Uniform continuity of heat equation with L 1 dataI came across the following remark in...

aggierabz2006zw

aggierabz2006zw

Answered

2022-07-07

Uniform continuity of heat equation with L 1 data
I came across the following remark in my reading.
Remark. If the initial data is in L 1 , then heat equation solutions approach 0 within the sup-norm. That is, if K(x,t) is the heat kernel and f L 1 ( R n ), then
lim t R n K ( x y , t ) f ( y ) d y = 0 ,
uniformly.
I can picture this working if f is uniformly continuous in R n and lim x f ( x ) = 0. Does f L 1 ( R n ) imply this somehow, and I'm not seeing this? I'm relatively new to L p -spaces.
If so, then for ϵ > 0, we can find some radius r ϵ > 0 such that | | f ( y ) | < ϵ / 2.
Then we can bound the heat kernel, and use the fact that, K ( x y , t ) d y = 1 to conclude, more or less.

Answer & Explanation

Alisa Jacobs

Alisa Jacobs

Expert

2022-07-08Added 13 answers

This is quite straightforward from the definition. In n dimensions, the heat kernel is given by
K ( x y , t ) = 1 ( 4 π t ) n / 2 e | x y | 2 / 4 t .
As e x 1 for x 0 this implies that
| K ( x y , t ) | ( 4 π t ) n / 2 and therefore
| R n K ( x y , t ) f ( y ) d y | ( 4 π t ) n / 2 R n | f ( y ) | d y ,
which converges to 0 as t uniformly in x, as soon as f is in L 1 ( R n ).

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