Reminder

1
Reminder
2
Fundamental Property
In general let us call a steady state function
(in case of IBV(1)-(3) solution of the problem
(4)-(5)) an equilibrium if it attract solutions
for certain class of Initial data It extremely
important to notice that long-time behavior not
necessary should be associate to steady state
part of the dynamic equation but may be
periodic, A periodic, or even chaotic. Any
classified long time regime we will refer as
attracting if they are able dominate
In
In this case solution of the corresponding IBP
can be presented in the form
3
Solution of the problem (12)-(14)
4
Answer, in a sense yes
5
Back to KPP-Fisher
6
Equation for perturbation
Then F around equilibrium 1 can be extended as
As a result from (19), (20), and assumption (A)
about F
(21)-(23) is a linearized IBV around an
equilibrium with respect to KPP
Using method of separation variable, one can
easily obtain (see next slide)
7
Solution of Linearized KPP around equilibrium 1
Linear stability analyzes around equilibrium 0
lead as to study a behavior of the IBV
8
Home work (a) Check that 1 is a stable and 0 is
unstable equilibrium for classical KPP (
F(u)u(1-u) ) (b) Perform a stability analyzes
for KPP with a mixed boundary conditions (8)