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A new partial differential nonlinear system containing quasivariational and parabolic variational inequalities and its application

Wei Li, Zhenghui Tang, Zengbao Wu, Chunyan Yang

TL;DR

The paper introduces a novel three-coupled system in Banach spaces that combines a partial differential equation, a quasivariational inequality, and a parabolic variational inequality, with unknowns $\eta$, $\xi$, and $w$ on a finite interval $I=[0,T]$. By exploiting estimates for parabolic equations and fixed-point arguments in Sobolev spaces, it proves unique solvability under moderate, verifiable conditions, contrasting with semigroup-based approaches. The authors then apply the abstract results to a viscoelastic frictional contact problem with long-memory effects, wear diffusion, and material damage on curved contact surfaces, establishing the same uniqueness properties. This work advances the theory of differential-variational systems and provides a rigorous framework for curved-contact frictional models with memory, wear, and damage phenomena.

Abstract

We study a new nonlinear system which contains a partial differential equation, a quasivariational inequality and a parabolic variational inequality in Banach spaces. We obtain the unique solvability of the coupled system under moderate conditions by using the Banach's fixed point theorem. We employ the main results to investigate a viscoelastic frictional contact problem with long-memory effects, wear processes, and damage phenomenon.

A new partial differential nonlinear system containing quasivariational and parabolic variational inequalities and its application

TL;DR

The paper introduces a novel three-coupled system in Banach spaces that combines a partial differential equation, a quasivariational inequality, and a parabolic variational inequality, with unknowns , , and on a finite interval . By exploiting estimates for parabolic equations and fixed-point arguments in Sobolev spaces, it proves unique solvability under moderate, verifiable conditions, contrasting with semigroup-based approaches. The authors then apply the abstract results to a viscoelastic frictional contact problem with long-memory effects, wear diffusion, and material damage on curved contact surfaces, establishing the same uniqueness properties. This work advances the theory of differential-variational systems and provides a rigorous framework for curved-contact frictional models with memory, wear, and damage phenomena.

Abstract

We study a new nonlinear system which contains a partial differential equation, a quasivariational inequality and a parabolic variational inequality in Banach spaces. We obtain the unique solvability of the coupled system under moderate conditions by using the Banach's fixed point theorem. We employ the main results to investigate a viscoelastic frictional contact problem with long-memory effects, wear processes, and damage phenomenon.
Paper Structure (4 sections, 11 theorems, 87 equations)

This paper contains 4 sections, 11 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unique solvability for the new nonlinear system
  4. Application to frictional contact problem

Key Result

Lemma 2.1

sofonea2017variational Let $\mathcal{X}$ and $\mathcal{Y}$ be nonempty sets, $\mathcal{P}\subset\mathcal{X}\times\mathcal{Y}$, and $Ab=\{a\in\mathcal{X} \mid (a,b)\in\mathcal{P}\}$ for all $b\in\mathcal{Y}$. We assume $S\colon \mathcal{X}\to\mathcal{Y}$ is a single valued map, and for each $b \in \m

Theorems & Definitions (15)

  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 3.1
  • Theorem 3.2
  • ...and 5 more