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For the stability of degenerate parabolic equations, the usual boundary value condition may be overdetermined. How to find an optimal partial boundary value condition has been a long-standing problem. In this paper, we develop the weak characteristic function method to establish a reasonable partial boundary value condition, no matter whether the spatial variable domain Ω is bounded or unbounded. By means of the Kruzkov bi-variables method, the stability of entropy solutions dependent on this partial boundary value condition is proved.

Degenerate parabolic equations with partial boundary value conditions