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

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.