# On the Solvability of a Class of Degenerate or Singular Strongly Coupled Parabolic Systems

## TL;DR

This paper establishes the existence of strong solutions for a broad class of strongly coupled parabolic systems, including degenerate and singular cases with quadratic gradient growth, using a novel unified approach and relaxed assumptions.

## Contribution

It introduces a unified proof for degenerate and singular systems with quadratic growth, replacing the VMO assumption with a more versatile BMO-based condition.

## Key findings

- Improved existence results for strongly coupled parabolic systems.
- Applicable to degenerate and singular models in biology.
- Utilizes a new local weighted Gagliardo-Nirenberg inequality.

## Abstract

The existence of strong solutions to general class of strongly coupled parabolic systems will be discussed. These systems can be degenerate or singular as boundedness of theirs solutions are unavailable and not assummed. The results greatly improve those in a recent papers \cite{letrans,dleJFA,dleANS} as the systems can have quadratic growth in gradients. A unified proof for both cases is presented. Most importantly, the VMO assumption in \cite{dleJFA,dleANS} will be replaced by a much versatile one thanks to a new local weighted Gagliardo-Nirenberg involving BMO norms. Degenerate and singular generalized SKT models in biology will be presented as a nontrivial application of the main theorem.