Theoretical Study On Partial Differential Equations Of Electrical Impedance Tomography(EIT)

Electrical impedance tomography(EIT)is a new medical imaging technology,which is widely used in clinical diagnosis,such as lung ventilation in patients with ARDS.It is also closely related to cloaking technology.Its mathematical model is electric conductivity equation of electromagnetic field.There are only a few numerical results in the study of the spatio-temporal EIT problem,and there is no mathematical theory about the properties of the solution.This paper mainly discusses the properties of the solution to the forward problem of a special spatio-temporal EIT continuous model in an open region,and the conductivity equation with singular dynamical boundary condition and the spectral fractional pseudo-parabolic equation with singular potential from the mathematical point of view.In Chapter 1 and Chapter 2,we introduce the research backgrounds and preliminaries.In Chapter 3,we consider the forward problem for open spatio-temporal EIT continuous models with subcritical and critical Sobolev growth respectively.First,by means of the fractional DtN operator,we obtain that this problem is equivalent to a initial boundary value problem for spectral fractional parabolic equation.The results of this chapter are a general generalization of the work of F.Fang and Z.Tan(Advances in Mathematics,2018,328:217-247).We analyze the finite time blow-up of the solution,the decay estimates of the global solution and the long time asymptotic behavior of solution.In the critical Sobolev growth case,due to the lack of compactness of the fractional Sobolev trace inequality,we give a bubble description of the solution sequence by means of the concentration compactness principle.Finally,on the basis of proving the boundedness of the global solution in subcritical case and the exponential decay of the solution in critical case,we improve the regularity of the low-energy initial value solution.In Chapter 4,we study the conductivity equation with dynamical boundary condition coupled with singular potential and subcritical Sobolev growth.The difficulty is to deal with the singular potential.First,by Gagliardo-Nirenberg inequality,we prove the global existence of the solution using Galerkin method.Then,by means of fractional HardySobolev trace inequality,introduce the stable set and unstable set,we study the global existence and finite time blow-up of solutions,and further prove that the global solution decays exponentially and tends to stationary solution in long time series.The results of this chapter are the extension from integer diffusion equation with singular potential to fractional diffusion equation.In Chapter 5,we consider the initial boundary value problem of spectral fractional pseudo-parabolic equations with singular potential.First,the local existence of the approximate solution to the equation is proved by the Galerkin method and the contraction mapping principle.Then,using the potential well method,we analyze the sufficient conditions for global existence and finite time blow-up of solutions under three different initial energies.Differ from the Riesz fractional problem considered by predecessors,which requires a given external boundary condition,the spectral fractional problem we consider in this paper can give the same type of local boundary condition on the boundary as the standard local Laplacian.