Unstable Periodic Solutions of a Parabolic Problem in the Presence of Non-well-ordered Lower and Upper Solutions

We prove the solvability of the Dirichlet-periodic problem for a semilinear parabolic equation, assuming the existence of a lower solution α and an upper solution β, which do not satisfy the ordering condition α⩽β. Our results yield the existence of unstable solutions, localized by means of α and β, requiring weak regularity of the coefficients of the equation.