Let $\Omega =(0,1)$ and $\nu>0$ . We consider the one dimensional heat equation
$(1)$ $\dfrac{\partial u\left(x,t\right)}{\partial t}-\nu\dfrac{\partial^{2}u\left(x,t\right)}{\partial x^{2}}=0,x\in\Omega,t>0$
together with homogeneous Neumann boundary conditions
$(2)$ $ \dfrac{\partial u}{\partial x}\left(0,t\right)=\dfrac{\partial u}{\partial x}\left(1,t\right)=0,\forall t>0$
and the initial condition
$(3)$ $u\left(x,t=0\right)=u_{0}$
where $u_{0}$ is a given function belonging to $L^{2}\left(\Omega\right)$ .
Prove that for $v(x,t)$ smooth enough and $v$ is a solution of that equation
$2{\displaystyle \int}_{\left[0,1\right]}\dfrac{\partial v(x,t)}{\partial t}v(x,t)dx={\displaystyle \int}_{\left[0,1\right]}\dfrac{\partial}{\partial t}v^{2}(x,t)dx=\dfrac{d}{dt}\displaystyle \int_{0}^{1}v^{2}(x,t)dx$
Help me some hints, pls. Thank you in advance.
Multiply the equation by $v$ and integrate by parts. $$ v\,v_t-\nu\,v\,v_{xx}=0. $$ Integrate between $0$ and $1$: $$ \int_0^1v\,v_t\,dx-\nu\int_0^1\,v\,v_{xx}\,dx=0. $$ Integrate by parts: $$ \frac12\int_0^1(v^2)_t\,dx-\nu\Bigl(v\,v_x\Bigr|_0^1-\int_0^1(v_{x})^2\,dx\Bigr)=0. $$