Qualitative Analysis Of Some Von-Karman-plate Models

In this thesis,we focus on the qualitative analysis of several types of von-Karmanplate models with nonlinear source terms.The influence of source term(power function source,logarithmic source,general nonlinear source term,nonlinear diffusion term)on the global existence,blow-up and the decay rate of energy is discussed.Specifically,this thesis is divided into five chapters:Chapter 1 is an introduction.First of all,we introduce the background of the problems studied and the related works both in China and abroad.Furthermore,we state our main problems and some difficulties as well as some methods used.Finally,some necessary inequalities and definitions are given.In Chapter 2,we are devoted to study of the following von-Karman-plate model with power function source where Ω?Rn(n≥1)is a bounded domain with smooth boundary ?Ω,u0∈H,u1∈L2(Ω),θ0∈H01(Ω),H={u∈H2(Ω);u=Δu=0,x∈?Ω},p satisfies 2<p≤2n/n-4,if n≥5;2<p<∞,if 1≤n<5.Notice that this system is coupled by a fourth order wave equation and a second order parabolic equation,the structure of system(5)is asymmetrical,so,a natural question is how to balance this asymmetry.To do this,for low initial energy,we apply modified potential well method to establish two invariant sets(stable and unstable sets),construct some new Lyapunov functionals and employ differential inequality arguments to prove that the solution globally exists when the initial energy starts from the stable set and fails to globally exist when the initial energy starts from the unstable set.In addition,we also use the Komornik inequality to give uniform decay estimate of energy functional.For high initial energy,we find the new condition that the quotient of the L2 norm of the initial and the initial energy is bigger than a positive constant,which only depends on the exponent p rather than the Sobolev embedding constant.This,together with levine’s concave method,guarantees the solution blowing up in finite time.Finally,we construct a suitable controlled functional combined energy estimation to give some lower estimates for blow-up time.In Chapter 3,the following von-Karman-plate model with logarithmic nonlinear source or general nonlinear source is discussed where f(u)=|u|p-2 u log |u| or general source satisfies structural conditions,p satisfies 2<p≤2n/n-4,if n≥5;2<p<∞,if 1≤n<5.For von-Karman-plate model with logarithmic nonlinear source,we first construct new energy functional and apply potential well method and Galerkin method to establish a stable invariant set which helps us to complete the proof of global existence of solutions,further,by constructing new functional equivalent to E(t)and combining with differential inequalities technique,we give an uniform decay estimate of the energy.Secondly,for low initial energy,we find out the invariant set determined by Nehari manifold and explore some differential inequalities to end our proof of main results.Additionally,for high initial energy,we find out some sufficient conditions for the solution blowing up in finite time.At the same time,the inverse Holder inequality with correction constants is given by means of a new estimate of the energy inequality and an interpolation inequality,which in turn gives the lower bound estimate of blow-up time.By developing and modifying previous methods,we further investigate the influence of a more general source term f(u)on the properties of weak solution of the von-Karman-plate model,where the nonlinear force f(u)satisfies condition:? ?f>0,s.t.∫Ω uf(u)dx>(2+?f)∫Ω F(u)dx(F is the primitive of f).Motivated by some previous methods,we construct an explicit blow-up sub-solution and apply comparison principle to give a positive answer to open problems of wave equation with overdamping force proposed by Ikeda and Wakasugi.Namely,At the same time,the lower bounded estimation is also given.In Chapter 4,the following quasi-linear von-Karman-plate model with source term with power function is considered where p satisfies 2<p≤4n/n-8,if n≥ 9;4<p<∞,if 1≤n<9.Firstly,with the help of theory of semigroup,we obtain the global existence of the solution to the problem.Secondly,for low initial energy,we apply precise analysis and Sobolev embedding inequality to establish the two-sides controlled relations between the principle term and nonlinear source and utilize energy estimate and differential inequality argument to prove blow-up of solutions.At the same time,the upper bound of blow-up time is also given.Due to the failure of Sobolev embedding theorem W2,4(Ω)(?)L2p-2(Ω)with p>3n-8/n-8,with the aid of inverse Holder inequality which follows from energy inequality and Gagliardo-Nirenberg inequality and first order differential inequality technique,we obtain the lower bound of blow-up time.In Chapter 5,we summarize the innovations and main results of this thesis,and state a number of problems to be further researched.