann condition =Robin condition. Is this result worth proving?

Here is a sketch of proof:

Since a positive periodic solution is close to the positive stationary solutions in our model, if the solution is stable. We have

limsup|V(t,x)-V_i(x)|=0.

We also have

limsup|u(t,x)-V(t,x)|=0

and

limsup|u_(x,t；phi_{A}(x)-V_i(x)|=0 for i is an odd integer.

By triangle inequality,

we have limsup|u(t,x)-u_(x,t；phi_{A}(x)|=0. Then we prove it as desired.