Inverse Time-dependent Source Problem For The Nonlinear Parabolic Equations

This paper focuses primarily on the parabolic equation with an unidentified source function,where the unknown source term is derived from the space integral and this factor is only depends on time.And the article studies this problem with different methods,in the first part of each chapter,the existence and uniqueness of the solution of the inverse problem are obtained,and in the second part,the prior estimation and data dependency analysis are made respectively.The main contents of this paper are as follows:The first chapter mainly introduces the research background and current situation of the inverse problem of parabolic equation,as well as the main work of each chapter.In the second chapter,based on the variational theory,the existence and uniqueness of the solution are proved by using the variational formula.The backward Euler method is used to discretize the equation in time,and a series of prior estimates of the approximate scheme are derived.Finally,the existence of the weak solution is proved and the error analysis is carried out.In the third chapter,a fractional parabolic equation is studied.Based on the variational method,it is analyzed by using a different test function from that in the second chapter.Important conclusions are drawn mainly in the estimation of the uniqueness and stability of the solution to the inverse question.In the fourth chapter,we study the problem of equations with nonlocal boundaries.The boundary conditions of this problem are regular but not strongly regular.The main difference of this problem is that the eigenfunction system is not complete,However,the system of the characteristic function and the correlation function forms a Riesz basis.The existence and uniqueness of the solution of the inverse problem and the continuous dependence on the input data are proved by using the spectral problem of the generalized Fourier method and the Volterra equation theory.In the fifth chapter,we summarize the current work of the article and look forward to the future work.