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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 $u_0(x) \in C^1(\mathbb{T}^1).$

For the regularity of $f$ and $g$, I would need $f,g \in L^{\infty}((0, T); L^1(\mathbb{T}^1)),$ but "smoother" assumptions like $L^{\infty}(\mathbb{T}^1 \times (0,T)) $ would work too.

I am looking for a well-posedness result (existence, uniqueness and positivity of the solution in the mild sense) for this equation on the torus. Could someone recommend a literature reference?

I already tried Elliptic and Parabolic Equations. by Z. Wu, J. Yin, and C. Wang, but there are no results on the torus/periodic boundary conditions. Unfortunately, I only ever find results with Dirichlet boundary conditions and not with periodic boundary conditions.

Thanks for help in advance!

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    $\begingroup$ The heat kernel maps $L^1$ to $L^\infty$ with norm of order $t^{-1/4}$, so it is immediate that the Picard iteration yields a contraction for small enough times. $\endgroup$
    – Martin Hairer
    Commented Aug 18 at 14:20
  • $\begingroup$ Sorry, I meant $t^{-1/2}$... $\endgroup$
    – Martin Hairer
    Commented Aug 18 at 17:53
  • $\begingroup$ @MartinHairer Thanks, that makes sense. Then it should be possible to extend to a global solution. But that would only give existence? I suppose, uniqueness and positivity can be shown a maximum principle, because maximum principles do not depend on boundary conditions? $\endgroup$
    – kumquat
    Commented Aug 18 at 20:38

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