For Cauchy Problem, is a bounded interval, and ,
where , I wonder the stability for this problem. Thus I looked up the definition for the stability in the sense of Liapunov(zero stable).
Note: I didn’t give the Cauchy data for this problem, we had assumed all the initial data were given.
Definition: The stability means:
for any perturbation satisfying
with sufficient small to ensure that the solution to the perturbed problem does exist., then
independent of such that
Claim 1: If our force term is uniformly continuous, then we can assure the stability(Liapunov).
Proof: Let , we have
Thanks to the uniform-continuity.