Variational formulation of a parabolic equation

I have the following problem $$\dfrac{\partial u}{\partial t} - \Delta u + F(u)= f(x,t)$$ where $u \in L^2([0,T[;H^1(\mathbb{R}^n))$ with $\partial_t u \in L^2([0,T[;(H^1(\mathbb{R}^n))')$

I want to write the variational formulation associated with this problem, but without using $\dfrac{\partial u}{\partial t}$.

For this, I took a test function $v \in C^1([0,T[;H^1(\mathbb{R}^n))$ with $\partial_t v \in L^2([0,T[;(H^1(\mathbb{R}^n))')$ where $v(x,T)=0$.

Then we have $$\left\langle \dfrac{\partial u}{\partial t},v \right\rangle - \langle \Delta u,v \rangle + \langle F(u),v \rangle = \langle f,v \rangle$$ My difficulty is: how would we write $\left\langle \dfrac{\partial u}{\partial t},v \right\rangle$ as a function of $\langle u,\partial_t v \rangle$? Can we use integration by parts?

• In lieu of weak formulation for the first-order transport equation, you might want to integrate over time too if you want to move the time-derivative over the test function. – Chee Han Jul 27 '18 at 22:59