Carleman Estimate And Null Controllability For Degenerate Parabolic Equations

This thesis aims to study the controllability theory of two kinds of boundary de-generate parabolic problems.One is the degenerate parabolic problem with convection term,the other is the degenerate parabolic problem with superlinear source.This thesis is mainly divided into two parts.In the first part,we study the null controllability of the degenerate parabolic problem with convection term.Firstly,we linearize the nonlinear problem and prove the null controllability of the linear problem.At the same time,we estimate the solution of the linear problem.And then the null controllability of the nonlinear problem is obtained by the fixed point theorem.As we all know,the key to the proof of the null controllability is to establish the Carleman estimate of the conjugate equation of the degenerate parabolic equation.In the process of establishing the Carleman estimate,the boundary has a certain degeneracy and the classical solution may not exist,so we need to consider its weak solution.However,for the weak solution,some estimates of classical solution may no longer hold true.Therefore,we need to establish the correlative compactness estimate of the weak so-lution,which is also the key to establishing null controllability.At present,the null controllability of degenerate parabolic equation with convection is studied only when the convection depends on the diffusion,but we study the problem that the convection is independent of the diffusion.To get the uniform Carleman estimate,we introduce an auxiliary function by which the diffusion and the convection are transformed into a union.In the second part,we study the null controllability of degenerate parabolic equa-tions with the superlinear reaction term.Firstly,in Chapter 2,we consider the null controllability of the degenerate parabolic equation which is relatively weak,and the reaction term grows relatively slow at the infinity.The key to the null controllability of the equation is to establish the Carleman estimate of the corresponding problem.In the process of establishing the Carleman estimate,we need to overcome the technical diffi-culties of the degenerate and the surperlinear reaction term.So far,the controllability of nondegenerate parabolic equation with superlinear reaction term has been studied.But we research the controllability of the degenerate parabolic equation.Therefore,we can not directly apply the results of the nondegenerate parabolic equation.Since the equation may be degenerate on a portion of the lateral boundary,the regularity of the weak solution is the poor and some compactness estimates for solutions to nondegen-erate equations may no longer valid.Thus,we make some compactness estimates for the weak solution.This is our key to proving that the problem is null controllability.We need to regularize the degenerate parabolic problem,then we study the Carleman estimate of the regularized parabolic problem.We introduce the auxiliary function to establish the Carleman estimate of the correspond problem.The reaction term grows relatively slow at the infinity,so we can only establish the Carleman estimate in the weaker degenerate.Using the Carleman estimate,we can get the desired observability inequality.Since the reaction term is superlinear,one must establish a more precise observability inequality in which the dependence relation of various factors should be described.Then we can show the approximate controllability of the linear equation.What is more important is that we can get the uniformly bounded estimate of the con-trols by means of the observability inequality.By this uniformly bounded estimate of the controls and the Kakutani fixed point theorem,we prove the approximate control-lability of the nonlinear equation.Furthermore,the controls are uniformly bounded.Using the approximate controllability of the regularized problem and the uniformly bounded estimate of the controls,together with priori estimates,we can prove the problem is null controllability.In Chapter 3,we study the null controllability of more general degenerate parabolic equation with the superlinear reaction term.Specifically,the degenerate of the equation can be divided into the weakly degenerate case and the strongly degenerate case.The boundary value condition depends on its degeneracy.Different degenerate cases require different boundary values for the problem.Thus,we need to overcome the difficulties caused by degenerate and the superlinear reaction term.We first consider the linear case of the equations with the reaction term,do the estimates for the solution of the linear problem and prove the null controllability of the linear equation.Then,we prove the null controllability of the semilinear problem by the Kakutani fixed point theorem.In the process of establishing the Carleman estimate,we need to overcome the technical difficulties presented by degenerate and superlinear reaction term.Thus,we need to regularize the degenerate parabolic problem,then we study the Carleman estimate of the regularized parabolic equation.We know that the key to the proof of null controllability is to establishing the Carleman estimate of the conjugate problem of the corresponding problem.As we know,it is necessary to establish Carleman estimate under the different degenerate conditions.The superlinear reaction term of the equation bring new difficulties to the Carleman estimate.In the process of establishing the Carleman estimate,we selected an appropriate energy weight function.Under different degeneracy cases,we add some constraints to the reaction term.Further we get the observability inequality of the corresponding problem.Since the reaaction term is superlinear,one must establish a more precise observability inequality in which the dependence relation of various factors should be described.In the proof process,we mainly applied Hardy-Poincare inequality,Young inequality,Kakutani fixed point theorem,the uniform bounded estimate of the controls in the process.