Reference : Special case of Banach-valued function integration by parts

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.

• What exactly is the definition of $\partial_t u$ here? – Nate Eldredge Jan 27 '15 at 13:53
• @NateEldredge $\partial_t u$ is the derivative (in the sense of the distribution) of the function $u : (0,T)\times \Omega \to \mathbb{R}$. $u$ is considered as a function of $n+1$ variables (time + space). What definition do you have in mind ? – user37238 Jan 27 '15 at 15:22
• I would start looking in Zeidler's books on functional analysis but I don't have access right now. – Dirk Jan 27 '15 at 16:19
• @Dirk Zeidler wrote at least 5 books on functional analysis. Can you specify which one you have in mind? – user37238 Jan 28 '15 at 8:22
• @NateEldredge The derivative can also be viewed as the distributional derivative (for $L^1(\Omega)$-valued functions). – user37238 Jan 29 '15 at 13:29