ScholarMate
WeChat
Sina
Facebook
Share link
Summary
In this work, we study the following Neumann-initial boundary value problem for a three-component chemotaxis model describing tumor angiogenesis: @@@ {u(t) = Delta u-chi del center dot(u del v )+ xi(1)del center dot (u del w)+ u(a - mu u(theta)), x is an element of Omega, t > 0, @@@ v(t) = d Delta v + xi(2)del center dot (v del w)+ u- v, x is an element of Omega, t > 0, @@@ 0 = Delta w + u-(u) over bar, integral(Omega) w= 0, (u) over bar:= 1/vertical bar Omega vertical bar integral(Omega) u, x is an element of Omega, t > 0 @@@ partial derivative u/partial derivative v = partial derivative u/partial derivative v = partial derivative w/partial derivative v = 0, x is an element of partial derivative Omega, t > 0 @@@ u(x, 0) = u(0)(x) v(x, 0) = v(0)(x) x is an element of Omega, @@@ in a bounded smooth but not necessarily convex domain Omega subset of R-n(n >= 2) with model parameters xi(1), xi(2), d, theta > 0, a, chi, mu >= 0. Based on subtle energy estimates, we first identify two positive constants xi(0) and mu(0) such that the above problem allows only global classical solutions with qualitative bounds provided one of the following conditions holds: @@@ (1) xi(1) >= xi 0 chi(2); (2) theta = 1, mu >= max {1, chi(8+2n/5+n)} mu(0)chi(2/5+n); (3) theta > 1, mu > 0. @@@ Then, due to the obtained qualitative bounds, upon deriving higher order gradient estimates, we show exponential convergence of bounded solutions to the spatially homogeneous equilibrium (i) for mu large if mu> 0, (ii) for d large if a = mu= 0 and (iii) for merely d > 0 if chi = a = mu = 0. As a direct consequence of our findings, all solutions to the above system with chi = a = mu = 0 are globally bounded and they converge to constant equilibrium, and therefore, no patterns can arise.
