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).

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**Note**: I didn’t give the Cauchy data for this problem, we had assumed all the initial data were given.

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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

.

Hence,

Thanks to the uniform-continuity.

Which is,

equivalent to:

By Gronwall lemma,

Where .

Here

,

.

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Here I have to make some comment: the initial data, we say with pertubation , thus the integral need another term , which doesn’t matter much on this problem.