Formulas in an inverse problem for an evolution equation (Q1357986)

We consider the following inverse problem for a multidimensional evolution equation: find complex-valued functions \[ w(x,t),\;\lambda(x),\;{\partial\lambda \over\partial t} =0, \quad x\in D\subset\mathbb{R}^n,\;a\leq t\leq b, \] such that \[ {\partial w\over \partial t} =A(t)w+ \lambda(x)f(t), \quad w|_{t=a} =w_a(x),\;w|_{t=b} =w_b(x). \] Here \(A(t)\) is a linear operator acting in the variables \(x=(x_1,x_2, \dots, x_n)\) and \(x\) belongs to a domain \(D\) of the Euclidean real space \(\mathbb{R}^n\), \(n\geq 1\). The functions \(f(t)\not\equiv 0\), \(w_a(x)\), and \(w_b(x)\) and the constants \(a\) and \(b\), \(a<b\), are assumed to be known. In the present article, we give formulas for a solution \(w(x,t)\), \(\lambda(x)\) in various function classes.