Study On The Properties Of Solutions Of Equations Related To Electrorheological Theory

The variational exponential evolutionary equation p(x)-Laplace comes from the application of magnetorheological and other fields.This paper mainly studies the related properties of a class of variational exponential p(x)-Laplace equation with boundary degeneracy.In some cases,we do not need the boundary condition to obtain the uniqueness of the solution.This paper first discusses the existence and uniqueness of the solution of the equation,and mainly studies the existence of the global attractor of the equation in the L2(Ω),to prove that there exists a global attractor in the space Lp(x)(Ω).In the first chapter of this paper,we mainly introduce some background and research status of partial differential equations,and give the basic definition and prop-erties of space.Then we introduce the main contents of this paper.In the second chapter of this paper,we study the existence and uniqueness of solutions of the following parabolic equations where QT = Ω ×(0,T]:Ω∈ RN.Variable index p(x)is a measurable function,and p->1,a(x)≥ 0,and a(x)= 0 on(?)Ω.This section proves that in the case of u0∈L∞(Ω),p->1,a(x)|▽u0|p(x)∈L1(Ω),(1)While,then the equation admits a unique solution with the fixed initial boundary value conditions.(2)While then the solution of equation is completely controlled by the given initial value.In the third chapter of this paper,we shall consider the asymptotic behavior of solution to following equationwhere log-Holder continuous.First,we prove that the semigroup associated with equation has a global attractor in L2(Ω).Then,we are to obtain compactness of the semigroup in Lp(x)(Ω)by the compactness of the semigroup in L2(Ω).