For notations, let $p \in (1,\infty)$ and $p':=\frac{p}{p-1}$. For $n \in \mathbb{N}$, define $\mathbb{T}^n:= (\mathbb{R} / \mathbb{Z})^n$.
Let us consider the following cauchy problem on $[0,T] \times \mathbb{T}^n$: \begin{equation} \partial_t u = \Delta u + f \text{ with } u(0,x):= u_0(x) \end{equation} where $f \in L^{p'}_t H^{-1}_x$ and $u_0 \in L^2(\mathbb{T}^n)$.
We say that $u$ is a weak solution of the above Cauchy problem if \begin{equation} u \in L^p_t H^1_x \text{ while } \partial_t u \in L^{p'}_t H^{-1}_x \end{equation} and \begin{equation} \langle u_t, v \rangle_{H^{-1} \times H^1} + \langle \nabla u, \nabla v \rangle_{L^2} = \langle f, v \rangle_{H^{-1} \times H^1} \text{ for all } v \in H^1(\mathbb{T}^n) \text{ and almost every } t \in [0,T] \end{equation} and $u(0)= u_0$.
For $p=p'=2$ and $f \in L^2([0,T] \times \mathbb{T}^n)$, unique existence of a weak solution is presented in detail in Section 7.1 of the textbook "Partial Differential Equations - 2nd Edition" by Evans.
However, I wonder if there is any reference dealing with general $p \in (1,\infty)$ and $f \in L^{p'}_t H^{-1}_x$. Unique existence of a weak solution in this case seems quite plausible to me at least. For example, the notion of Gelfand triple still tells us that $u \in C\bigl([0,T], L^2(\mathbb{T}^n) \bigr)$ so that the condition $u(0)=u_0$ makes sense.
Could anyone please provide any reference on this topic?