BOUNDEDNESS OF A CHEMOTAXIS-CONVECTION MODEL DESCRIBING TUMOR-INDUCED ANGIOGENESIS

This paper is concerned with the parabolic-parabolic-elliptic system {u(t) = Delta u - chi del . (u del v) + xi(1)del . (u(m)del w), x is an element of Omega, t > 0, {v(t) = Delta v - xi(2)del . (v del w) + u - v, x is an element of Omega, t > 0, 0 = Delta w + u - 1/vertical bar Omega vertical bar integral(Omega)u, integral(Omega)w = 0, x is an element of Omega, t > 0, partial derivative u/partial derivative nu = partial derivative v/partial derivative nu = 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 domain Omega subset of R-n with a smooth boundary, where the parameters chi, xi(1), xi(2) are positive constants and m >= 1. Based on the coupled energy estimates, the boundedness of the global classical solution is established in any dimensions (n >= 1) provided that m > 1.