Stability of Weak Solutions to Parabolic Problems with Nonstandard Growth and Cross–Diffusion

André H Erhardt

doi:10.3390/axioms10010014

Abstract

We study the stability of a unique weak solution to certain parabolic systems with nonstandard growth condition, which are additionally dependent on a cross-diffusion term. More precisely, we show that two unique weak solutions of the considered system with different initial values are controlled by their initial values.

Highlights

We will prove that two weak solutions with different initial values are controlled by their initial values
Summarising, we were able to show that the unique weak solution to system (1)
Satisfies certain stability estimates, i.e., we proved that two weak solutions to system (1)

Summary

Introduction

In [1], we have recently established the existence of a unique weak solution to the following parabolic problem involving nonstandard p( x, t)-growth conditions and a crossdiffusion term δ∆u with δ ≥ 0:. One uses such diffusion models in the context of the restoration in image processing [10,11,12] and applications include models for flows in porous media [13] or parabolic obstacle problems [14]. The cross-diffusion term δ∆u for δ > 0 will complicate the derivation of the desired stability estimate and requires certain additional assumption, which we will discuss later in detail. A(z, w) satisfy the following nonstandard growth and monotonicity conditions with variable exponents p : Ω T → [2, ∞), μ ∈ [0, 1].

Main Result

Conclusions and Discussion