Descret Solution for a Nonlinear Parabolic Equations with Diffusion Terms in Museilak-Spaces

TL;DR

This paper develops a numerical approach for solving nonlinear parabolic equations with complex growth conditions in Musielak spaces, proving existence, uniqueness, and error estimates for solutions.

Contribution

It introduces a discretization method combining internal approximation and backward Euler scheme for equations in Musielak spaces, with proven error bounds.

Findings

Existence and uniqueness of weak solutions established.

A priori error estimates for temporal semi-discretization derived.

Applicable to nonlinear equations with non-standard growth in fluid dynamics.

Abstract

In this paper, we study a class of nonlinear evolution equations with damping arising in fluid dynamics and rheology. The nonlinear term is monotone and possesses a convex potential but exhibits non-standard growth. The appropriate functional framework for such equations is the modular Museilak-Spaces. We prove the existence and uniqueness of a weak solution using an approximation approach by combining an internal approximation with the backward Euler scheme, also a priori error estimate for the temporal semi-discretization is given.