Given: Heat equation: $(1)\begin{cases}\frac{\partial u}{\partial t }-\frac{\partial^2 u}{\partial x^2}=f\quad pp. \text{in}\quad \Omega\times]0,T[\\ u=0\quad pp. \text{in}\quad \partial\Omega\times]0,T[\\ u(x,0)=u_0(x)\quad pp. \text{in}\quad \Omega \end{cases}$
Letting $u$ the solution of $(1)$ such that is a regulier in $\Omega\times]0,T[$ , I want to prove the following equality $(2)$ which is called "energy equality" of $(1)$: $$\frac12\int_{\Omega} u(x,t)^2+\int_{0}^{t}\int_{\Omega}|\nabla(x,s)|^2 dx ds =\frac12\int_{\Omega} u_0(x)^2+\int_{0}^{t}\int_{\Omega} f(x,s)u(x,s)dxds\tag{2}$$. In my attempt I multipliedboth sides of $(1)$ by $u$ then using integration by part but i didn't come up to the desired equality ,I want to know its proof using clear steps because this equality is so important in my studies for evolution equation topics .However I made some search about this energy equality i didn't find its proof .Thanks in advanced
The energy functional is : $$E(t) = \frac 12 \int_\Omega u(x,t)^2$$ Assuming $u$ is regular enough, we have : \begin{align} \frac{\text dE}{\text dt} &= \int_\Omega u \frac{\partial u}{\partial t} \\ &= \int_\Omega u \left(f +\nabla^2 u\right)\\ &= \int_\Omega uf - |\nabla u|^2 \end{align} were the last step is integration by part over $\Omega$, using $u= 0$ on $\partial \Omega$.
Therefore, we have : $$E(t) + \int_0^t\int_\Omega |\nabla u|^2 = E(0) + \int_0^t\int_\Omega uf$$