On the Cauchy problem for a one-dimensional loaded parabolic equation of a special form

In this paper, we consider a loaded parabolic equation of a special form in an unbounded domain with Cauchy data. The equation is one-dimensional and its right-hand side depends on the unknown function $u(t,x)$ and traces of this function and its derivatives by the spatial variable at a finite number of different points of space. Such equation appear after the reduction of some identification problems for coefficients of one-dimensional parabolic equations with Cauchy data to auxiliary direct problems. We obtain sufficient conditions of the global solvability and sufficient conditions of the solvability of the problem considered in a small time interval. We search for solutions in the class of sufficiently smooth bounded functions. We examine the uniqueness of the classical solution found and prove the corresponding sufficient conditions. We also obtain an a priori estimate of a solution that guarantees the continuous dependence of the solution on the right-hand side of the equation and the initial conditions.