The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

Abstract We study the singular set in thin obstacle problem for degenerate parabolic equations with weight $$|y|^a$$ |y|a $$a \in (-1,1)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">a?(-1,1) . Such arises as local extension of fractional heat operator $$(\partial _t - \Delta _x)^s$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">(?t-?x)s $$s (0,1)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">s?(0,1) Our main result establishes complete structure and regularity free boundary. To achieve it, we prove Almgren-Poon, Weiss, Monneau type monotonicity formulas which generalize those case equation ( $$a=0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">a=0 ).