变系数非线性抛物型方程的辐射系数识别问题

Identification of radiation coefficients for nonlinear parabolic equations with variable coefficients

摘要

研究一类利用附加条件重构并带有变系数的非线性抛物型方程辐射系数的反问题,其中方程的变系数依赖于解的梯度。首先由能量估计证明相应定解问题的解的唯一性与稳定性,然后基于最优控制理论并利用Tikhonov正则化方法将原问题转化为一个优化问题,最后利用极小元所满足的必要条件证明极小元的存在性、唯一性和稳定性。

Abstract

In this paper, we study a class of inverse problems that use additional conditions to reconstruct the radiation coefficients of nonlinear parabolic equations with variable coefficients, where the variable coefficient of the equation depends on the gradient of the solution. Firstly, the uniqueness and stability of the corresponding solution to definite solutions problem are proved by energy estimation. Secondly, based on the optimal control theory, the original problem is transformed into an optimization problem by using Tikhonov regularization method. Finally, the existence, uniqueness and stability of the minimum element are proved by using the necessary conditions satisfied by the minimum.

关键词

反问题 / 非线性抛物型方程 / 最优控制

Key words

inverse problem / nonlinear parabolic equations / optimal control

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基金资助

国家自然科学基金资助项目(61663018)

国家自然科学基金资助项目(11961042)

甘肃省自然科学基金资助项目(22JR5RA341)