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
Proof. First, we try to estimate .
Now, it is sufficient to prove that
We now have to show that
Using Hölder’s inequality and , we have
This completes the proof.