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

with

on the boundary . Assume that

then

.

*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

which implies

.

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 satisfieswe then have

.

The proof of this statement is quite simple. We first try to write

.

Clearly

which gives, after integrating with respect to ,

.

Thus

.

Hence

.

The proof follows.

Rất hay. Cảm ơn bạn nhiều.

Bạn có thể đưa thêm bài ứng dụng của bất đẳng thức Gronwall vào cũng rất hay.

Comment by kakro — June 10, 2010 @ 10:08

doi chieu lai voi kqua dpcm thi khac nhau dau tru cua e^

http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cpsi+%28x%2Ct%29+%5Cleqslant+%7Be%5E%7B+-+%5Cint_0%5Et+%7B%5Calpha+%28%5Ctau+%29%7D+d%5Ctau+%7D%7D%5Cpsi+%28x%2C0%29+%2B+%5Cint_0%5Et+%7B%5Cbeta+%28%5Ctau+%29%7Be%5E%7B%5Cint_%5Ctau+%5Et+%7B%5Calpha+%28s%29%7D+ds%7D%7Dd%5Ctau+%7D&bg=ffffff&fg=333333&s=0

Comment by toan — September 20, 2012 @ 17:05

Xin bạn làm rõ thêm được ko ? mình thấy khó hiểu chỗ suy ra kqua dpcm, kqua của ta là tích phân từ 0 -t của e^{ – tich phân từ \tau đến t ..} mà ??? xin giải thích rõ làm sao để có dấu trừ của e^ , vì đang là e^ tích phân từ 0 đến s mà ….

Comment by toan — September 20, 2012 @ 17:30

Ở dòng nào thế bạn?

Comment by Ngô Quốc Anh — September 20, 2012 @ 17:33