# Note on the existence theory for pseudo-monotone evolution problems

## TL;DR

This paper introduces a new framework with extended concepts of pseudo-monotonicity and coercivity, enabling the proof of existence for a broad class of non-linear evolution problems involving time-dependent operators.

## Contribution

It develops the notions of Bochner pseudo-monotonicity and Bochner coercivity, extending existing theories to handle more complex evolution problems where standard methods fail.

## Key findings

- Established a new existence theorem for non-linear evolution problems
- Introduced the concepts of Bochner pseudo-monotonicity and coercivity
- Provided conditions applicable to a wide range of problems

## Abstract

In this note we develop a framework which allows to prove an existence result for non-linear evolution problems involving time-dependent, pseudo-monotone operators. This abstract existence result is applicable to a large class of concrete problems where the standard theorem on evolutionary pseudo-monotone operators is not applicable. To this end we introduce the notion of Bochner pseudo-monotonicity, and Bochner coercivity, which are appropriate extensions of the concepts of pseudo-monotonicity and coercivity to the evolutionary setting. Moreover, we give sufficient conditions for these new notions, which are easily and widely applicable.