# Interfaces and Free Boundaries

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**Volume 15, Issue 2, 2013, pp. 167–180**

**DOI: 10.4171/IFB/299**

Published online: 2013-09-17

A free-boundary problem for concrete carbonation: Front nucleation and rigorous justification of the $\sqrt{{t}}$-law of propagation

Toyohiko Aiki^{[1]}and Adrian Muntean (1) Japan's Women's University, Tokyo, Japan

We study a one-dimensional free-boundary problem describing the penetration of carbonation fronts (free reaction-triggered interfaces) in concrete. Using suitable integral estimates for the free boundary and involved concentrations, we reach a twofold aim:

(1) We fill a fundamental gap by justifying rigorously the experimentally guessed $\sqrt{t}$ asymptotic behavior. Previously we obtained the upper bound $s(t)\leq C'\sqrt{t}$ for some constant $C'$; now we show the optimality of the rate by proving the right nontrivial lower estimate, i.e. there exists $C''>0$ such that $s(t)\geq C''\sqrt{t}$.

(2) We obtain weak solutions to the free-boundary problem for the case when the measure of the initial domain vanishes. In this way, we allow for the {\em nucleation of the moving carbonation front} – a scenario that until now was open from the mathematical analysis point of view.

*Keywords: *Large-time behavior; free-boundary problem; concrete carbonation; integral estimates

Aiki Toyohiko, Muntean Adrian: A free-boundary problem for concrete carbonation: Front nucleation and rigorous justification of the $\sqrt{{t}}$-law of propagation. *Interfaces Free Bound.* 15 (2013), 167-180. doi: 10.4171/IFB/299