Today we give an inequality, which is very useful in proving the convergence of a solution to PDEs tends to the stationary solution as time tends to infinity.

**Lemma. **Let be a domain and let satisfy and . Furthermore, let satisfy

for all and some , where is the characteristic function on . Finally, let

.

Then

**Proof. **First, we try to estimate .

(since )

Now, it is sufficient to prove that

or equivalently

We have

(since ).

We now have to show that

.

Using Hölder’s inequality and , we have

Thus,

.

This completes the proof.

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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