In the last topic we consider a pointwise estimate for homogeneous heat equation. The conclusion is the following: if the initial data is -bounded then the solution decays as . Today, we consider another phenomena. Let assume be an open set with smooth boundary and suppose is a solution of
on the boundary . Assume that
Proof. The method used here is very standard when we deal with energy estimate or if we want to estimate -norm of the solution.
Multiplying the equation by and then integrating the resulting equation on , we can get
The Poincare inequality gives
where is constant. For any , by the Young inequality, it holds that
Taking we obtain
Solving this differential inequality, we have
Theorem ( estimates).
It is not difficult to verify that
This completes the proof.
Note that in the above proof, we use a result which is similar to the Gronwall inequality. Clearly, the statement is as follows.
Lemma. If the function satisfies
we then have
The proof of this statement is quite simple. We first try to write
which gives, after integrating with respect to ,
The proof follows.