On weak solution of a hyperbolic differential inclusion with nonmonotone discontinuous nonlinear term

Guo Xingming

doi:10.1155/s0161171299225872

Abstract

In this paper, a hyperbolic differential inclusion with nonmonotone discontinuous and nonlinear term, which the generalized velocity acts as its variable, is studied and the existence and decay of its weak solution are obtained.

Highlights

We investigate the initial boundary value problem of the following degenerate multi-valued hyperbolic differential inclusion: u(t) + B(u)(t) + φ(u)(t) f (t), a.e. t ∈ [0, T ], u(x, t) = 0, a.e. (x, t) ∈ = ∂Ω × [0, T ], u(0) = u0, u(0) = u1, (1.1)
Physical motivations for studying equation (1.1) come partly from problems of continuum mechanics, where nonmonotone, nonlinear, discontinuous, and multi-valued constitutive laws and boundary constraints lead to the above variational inequalities
When elastobody is constrainted by boundary friction, (1.1) denotes its control equation; if we study viscoelastical body and the unilateral problem of plate, (1.1) is their control equation, etc. [10, 8, 5]

Summary

Introduction

We investigate the initial boundary value problem of the following degenerate multi-valued hyperbolic differential inclusion: u(t) + B(u)(t) + φ(u)(t) f (t), a.e. t ∈ [0, T ], u(x, t) = 0, a.e. Physical motivations for studying equation (1.1) come partly from problems of continuum mechanics, where nonmonotone, nonlinear, discontinuous, and multi-valued constitutive laws and boundary constraints lead to the above variational inequalities (differential inclusions). We investigate the existence and decay of the weak solutions of the hyperbolic in equation (1.1), with φ and B satisfying adequate conditions under zero boundary conditions. Consider the following initial boundary value problem of a hyperbolic variational inequation (inclusion): u(t) + Bu(t) + g(t) = f (t), a.e. t ∈ [0, T ], g(x, t) ∈ φ u(x, t) , a.e.

Existence of solution

Decay of solution