Abstract
We consider the following repulsive-productive chemotaxis model: find u ≥ 0, the cell density, and v ≥ 0, the chemical concentration, satisfying1∂tu−Δu−∇⋅(u∇v)=0inΩ,t>0,∂tv−Δv+v=upinΩ,t>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\left\\{ \\begin{array}{l} \\partial_t u - {\\Delta} u - \\nabla\\cdot (u\\nabla v)=0 \\ \\ \\text{ in}\\ {\\Omega},\\ t>0,\\\\ \\partial_t v - {\\Delta} v + v = u^p \\ \\ { in}\\ {\\Omega},\\ t>0, \\end{array} \\right. $$\\end{document}with p ∈ (1, 2), {Omega }subseteq mathbb {R}^{d} a bounded domain (d = 1, 2, 3), endowed with non-flux boundary conditions. By using a regularization technique, we prove the existence of global in time weak solutions of (1) which is regular and unique for d = 1, 2. Moreover, we propose two fully discrete Finite Element (FE) nonlinear schemes, the first one defined in the variables (u,v) under structured meshes, and the second one by using the auxiliary variable σ = ∇v and defined in general meshes. We prove some unconditional properties for both schemes, such as mass-conservation, solvability, energy-stability and approximated positivity. Finally, we compare the behavior of these schemes with respect to the classical FE backward Euler scheme throughout several numerical simulations and give some conclusions.
Highlights
Chemotaxis is the biological process of the movement of living organisms in response to a chemical stimulus, movement that can be addressed towards a higher or lower concentration of a chemical substance
We prove some unconditional properties for both schemes, such as mass-conservation, solvability, energy-stability and approximated positivity
From the biological point of view, the nonlinear signal production considered in model (3) is justified and explains the saturation effects of chemotactic signal production at large densities of cells
Summary
Chemotaxis is the biological process of the movement of living organisms in response to a chemical stimulus, movement that can be addressed towards a higher (chemo-attraction) or lower (chemorepulsion) concentration of a chemical substance. In the case of linear ( 1) or quadratic ( 2) production term, problem (3) is well-posed (see [9, 17] respectively) in the following sense: there exist global in time weak solutions in 3 domains, which are regular (and unique) for 1 and 2 domains. Energy-stability of time-discrete numerical approximations and fully discrete FE schemes for a chemorepulsion model with quadratic production have been analyzed in [17] and [18, 19], respectively; while, in the case of linear production, we refer [20]. The positivity of only time-discrete schemes and approximated positivity of a fully discrete FE scheme associated with a chemorepulsion model with quadratic signal production were proved in [17] and [19], respectively; while, for the case of linear production, we refer to [20]. By taking 1 in (6): i.e., In addition, integrating (7) in , one has
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