Nonlinear Evolution Equations And Variational Inequalities With Variable Exponent Growth

After M. Ruzicka first proposed the electrorheological fluid motion model, many re-searchers have devoted to studying the problems with variable exponent growth. Besides the important applications of the problems with variable exponent growth in nonlinear flu-id mechanics, image processing and so on, some new theories and methods arising from the study of these problems have greatly enriched the mathematical fields. At present, there are many important results concentrated in the elliptic partial differential equations while only a few results about parabolic and hyperbolic problems with variable exponent growth.In this thesis, the existence of solutions for several kinds of nonlinear evolution equations and evolution variational inequalities with variable exponent growth is studied under the framework of variable exponent spaces.Firstly, the basic properties of a class of variable exponent spaces of parabolic type X(QT) as reflexivity, completeness, density, etc, are studied. A compact embedding the-orem of X(QT) is obtained. Then, the basic characterization of the dual space X’(QT) of X(QT) is given and a new class of spaces named W(QT), which is constituted by X(QT) and X’(QT), is introduced. In the meantime, the basic properties of W(QT) and an inte-gration by parts formula are given.Secondly, on the basis of a population model, the existence of weak solutions for a class of parabolic equations of Kirchhoff type with variable exponent growth is studied. This problem has been discussed in two cases. In the first case, the existence of weak solutions of nonlocal parabolic equation (Kirchhoff type) with a linear growth term on the right of the equation is studied. A Galerkin approximate sequence of solutions with well regularity is obtained by picking a smooth orthogonal basis in L2(Ω) and using the existence theory of ordinary differential equations. Then with a priori estimates and the theory of monotone operators, the existence of weak solutions is obtained. In the second case, the parabolic equation with nonlinear growth condition is dicussed. The Galerkin approximation is used to investigate the existence of weak solutions. However, under the given conditions, only the local existence of weak solutions is obtained, i.e., there exits a constant T0>0such that the parabolic equation has a weak solution as t<T0. Thirdly, in variable exponent spaces, the existence and extinction behavior of solu-tions for two class of nonlinear evolution variational inequalities of parabolic type with variable exponent growth are investigated. For the evolution variational inequalities of parabolic type with nonzero initial value, the problem of inequalities is converted into the problem of parabolic equations by introducing a suitable penalty term. On the basis of existence of solutions for parabolic equations, the existence of solutions of evolution variational inequalities is obtained by using a priori estimates, the theory of monotone operators and so on. For the variational inequalities with zero initial value and a gradient constraint, a suitable penalty term is introduced according to the gradient constraint. The problem of inequalities is converted into the problem of parabolic equations. Further-more, the existence of weak solutions of evolution variational inequalities is obtained. On the basis of existence of weak solutions for variational inequalities of parabolic type, the extinction behavior of weak solutions is studied.Last, in the variable exponent space X(QT), the existence of weak solutions for a class of differential inclusion of hyperbolic type and a class of pseudoparabolic equations with variable exponent growth is studied, respectively. Under some appropriate condition-s, the existence of weak solution of differential inclusion is obtained by using Galerkin approximation, Alzela-Ascoli’s theorem, Egoroff’s thorem and Rusin’s thorem and so on. On the basis of existence of weak solutions for hyperbolic problems, the existence of weak solutions of a class of pseudoparabolic equations is discussed.