A local monotonicity formula for an inhomogenous singular perturbation problem and applications

Abstract

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 $${{u^\varepsilon}}$$ to the singular perturbation problem and for $${u=\lim{u^\varepsilon}}$$ , assuming that both $${{u^\varepsilon}}$$ and u were defined in an arbitrary domain $${\mathcal{D}}$$ in $${\mathbb{R}^{N+1}}$$ . In the present work we obtain global monotonicity formulae for limit functions u that are globally defined, while $${{u^\varepsilon}}$$ 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.