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

TL;DR

This paper proves existence, uniqueness, and representation of weak solutions for non-autonomous parabolic PDEs with rough initial data in Hardy–Sobolev and Besov spaces, under minimal regularity assumptions on coefficients.

Contribution

It extends well-posedness results to equations with complex, measurable, and non-smooth coefficients and initial data in advanced functional spaces.

Findings

Established existence and uniqueness of weak solutions.

Represented solutions using integral formulas.

Extended results to initial data in Besov spaces.

Abstract

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 $T^{p}_{s/2}$. Analogous results are also exhibited for initial data in homogeneous Besov spaces $\dot{B}^{s}_{p,p}$.