This research demonstrates the existence and uniqueness of weak solutions for non-autonomous parabolic Cauchy problems in weak spaces, indicating important implications for divergence-type equations.
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 Ḣs,p, and source terms belong to certain scales of weighted tent spaces. Weak solutions are constructed with their gradients in weighted tent spaces Tᵖs/2. Analogous results are also exhibited for initial data in homogeneous Besov spaces Ḃˢp,p.
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Hedong Hou (2025) studied this question.