Motivated by the application of Lyapunov methods to partial differential equations (PDEs), we study functional inequalities form f(I1(u), . ., Ik(u)) ? 0 where f is a polynomial, u any function satisfying prescribed constraints, and I1(u), .,Ik(u) are integral functionals whose integrands polynomial in u, its derivatives, integration variable. We show that such can be strengthened into sufficie...