On a parabolic–elliptic chemotactic system with non-constant chemotactic sensitivity

Abstract

We study a parabolic-elliptic chemotactic system describing the evolution of a population's density "u" and a chemoattractant's concentration "v". The system considers a nonconstant chemotactic sensitivity given by "/(JV -u)", for JV > 0, and a source term of logistic type "Xu(\ -u)". The existence of global bounded classical solutions is proved for any / > 0, JV > 0 and X > 0. By using a comparison argument we analyze the stability of the constant steady state u = \,v = 1, for a range of parameters.-For JV > 1 and NX > 2/, any positive and bounded solution converges to the steady state. -For JV < 1 the steady state is locally asymptotically stable and for /JV < X, the steady state is globally asymptotically stable.

On a competitive system under chemotactic effects with non-local terms

“…Similar comparison arguments have been used in the context of chemotaxis in other papers, see for instance [26,28] or [23]. In [23], the method is applied to a case of non-constant chemotaxis coefficient, defined by x(w) = xo(N -u), which becomes negative for a large concentration of u.…”

Asymptotic stability of a mathematical model of cell population

“…Comparison arguments have been widely used in parabolic systems, see for instance [4][5][6][7]. In the next section we construct a coupled system of ODE's to obtain a sub-and super-solutions of the hyperbolic equation (Lemma 2.4).…”

On a comparison method to reaction-diffusion systems and its applications to chemotaxis

“…Problem 1. First we study a system of type (7), (8) and (9) consisting on three partial differential equations modelling the spatio-temporal behavior of two competitive populations of biological species (u, v), both of which are attracted chemotactically by the same signal substance "w".…”

“…with the boundary conditions given by (8) for any f 1 ∈ L p (0, T : W −1,p (Ω)) and f 2 ∈ L q (0, T : W −1,q (Ω)). Since A i satisfy assumptions (H1)-(H4) we obtain the existence and uniqueness of solutions (u, v) ∈ (L p (0, T : W 1,p (Ω)), L q (0, T : W 1,q (Ω))).…”

“…Comparison methods based on upper and lower solutions have been applied to a large number of reaction-diffusion systems as the extensive literature shows (see for instance [10] and reference therein). Comparison methods based on ODE s systems have already used in the last decades, see for instance [7], [15] and [8] for chemotactic systems of two PDE s and extended to parabolic-parabolic-elliptic chemotactic systems in [16], [9] and [14]. We study a reaction-diffusion parabolic system "weakly coupled" (i.e.…”