Investigation of mathematical models of one-phase Stefan problems with unknown nonlinear coefficients

One-phase models of inverse Stefan problems with unknown temperature-dependent convection coefficients are considered. The final observation is considered as an additional information on the solution of the direct Stefan problem. For such inverse problems we justify the corresponding mathematical statements allowing to determine coefficients multiplying the lowest order derivatives in quasilinear parabolic equations in a one-phase domain with an unknown moving boundary. On the basis of the duality principle conditions for the uniqueness of their smooth solution are obtained. The proposed approach allows one to clarity a relationship between the uniqueness property for coefficient inverse Stefan problems and the density property of solutions of the corresponding adjoint problems. It is shown that this density property follows, in turn, from the known inverse uniqueness for linear parabolic equations.