On a family of initial-boundary value problems for the heat equation

We consider the heat problem with nonlocal boundary conditions containing a real parameter. For the zero value of the parameter, this problem is well known as the Samarskii-Ionkin problem and has been comprehensively studied. We analyze the spectral problem for the operator of second derivative subjected to the boundary conditions of the original problem. By separation of variables, we prove the existence and uniqueness of a classical solution for any nonzero value of the parameter. The obtained a priori estimates for a solution imply the stability of the problem with respect to the initial data.