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