Nonlinear Analysis: Real World Applications 21 (2015) 1–12 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa Global existence for a 1D parabolic–elliptic model for chemical aggression in permeable materials Giuseppe Alì a,b , Roberto Natalini c , Isabella Torcicollo d,∗ a Dipartimento di Matematica, Università della Calabria, via P. Bucci 30/B, Arcavacata di Rende I-87036, Cosenza, Italy b INFN, Gruppo Collegato di Cosenza, Arcavacata di Rende I-87036, Cosenza, Italy c IAC, Consiglio Nazionale delle Ricerche, via dei Taurini, 19, I-00185, Roma, Italy d IAC, Sez. di Napoli, Consiglio Nazionale delle Ricerche, via P. Castellino 111, I-80131, Napoli, Italy article info Article history: Received 22 July 2013 Received in revised form 23 May 2014 Accepted 26 May 2014 Available online 20 June 2014 Keywords: Reaction–diffusion Existence and uniqueness of solutions Porous media Convective and diffusive flows Chemical reactions Carbonate rocks abstract We prove the global existence and uniqueness of smooth solutions to a nonlinear system of parabolic–elliptic equations, which describes the chemical aggression of a permeable ma- terial, like calcium carbonate rocks, in the presence of acid atmosphere. This model applies when convective flows are not negligible, due to the high permeability of the material. The global (in time) result is proven by using a weak continuation principle for the local solu- tions. © 2014 Elsevier Ltd. All rights reserved. 1. Introduction We consider the analytical study of the 1D version of a reaction–diffusion system of the form  ∂ t (ϕ(x, c )s) = div(D(x, c , v)∇s − sv) + F (x, s, c ), ∂ t c = F (x, s, c ), div v = µF (x, s, c ) − ∂ t ϕ(x, c ), (1.1) for two concentrations of reactants, s, c , and a pressure p, where the velocity v is given by v =−κ(x, c )∇p. (1.2) The model (1.1)–(1.2) was introduced in [1,2]. The first reactant, with concentration s, is a fluid or gaseous species, which sat- isfies a modified porous media equation, with several distinguished features. To start with, the porosity ϕ is not constant in time, since it may depend also on the concentration of the second reactant, c , which corresponds to a solid species. Moreover, the equation incorporates two components for the velocity: a diffusive term proportional to the gradient of s, with diffusivity D which may depend on c and on the velocity v of the fluid where the fluid species lives, and a drift term proportional to v. The velocity v is proportional to the gradient of the fluid pressure, according to a Darcy law, with permeability κ which may de- pend on the concentration c . Finally, the term F is a reaction term which describes the chemical interaction between s and c . ∗ Corresponding author. Tel.: +39 0816132388; fax: +39 0816132597. E-mail addresses: giuseppe.ali@unical.it (G. Alì), roberto.natalini@cnr.it (R. Natalini), i.torcicollo@iac.cnr.it, i.torcicollo@cnr.it (I. Torcicollo). http://dx.doi.org/10.1016/j.nonrwa.2014.05.006 1468-1218/© 2014 Elsevier Ltd. All rights reserved.