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 $W^{2,p}$.

notgiven by a convolution (although this is true if $\Omega=R^n$). $\endgroup$ – Delio Mugnolo Dec 12 '12 at 21:24