In this article we are concerned with an inverse initial boundary value problem for a non-linear wave equation in space dimension $n\geq 2$. In particular we consider the so called interior determination problem. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear wave equation with a time independent potential. For any small solution $u=u(t,x)$ of this non-linear equation, it is the perturbation of linear wave equation with time-independent potential perturbed by a divergence with respect to $(t,x)$ of a vector whose components are quadratics with respect to $\nabla_{t,x} u(t,x)$. By ignoring the terms with smallness $O(|\nabla_{t,x} u(t,x)|^3)$, we will show that we can uniquely determine the potential and the coefficients of these quadratics by many boundary measurements at the boundary of the spacial domain over finite time interval and the final overdetermination at $t=T$. In other words, the measurement is given by the so-called the input-output map (see (1.5)).

中文翻译：

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非线性双曲偏微分方程的逆初边值问题

在本文中，我们关注空间维数为 $n\geq 2$ 的非线性波动方程的逆初边值问题。我们特别考虑所谓的内部确定问题。这个非线性波动方程有一个简单的解，即零解。通过在平凡解上对该方程进行线性化，我们得到了具有时间无关势的常用线性波动方程。对于这个非线性方程的任何小解 $u=u(t,x)$，它是线性波动方程的扰动，其具有时间无关的势，被一个关于 $(t,x)$ 的散度扰动向量，其分量是关于 $\nabla_{t,x} u(t,x)$ 的二次方。通过忽略较小的项 $O(|\nabla_{t,x} u(t,x)|^3)$，我们将证明我们可以通过在有限时间间隔内空间域边界处的许多边界测量以及在 $t=T$ 处的最终超定来唯一地确定这些二次方程的势能和系数。换句话说，测量是由所谓的输入输出图给出的（见（1.5））。