I am looking for existence results on inhomogeneous linear heat equations. Concretely, I have the equation $$ \frac{\partial}{\partial t} u(t, x) = \Delta_t u(t, x), ~~~~~u(0, x) = u_0(x)$$ where $\Delta_t$ is a family of elliptic operators.
In my application, I would like to consider this equation on a compact Riemannian manifold $M$ with a (smooth) family of Riemannian metrics $g_t$, and for sections $u$ of a vector bundle $V$ over $M$. Moreover, $\Delta_t = \mathrm{tr}_{g_t}((\nabla^t)^2)$ is the Laplacian associated to this Riemannian metric and a smooth family of connections $\nabla^t$ on $V$.
There will be a strongly continuous family of solution operators $E(t, s)$, where $t \geq s$, such that $u(t, x) := E(t, 0)u_0(x)$ solves the above equation, and such that the inhomogeneous equation $$ \frac{\partial}{\partial t} u(t, x) = \Delta_t u(t, x) + f_t(x), ~~~~~u(0, x) = u_0(x)$$ is solved by $$ u = E(t, 0)u_0 + \int_0^t E(t, s)f_s \mathrm{d}s. $$ Moreover, given $T>0$, there will be constants $C_1, C_2 >0$ such that for all $0 \leq s \leq t \leq T$, we have $$\|E(t, s)u\|_{\infty} \leq C_1\|u\|_\infty, ~~~~~~\|\nabla E(t, s)u\|_\infty \leq \frac{C_2}{\sqrt{t-s}}\|u\|_\infty.$$ These statements can also be formulated in terms of the integral kernel of $E(t, s)$.
All these results should follow in a similar way as the time-homogeneous case (where $g_t$ and $\nabla^t$ are constant in $t$). However, I cannot seem to find a reference for these statements. So my question is: What is a good reference for these statements?
