On a sharp in Lp a priori estimate for the solution to the heat transfer problem in a rod with heat capacities concentrated at the ends

A mathematical model of heat transfer in a homogeneous rod with heat capacities concentrated at the ends is described. The coefficients of the problem are positive constants and the initial function is in the Hölder class. A spectral problem is obtained while solving a problem by separation of variables. The functions in the mathematical formulas of the analysis are the solutions to the equation with a spectral parameter in the boundary conditions. Under the assumptions made from the mathematical analysis, the classical solution to the problem satisfies the estimate, which is a consequence of the inequalities. The problem generated by the boundary value problem for a parabolic-hyperbolic equation with two type-change lines are also formulated. Two functions with indices of different parity thereby gives a complete and minimal system, while eliminating two functions with indices of the same parity yields a system that is neither complete nor minimal.