Abstract: In this paper, the artificial boundary method and the optimal control theory are adopted to investigate the forward problem of the semi-infinite degenerate parabolic equation and the inverse problem of determining the source term under additional conditions. First, for the forward problem, the artificial boundary method is used to transform the problem defined on a semi-infinite domain into an equivalent problem on a bounded domain, and  exact boundary conditions at the right boundary are established. Then, the existence and uniqueness of the weak solution are demonstrated through  energy estimates. Second, based on the optimal control theory, the original inverse problem is transformed into an optimal control problem, and the existence of an optimal solution and the necessary conditions it satisfies are proved. Finally, the stability of the optimal solution is proven by using the necessary conditions. This work provides both theoretical analysis and numerical solution support for the forward and inverse problems of such equations.

Key words: semi-infinite, degenerate parabolic equation, optimal control, necessary condition, well-posedness