A class of non-linear non-classical parabolic equations

Abstract In this paper, we deal with a class of non-classical parabolic equations which arise from some practical problems such as the identification of unknown parameters, free boundary problems, etc. One of the characteristics of this kind of equation is that the maximum principle is no longer valid. We will first study the local solvability for a general equation of this kind via an auxiliary problem and the Schauder fixed point principle. Then, some a priori estimates are derived to establish the global existence of the solution. Moreover, uniqueness and continuous dependence of the solution are proved. Finally, some examples are given to illustrate the applications of the theory.