Beyond the classical strong maximum principle: Forcing changing sign near the boundary and flat solutions

We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on boundary domain $ \Omega can be extended, under suitable conditions, to case in which forcing term f(x) is changing sign. In addition, for solutions normal derivative may also vanish (definition flat solutions). This leads examples unique continuation property fails. As a first application, we existence sublinear semilinear problem indefinite A second positivity heat equation, some large values time, with and/or initial datum sign given.