A free-boundary problem for concrete carbonation : rigorous justification of the $\sqrt t$-law of propagation

We study a one-dimensional free-boundary problem describing the penetration of carbonation fronts (free reaction-triggered interfaces) in concrete. A couple of decades ago, it was observed experimentally that the penetration depth versus time curve (say s(t) vs. t) behaves like $s(t) = C \sqrt t$ for sufficiently large times t > 0 (with C a positive constant). Consequently, many ??fitting arguments solely based on this experimental law were used to predict the large-time behavior of carbonation fronts in real structures, a theoretical justification of the $\sqrt t$-law being lacking until now. The aim of this paper is to fi??ll this gap by justifying rigorously the experimentally guessed asymptotic behavior. We have previously proven 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 $s(t) \geq C'' \sqrt t$. Additionally, we obtain weak solutions to the free-boundary problem for the case when the measure of the initial domain vanishes. In this way, our mathematical model is now allowing for the appearance of a moving carbonation front -- a scenario that until was hard to handle from the analysis point of view.