On well-posedness for parabolic Cauchy problems of Lions type with rough initial data

We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of $p$ for which tempered distributions in homogeneous Hardy–Sobolev spaces ${\dot{H}}^{s,p}$ with regularity index $s\in (-1,1)$ are initial data. Source terms of Lions’ type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces ${\dot{B}}_{p,p}^s$.