Pseudo-monotone operator theory for unsteady problems in variable exponent spaces

TL;DR

This paper develops a new theoretical framework using pseudo-monotonicity methods to establish existence results for unsteady parabolic PDEs with variable exponent nonlinearities, introducing Bochner pseudo-monotonicity and coercivity.

Contribution

It introduces the concepts of Bochner pseudo-monotonicity and Bochner coercivity for unsteady problems in variable exponent spaces, extending classical notions to this context.

Findings

Established an abstract existence theorem for parabolic PDEs with variable exponent nonlinearities.

Developed new parabolic embedding and compactness results involving the symmetric gradient.

Applied the Hirano-Landes approach to derive verifiable conditions for the new notions.

Abstract

We prove by means of advanced pseudo-monotonicity methods an abstract existence result for parabolic partial differential equations with $\log$-H\"older continuous variable exponent nonlinearity governed by the symmetric part of a gradient only. To this end, we introduce the notions Bochner pseudo-monotonicity and Bochner coercivity, which are appropriate extensions of the concepts of pseudo-monotonicity and coercivity to unsteady problems in variable exponent spaces. In this context, we apply the so-called Hirano-Landes approach, which enables us to give general and easily verifiable conditions for these new notions. Moreover, we prove essential parabolic embedding and compactness results involving only the symmetric part of the gradient.