Why does $\int_{\Omega}\Delta u \frac{\partial u}{\partial t}dx=-\frac{1}{2}\frac{d }{d t}|u|^2_{H^1_0}$ hold?

Question

In his book, Roger Temam, considered the following reaction diffusion system: He considered the norms:

What I don't get is what's written here How does he obtain the equality:

It seems that to have this equality using Green, we should have $\frac{\partial u}{\partial t}=0$ on the boundary of $\Omega$.

Answer 1

You are right, $\frac{\partial u}{\partial t}=0$ on $\partial\Omega$, since (see eq. (1.3)) $$\forall x\in \partial\Omega, \forall t\in\mathbb{R}^+\ \ u(x,t)=0.$$ Therefore in these $x$, $u$ is independent of $t$ and the partial derivative satisfies $$\forall x\in \partial \Omega,\ \frac{\partial u}{\partial t}(x,\cdot)=0.$$