Existence of solutions and stability analysis for a Darcy flow with extraction

Competing through altering the environment: A cross-diffusion population model coupled to transport–Darcy flow equations

“…We assume the subsurface region, Ω ⊂ R 3 , to be an open and bounded set that, after the introduction of dimensionless variables, see [11], takes the form Ω = Γ D ×(0, 1). We denote a point in Ω by x = (x, y, z), z being the depth.…”

“…In [11] we deduced the following dimensionless model. Let c ∈ [0, 1] be the salt concentration, q the water flow discharge and p the pressure, and consider the domain…”

“…The positive parameter R is a Rayleigh number. The usual example considered in the literature [8,11] for the extraction function is of the form…”

“…However, only general assumptions are needed to prove the existence of solutions, see [11], which read…”

“…Passioura et al [8] provided an analytical approach to the mechanisms of soil salinization produced by mangroves and investigated the consequences of salt concentration increase on mangroves' transpiration rate. In their work, later generalized in a series of papers [9][10][11], the authors presented one-dimensional (depth) steady state equations governing the flow of salt and uptake of water in the root zone, for one mangrove species, assuming that there is an upper limit c 1 to the salt concentration at which roots can take up water, and that the rate of uptake of water is proportional to the difference between the local concentration c and the assumed upper limit c 1 . The general time-dependent N-dimensional model [11] reads…”