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Direct and inverse source problems for degenerate parabolic equations
Journal of Inverse and Ill-posed Problems ( IF 1.1 ) Pub Date : 2020-06-01 , DOI: 10.1515/jiip-2019-0046
M. S. Hussein 1 , Daniel Lesnic 2 , Vitaly L. Kamynin 3 , Andrey B. Kostin 3
Affiliation  

Abstract Degenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading coefficient make the PDE non-uniformly parabolic, and new theories need to be developed in the context of practical applications of such rather unstudied mathematical models arising in porous media, population dynamics, financial mathematics, etc. With this new challenge in mind, this paper considers investigating newly formulated direct and inverse problems associated with non-uniform parabolic PDEs where the leading space- and time-dependent coefficient is allowed to vanish on a non-empty, but zero measure, kernel set. In the context of inverse analysis, we consider the linear but ill-posed identification of a space-dependent source from a time-integral observation of the weighted main dependent variable. For both, this inverse source problem as well as its corresponding direct formulation, we rigorously investigate the question of well-posedness. We also give examples of inverse problems for which sufficient conditions guaranteeing the unique solvability are fulfilled, and present the results of numerical simulations. It is hoped that the analysis initiated in this study will open up new avenues for research in the field of direct and inverse problems for degenerate parabolic equations with applications.

中文翻译:

退化抛物线方程的正反源问题

摘要 前导系数为零或无界的简并抛物偏微分方程 (PDE) 使 PDE 非均匀抛物线,需要在多孔介质、种群动力学、金融数学等。 考虑到这一新挑战,本文考虑研究与非均匀抛物线偏微分方程相关的新制定的正反问题,其中允许领先的空间和时间相关系数在非空上消失,但零度量,内核集。在逆分析的背景下,我们考虑从加权主要因变量的时间积分观察中线性但不适定地识别空间相关源。对彼此而言,这个逆源问题及其相应的直接公式,我们严格研究适定性问题。我们还给出了满足保证唯一可解性的充分条件的逆问题的例子,并给出了数值模拟的结果。希望本研究中发起的分析将为退化抛物线方程的正反问题领域的研究开辟新的途径和应用。
更新日期:2020-06-01
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