Let $u \in L^p (0,T; L^1(\Omega))$ with $\partial_t u \in L^p(0,T; L^1(\Omega))$ and $v \in L^q (0,T; L^\infty(\Omega))$ with $\partial_t u \in L^q(0,T; L^\infty(\Omega))$ (with $1/p+1/q=1$ and $p \in (1,\infty)$). $\Omega$ is an open bounded subset of $\mathbb{R}^n$ with smooth boundary.

Is it true that $$ \int_0^T \int_\Omega \partial_t u v + \int_0^T \int_\Omega u \partial_t v = \int_\Omega u(t=T)v(t=T) - \int_\Omega u(t=0)v(t=0) $$ and where can I find a reference for such a result.