Inverse initial boundary value problem for a non-linear hyperbolic partial differential equation
Inverse initial boundary value problem for a non-linear hyperbolic partial differential equation
复制标题
非线性双曲偏微分方程的反初边值问题
DOI：
10.1088/1361-6420/abcd27
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Manmohan Vashisth and Michiyuki Watanabe
中科院分区：
文献类型：
--
作者：
Gen Nakamura;Manmohan Vashisth and Michiyuki Watanabe
在本文中，我们关注空间维度 n⩾ 2 中非线性波动方程的逆初始边值问题。特别是我们考虑所谓的内部确定问题。该非线性波动方程有一个平凡解，即零解。通过在平凡解处对该方程进行线性化，我们得到了具有与时间无关的势的通常线性波动方程。对于非线性波动方程的任何小解 u= u (t, x)，它是具有与时间无关的势的线性波动方程的扰动，该扰动受到向量 (t, x) 的散度的扰动，该向量的分量相对于 ∇ t, x u (t, x) 是二次的。通过忽略较小的项 O (|∇ 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 t= T 处的最终超定来唯一确定这些二次方程的势和系数。换句话说，我们的测量由所谓的输入输出映射给出（参见（1.5））。
In this article we are concerned with an inverse initial boundary value problem for a non-linear wave equation in space dimension n⩾ 2. In particular we consider the so called interior determination problem. This non-linear wave equation has a trivial solution, ie 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 our non-linear wave equation which 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∇ t, x u (t, x). By ignoring the terms with smallness O (|∇ 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, our measurement is given by the so-called the input-output map (see (1.5)).