Blow-up and global solutions to L^p norm preserving non-local flows

In this paper, we study global existence and blow up properties to $L^p$ norm preserving non-local heat flows. We first study two kinds of $L^p$ norm preserving non-local flows and prove that these flows have the global solutions. Finally, we give a example to show that one kind of this heat flow may blow up in $L^{\mathrm{\infty }}$ norm though its $L^p$ norm is preserved.