On one problem of Kufarev (Q1594401)

The authors announce some results on the solution of the Kufarev problem \[ \partial_tf_t(z)+zh_t(z)\partial_zf_t(z)=0,\quad t\in[0,T],\tag{1} \] \[ f_t(z)=g(z). \] The first statement (Theorem 1) concerns the linear case as given in (1) and represents some exactification and development of the result of P. P. Kufarev. Some corollaries are also given. Two other statements (Theorems 2 and 3) are given for a nonlinear case when \(h_t(z)\) in (1) is replaced by a continuous operator \(P_t(z,f_t)\).