Stability analysis for Allen-Cahn type equation associated with the total variation energy

In this paper, we shall discuss about the large-time behavior of solutions of an Allen-Cahn type equation generated by the total variation functional with constraints. In the one-dimensional case, the large time behavior of solutions has been studied in (Nonlinear Anal. 46 (2001) 435; Funkcial. Ekvac. 44 (2001) 119; J. Math. Anal. 47 (2001) 3195). According to the results, all steady-state patterns are represented as piecewise constant solutions of the equation, and any stable (steady-state) solution takes only values corresponding to pure phases. In the argument, the authors in (Funkcial. Ekvac. 44 (2001) 119; J. Math. Anal. 47 (2001) 3195) introduced an original concept, named as “local stability”, and discussed the stability of steady-state solutions by means of this concept. The main objective of this paper is to investigate the situation of multi-dimensional solutions. Referring to the results in the one-dimensional case, we target piecewise constant (steady-state) solutions as the object of consideration, and try to extend the theory of local stability to multi-dimensional cases. Consequently, some geometric conditions concerned with the structure of steady-state solutions and the stability will be shown.