与时间相关的非线性抛物型方程的反源问题  

Inverse Source Problems for theTime-dependent Nonlinear Parabolic Equations

作　　者：王泽慧 WANG Ze-hui(School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,China)

机构地区：[1]兰州交通大学数理学院,甘肃兰州730070

出　　处：《兰州文理学院学报（自然科学版）》2022年第4期1-9,共9页Journal of Lanzhou University of Arts and Science(Natural Sciences)

基　　金：国家自然科学基金项目(11461039、61663018、11961042);兰州交通大学“百名青年优秀人才培养计划”项目(61663018);甘肃省自然科学基金资助项目(18JR3RA122)。

摘　　要：研究了一类具有未知源函数的非线性抛物型方程问题,其中未知的源项是从空间积分中得到,并且此源项仅依赖于时间.与以往研究不同的是,文中在原有的抛物型方程中增加了Lipschitz连续的非线性项k(u).首先基于变分理论利用变分公式证明了解的存在唯一性;接着使用向后欧拉方法对方程进行时间离散化,推导出了此近似方案的一系列先验估计;最后证明了弱解的存在性并进行了误差分析.In this paper,a nonlinear parabolic equation with unknown source functions is studied,the unknown source term is derived from the space integral and the source term is only time-dependent in this equation.Different from previous studies,a Lipaschitz continuous nonlinear term k(u)is added to the equation.Firstly,the existence and uniqueness of solutions is proved by using variational formula based on variational theory.Then the backward Euler method is used to discretize the equation and a series of prior estimates of this approximate scheme are derived.Finally,the existence of the weak solution is proved and the error analysis is carried out.

关 键 词：非线性 抛物型方程 反源问题 唯一性 弱解存在性 

分 类 号：O175.26[理学—数学]