On well-posedness for non-autonomous parabolic Cauchy problems with rough initial data

We establish a complete picture for existence, uniqueness, and representation of weak solutions to non-autonomous parabolic Cauchy problems of divergence type. The coefficients are only assumed to be uniformly elliptic, bounded, measurable, and complex-valued, without any additional regularity or symmetry conditions. The initial data are tempered distributions taken in homogeneous Hardy–Sobolev spaces ${\dot{H}}^{s,p}$, and source terms belong to certain scales of weighted tent spaces. Weak solutions are constructed with their gradients in weighted tent spaces $T_{s\mathrm{/}2}^p$. Analogous results are also exhibited for initial data in homogeneous Besov spaces ${\dot{B}}_{p,p}^s$.