I'm working on some PDE problems and my biggest issue is vector calculus facts.
Let $u \in C^2(\Omega)$, where $\Omega$ is some bounded subset of $R^2$ with smooth boundary such that
$u_{t}-\Delta u=0$ in $\Omega$ and $t>0$,
$v\cdot\nabla u=u_{v}=0$ on $\partial\Omega$ and $t>0$.
I'd like to show $\frac{d}{dt}\int_{\Omega}u^2dx\leq 0$. Now I wish to do integration by parts on $\int_{\Omega}u\Delta udx$. Let $*$ denote multiplication and $\cdot$ dot product for clarity.Am I correct in saying that since $\Delta u=\nabla\cdot\nabla u$, we have
$\frac{d}{dt}\int_{\Omega}u^2dx=\int_{\Omega}2u*u_tdx=2\int_{\Omega}u\Delta udx=2\int_{\Omega}u*(\nabla\cdot\nabla u)dx=-2\int_{\Omega}\nabla u\cdot \nabla u dx+2\int_{\partial\Omega}u*(\nabla u\cdot v)dS=-2\int_{\Omega}\nabla u\cdot \nabla udx\leq 0$, since $\nabla u\cdot \nabla u=u_{x_1}^2+u_{x_2}^2\geq 0$ for all $x=(x_1,x_2)$. Is this correct? Just a bit confused with the differences of $\nabla\cdot\nabla u =\Delta u$ and $\nabla u\cdot \nabla u=u_{x_1}^2+u_{x_2}^2$.