On a stability estimate for a solution to an~inverse problem for a hyperbolic equation
Suppose that a function \(u(x,t)\) is a solution to the Cauchy problem \[ L_q u\equiv u_{tt}-\Delta u -q(x) u =8\pi\delta (x,t), \quad u|_{t<0} \equiv 0, \quad (x, t)\in {\mathbb R}^4, \] where \(\delta (x,t)\) is Dirac's delta-function supported at the point \((0,0)\). Let \(D\subset B(x^0,R)\) \(=\{x\in {\mathbb R}^3: |x^0-x|<R\}\), \(S_T=\{(x,t):\;x\in\partial D,|x|<t<T+|x|\}\), and \(|x^0|> R\). Suppose that the traces of the function \(u(x,t)\) and its first-order derivatives are known on the surface \(S_T\): \[ D^\alpha u\left|_{S_T}=f^\alpha (x,t),\right. \quad \alpha=(\alpha_1,\alpha_2, \alpha_3, \alpha_4), \quad |\alpha|\leq 1. \] The inverse problem is to reconstruct the coefficient \(q(x)\) for \(x\in D\). The author establishes stability estimates for \(q(x)\) and a uniqueness theorem for a solution to the inverse problem. In particular, the conditions guaranteeing stability are \(|x^0|>\sqrt{2}R\), \(4R/T\in(0,1)\), or a smallness condition on the number \(R\).
scientific article; zbMATH DE number 1431678