Uniqueness of the solution of an inverse problem for the kinetic equation (Q1109238)

We investigate the uniqueness of the solution of an inverse problem for the kinetic equation in the region \[ (1)\quad \ell u+\beta u=g(p,t)\mu (p), \] where \(\ell u\equiv u_ t+<v,\nabla_ xu>+<f,\nabla_ xu>\), \(<, >\) is the scalar product in \(R^ n\), \(f=(f_ 1,...,f_ n).\) Equation (1) is kinetic and used in plasma physics and astrophysics. Problem. In the region Q find a pair of functions (u,\(\mu)\) of (1), under the conditions \(u=\phi_ 1\) on \(\Gamma_-\), \(u_ t(p,0)=\phi_ 2\), \(u(p,T)=\phi_ 3.\) Theorem 1. Under the conditions (1) the functions f, \(\beta\), and g are continuously differentiable on \(\bar Q,\) and f and \(\beta\) do not depend on t; (2) \(f_ 1>0\) and \(g\neq 0\) on \(\bar Q,\) there is only one solution \((u,\mu)\in C^ 2(\bar Q)\times C^ 1(\bar {\mathcal D})\).