Some Topics On Viscoelastic Models

In this thesis, we study some mathematical models on viscoelastic materials and mainly discuss the well-posedness and some control problems which related to those models.The viscoelastic materials exhibit not only elasticity but also hereditary properties. The hereditary properties are described by viscoelasticity, where the mechanical response of the materials is taken to be influenced by the previous behavior of the materials themselves. In other words, the viscoelastic materi-als exhibit memory effects. This leads to a constitutive relationship between the stress and strain involving the convolution of the strain with a relaxation function. This term is so-called memory term. From the mathematical point of view, these hereditary properties are modeled by integro-differential operators where A is a differential operator such as △, △2, g is called memory kernel function. When t =+∞, it is the infinite memory; And when 0< t<+∞, it is the finite memory. So, in this thesis, the so-called viscoelastic model is some initial-boundary value problem of PDEs with the memory term (1).The mathematical theory on viscoelastic materials has been rapid devel-oped in half a century. At present, some research result on the viscoelastic model spring up in the world. But due to the complexity of these problems, many problems still are worth to study. And this will provide mathematicians a stage to show their talents.The main contents of this paper are as follows:In Chapter 1, we introduce the research background and the main work of this thesis from two aspects:the well-posedness and control.In Chapter 2, we give some preliminaries which focus on the mathematical terminology and mathematical tools in this thesis.In Chapter 3. we first discuss the existence and uniqueness and energy de-cay problem of a class of viscoelastic wave equation with internal feedback delay and solve an open problem put forward by M. Kirane and B. Said-Houari in Z. Angew. math. Phys. in 2011. And then, we extend these results to the case of the boundary feedback model and the fourth-order Euler Bernoulli viscoelastic plate model. Next, in order to discuss the competitive relationship between viscoelastic and source terms, we study the blow up properties of nonlinear Eu-ler Bernoulli plate model and obtain some intuitive understanding about the competition relationship of viscoelastic term and the source term.In Chapter 4, we study some control problems about a class of Euler-Bernoulli viscoelastic model. Specifically, we mainly discuss the approximate controllability and multi-point observation problems. For the discussion of the approximate controllability, we first define the function space of Hθ,k.And then, by using of the method of variables separation, we give the accurately solution of the dual system and analysed its properties in this space. At last, by using of the duality relation and Hahn-Banach theorem, we obtain the boundary ap-proximate controllability results of the system in the product space Hθ,k2. For the multi-point observation problem, we first write the solution of the system to the form of series and obtain an observability inequality of a single observa-tion point. And then, we obtain an observability inequality of multi observation points. Finally, by defining the so-called observation information, we give a rea-sonable interpretation of the results from the perspective of information theory. And we also explain the objective phenomenon of the experiment.