Free boundary problems arising in the freezing of soils in a bounded region. II: Analytical treatment of the nonlinear parabolic free boundary problem
[For Part I see the preceding review.] This paper treats the nonlinear heat conduction equation subject to the prescribed heat-flux boundary condition which was stated in Part I and referred to as problem I. By adopting Storm's method [\textit{M. L. Storm}, J. Appl. Phys. 22, 940-951 (1951; Zbl 0043.197)] the authors transform this equation subject to the transformed condition on the assumption that physical properties depend on the temperature in the form specified by certain equations. The resulting differential system is then reduced to an equivalent system of integral equations and a sequence of approximating solutions is introduced. Thereafter convergence of this sequence as well as the existence of solutions to the system of integral equations for sufficiently small times is proved. The authors also obtain uniqueness and stability of the solution to the system of integral equations. Lastly they go back to problem I and deduce its well- posedness.
scientific article; zbMATH DE number 3951109