Akademik Veri Yönetim Sistemi
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, cilt.44, sa.4, ss.882-904, 2024 (SCI-Expanded)
The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, {u(t) = Delta u - chi(1)del center dot (u/w del w)+ u(a(1) - b(1)u - c(1)v) , x is an element of Omega v(t) = Delta v - chi(2)del center dot (u/w del w) + v(a(2) - b(2)v - c(2)u) , x is an element of Omega (1) 0 = Delta w - mu w + nu u + lambda v, x is an element of Omega 2 partial derivative/partial derivative u = partial derivative v partial derivative n = partial derivative w/partial derivative n = 0 , x is an element of partial derivative Omega , where Omega subset of R-N is a bounded smooth domain, and chi(i), a(i), b(i), c(i) (i = 1 ,2) and mu, nu, lambda are positive constants. In [25], among others, we proved that for any given non-negative initial data u(0) , v(0) is an element of C-0 ((Omega) over bar) with u(0) + v(0) (sic) 0, (1) has a unique globally defined classical solution (u(t, x; u(0), v(0)) , v(t, x; u(0), v(0)) , w(t, x; u(0), v(0))) with u(0, x; u(0), v(0)) = u(0)(x) and v(0 , x; u(0) , v(0)) = v0(x) in any space dimensional setting with any positive constants chi(i) , a(i) , (b)i , c(i) (i = 1 ,2) and mu, nu, lambda. In this paper, we assume that the competition in (1) is weak in the sense that c(1)/b(2) < a(1)/a(2), c(2)/b(1) < a(2)/a(1). Then (1) has a unique positive constant solution (u*, v*, w*), where We obtain some explicit conditions on chi(1) , chi(2) which ensure that the positive constant solution (u*, v*, w*) is globally stable, that is, for any given nonneg-ative initial data u(0) , v(0) is an element of C-0((Omega) over bar) with u(0) (sic) 0 and v(0) (sic) 0, lim(t ->infinity) (parallel to u(t, center dot; u(0), v(0)) - u* parallel to(infinity)+parallel to v(t, center dot; u(0), v(0)) - v* parallel to infinity+parallel to w(t, center dot; u(0), v(0))-w*parallel to(infinity) = 0.