A multidimensional inverse problem for an equation with memory (Q1920769)

The author considers the problem \[ u_{zz}-u_{xx}-\Delta u=\int^t_0 k(\tau,x)u(x,z,t-\tau)d\tau,\;(x,t)\in\mathbb{R}^{n+1},\;z>0,\tag{1} \] \[ u|_{t<0}\equiv 0,\;u_z|_{z=0}=\delta'(t)+h(x,t)\theta(t),\;(x,t)\in\mathbb{R}^n,\tag{2} \] \[ u|_{z=0}=-\delta(t)+f(x,t)\theta(t),\;(x,t)\in\mathbb{R}^{n+1},\tag{3} \] with unknowns \(u(x,z,t)\) and \(k(x,t)\). Here \(\Delta\) is the Laplace operator in \((x_1,x_2,\dots,x_n)=x\), \(\delta'\) is the derivative of the Dirac function, \(\theta(t)\) is the Heaviside function; \(h\) and \(f\) are given smooth functions. A system of integro-differential equations for \(u\) and \(k\) is obtained and the existence and uniqueness of a solution in a class of analytical functions in the variable \(x\) is proved. A stability property and a global uniqueness theorem are also proved.