带阻尼的非线性波动方程的反问题

Pub Date : 2023-01-01 DOI:10.32523/2306-6172-2023-11-2-99-115

V. Romanov

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引用次数: 0

摘要

考虑半线性波动方程中两个系数的反演问题。该方程包含一个阻尼项和一个二次非线性项。逆问题包括在这些项下恢复系数作为空间变量x∈r3的函数。研究了带点源方程的正演问题。将反问题简化为两个问题，一个是众所周知的x射线层析成像问题，另一个是带有特殊权函数的几何积分问题。对后一个问题进行了研究，并给出了该问题解的稳定性估计。

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AN INVERSE PROBLEM FOR A NONLINEAR WAVE EQUATION WITH DAMPING

We consider an inverse problem of recovering two coefficients in a semi-linear wave equation. This equation contains a damping term and a term with a quadratic nonlinearity. The inverse problem consists in recovering coefficients under these terms as function of the space variable x ∈ R 3 . A forward problem for the equation with a point source is studied. As a result, the inverse problem reduce to two problems, one of them is the well known problem of X-ray tomography, the other one is the problem of the integral geometry with a with a special weight function. The latter problem is studied and a stability estimate for the solution of this problem is stated.

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