A Stefan problem for a non-classical heat equation with a convective condition

Keywords: Stefan problem Non-classical heat equation Free boundary problem Similarity solution Nonlinear heat sources Volterra integral equation a b s t r a c t We prove the existence and uniqueness, local in time, of the solution of a one-phase Stefan problem for a non-classical heat equation for a semi-infinite material with a convective boundary condition at the fixed face x = 0. Here the heat source depends on the temperature at the fixed face x = 0 that provides a heating or cooling effect depending on the properties of the source term. We use the Friedman–Rubinstein integral representation method and the Banach contraction theorem in order to solve an equivalent system of two Volterra integral equations. We also obtain a comparison result of the solution (the temperature and the free boundary) with respect to the one corresponding with null source term. The one-phase Stefan problem for a semi-infinite material for the classical heat equation requires the determination of the temperature distribution u of the liquid phase (melting problem) or of the solid phase (solidification problem), and the evolution of the free boundary x = s(t). Phase-change problems appear frequently in industrial processes and in other problems of technological interest [1,2,6,8–12,18,21]. A large bibliography on the subject was given in [28]. Motivated by [30] the free boundary problem which we want to consider consists in determining the temperature u = u(x, t) and the free boundary x = s(t) which satisfy the following conditions