Carleman Estimates For A Degenerate Grushin Equation And A Reaction-diffusion System With Cross Diffusion And Its Applications

With the development of science and technology,the inverse problems in math-ematical physics have been widely used in many fields,such as material science,en-gineering control theory and biomedical science.The theoretical research on inverse problems mainly focus on three aspects:existence,uniqueness and stability.In this paper,we study the Carleman estimates for the degenerate Grushin e-quation with singular term and the reaction-diffusion system with cross diffusion,re-spectively.As the applications of these two Carleman estimates,we then discuss the null controllability problem of the Grushin equation with singular term and the unique continuation of the reaction-diffusion system with cross diffusion.The structure of this article are as follows.In the first chapter,we first summarize the research fields on the inverse problems in mathematical physics.Then the physical background and research development of the Grushin equation and the cross diffusion system are described.Finally the main work of this paper is introduced.In the second chapter,we discuss the null controllability problem of the degenerate Grushin equation with singular term.In this chapter,we use a special weight function to prove the Carleman estimation of this equation.We then use this Carleman estimate to obtain the observation inequality for the degenerate Grushin equation with singular term.Based on the observation inequality,we finally prove the null controllability problem for this equation.In the third chapter,we study the unique continuation of a reaction-diffusion sys-tem with cross diffusion.In this chapter,under the condition that the coefficient matrix of principal terms has two different eigenvalues,we transfer the first two equations in the cross-diffusion system to two complex Ginzburg-Landau equations,and then prove the Carleman estimate for this reaction-diffusion system.Finally,the unique continuation of the cross diffusion system is proved by means of this Carleman estimate.