Consider the heat equation $u\_t=\Delta u$ with Neumann boundary condition in a bounded domain $\Omega$.


Is this true to say:


$$\|u(. , t)-v(. , t)\|\_p\leq \|u(. , 0)-v(. , 0)\|\_p$$ where $u$ and $v$ are two solutions of the heat equation in $W^{2,p}$.



Heat equation with Neumann BC
yes, with some regularity on the boundary. 


Theorem 3.2.9 p. 90 of E.B. Davies book, Heat Kernels and Spectral Theory
gives Gaussian bounds for the heat kernel of an elliptic operator with Neumann boundary 
conditions. These bounds imply that the heat flow preserves L^p. 


