Initial-Boundary Value Problems For A Class Of Nonlinear Thermoelastic Plate Equations

Recently, due to the development of mathematic and promotion of physics and other mechanics, the study of nonlinear developing equation have been become an important subject in the partial differential equation research fields. Now, plate equations are topic problem, especially, the thermoelastic plate equations in presence of thermal effect have become important to many mathematician.In this paper, we consider the initial-boundary value problems of a class of nonlinear thermoelastic plate equationsin a rectangular domainΩ=(0,l)×(0,l) of R² with the initial conditionsu(x,y,0)=u₀(x,y) (u|·)(x,y,0)=u₁ (x,y)θ(x,y,0)=θ₀(x,y) (3)and the four edges simply supported boundary conditionsAnd we prove the existence and uniqueness of global weak solution of the system (1)-(6), then we also prove the existence of strong solution and classical solutionwhen f=g=0. The details will go as follows:Firstly, the current study situation in the world about plate equations is introduced;Secondly, we put forward some important definitions and lemmas, andexplain some marks simultaneously;Thirdly, we prove the existence and uniqueness of the weak solution of the problem(1)-(6)by means of the Galerkin method;Fourthly, we prove the existence of the strong solution of the nonlinear thermoelastic plate equationswith the initial conditions (3) and the four edges simply supported boundary conditions (4)-(6);Fifthly, we study the existence of the classical solution of the problem(3)-(8).The main characteristic of this paper is that we discussed the initial-boundary value problems of a class of nonlinear thermoelastic plate equations in a rectangular domainΩof R², with the four edges simply supported conditions. Because of the boundary of the rectangularΩhas no smoothness, we meet some problems which are difficult to be overcomed when we prove the existence theorems of strong solution and classical solution. At last, we solve these difficulties by using the complicated and meticulous priori estimates.