Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$ $$ \partial_t u- a \cdot\partial_{x}^2u +g(x,t)\cdot u= f(x,t),$$ with the constant $a > 0,$ and the given functions $g,f \geq 0,$ subjected to the strictly positive initial condition $u_0(x) \in C^1(\mathbb{T}^1).$ Let $f,g \in L^{\infty}((0, T); L^1(\mathbb{T}^1)).$

I need a well-posedness result (existence, uniqueness and positivity of the solution in the mild sense) for this equation on the torus.

A mild formulation is given by (representing the flat torus by $[0,1]$ and periodic boundary conditions): $$ u(x,t) = \int_{0}^{1} G(x-y,t) u_0(y) \, \mathrm{d}y + \int_{0}^{t} \int_{0}^{1} G(x-y,t-s) (f - g \cdot u)(y,s) \, \mathrm{d}y\mathrm{d}s, $$ where $G = G(x,t) > 0$ is the heat kernel on the one-dimensional torus. The existence of a local solution $u \in L^{\infty}(\mathbb{T}^1 \times (0,T))$ would follow from a Picard-Iteration, at least for small enough times. Since the right-hand side of the mild formulation would define a contraction from $L^{\infty}(\mathbb{T}^1 \times (0,T))$ to $L^{\infty}(\mathbb{T}^1 \times (0,T)).$ (I supposed semigroup arguments would fail here since $g$ depends on time.)

But how can I show positivity for the local solution? (The problem is I don't have enough regularity for the solution to use a maximum principle.)

Would be very grateful for help!