A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II

In this paper we continue with our work in Lederman and Wolanski (Ann Math Pura Appl 187(2):197–220, 2008) where we developed a local monotonicity formula for solutions to an inhomogeneous singular perturbation problem of interest in combustion theory. There we proved local monotonicity formulae for solutions u^e{{u^\varepsilon}} to the singular perturbation problem and for u=limu^e{u=\lim{u^\varepsilon}} , assuming that both u^e{{u^\varepsilon}} and u were defined in an arbitrary domain D{\mathcal{D}} in \mathbbR^N+1{\mathbb{R}^{N+1}} . In the present work we obtain global monotonicity formulae for limit functions u that are globally defined, while u^e{{u^\varepsilon}} are not. We derive such global formulae from a local one that we prove here. In particular, we obtain a global monotonicity formula for blow up limits u ₀ of limit functions u that are not globally defined. As a consequence of this formula, we characterize blow up limits u ₀ in terms of the value of a density at the blow up point. We also present applications of the results in this paper to the study of the regularity of ∂{u > 0} (the flame front in combustion models). The fact that our results hold for the inhomogeneous singular perturbation problem allows a very wide applicability, for instance to problems with nonlocal diffusion and/or transport.