In this work we determine the second-order coefficient in a parabolic équation from the knowledge of a single final data. Under assumptions on the concentration of eigenvalues of the associated elliptic operator, and the initial state, we show the uniqueness of solution, and we derive a Lipschitz stability estimate of the inversion when the final time is large enough. The Lipschitz stability constant grows exponentially with respect to the final time, which makes the inversion ill-posed. The proof of stability estimate is based on a spectral decomposition of the solution to the parabolic equation in terms of the eigenfunctions of the associated elliptic operator, and an ad hoc method to solve a nonlinear stationary transport equation that is itself of interest.

在这项工作中，我们根据单个最终数据的知识确定抛物线方程中的二阶系数。在相关椭圆运算符特征值的集中度和初始状态的假设下，我们显示了解的唯一性，并且当最终时间足够长时，我们得出了反演的Lipschitz稳定性估计。Lipschitz稳定性常数相对于最终时间呈指数增长，这使反演不适当。稳定性估计的证明基于抛物线方程解的频谱分解（根据相关的椭圆算子的本征函数），以及一种求解非线性平稳输运方程本身的自组织方法。