Boundedness in a two-dimensional chemotaxis-haptotaxis system

Publication date: 28 July 2014

Abstract: This work studies the chemotaxis-haptotaxis system

left{ �egin{array}{ll} u_t= Delta u - chi

abla cdot (u abla v) - xi abla cdot (u

abla w) + mu u(1-u-w), &qquad xin Omega, , t>0, \[1mm] v_t=Delta v-v+u, &qquad xin Omega, , t>0, \[1mm] w_t=-vw, &qquad xin Omega, , t>0, end{array} ight.

in a bounded smooth domain

O m e g a s u b s e t m a t h b b R^{2}

with zero-flux boundary conditions, where the parameters

c h i, x i

and

m u

are assumed to be positive. It is shown that under appropriate regularity assumption on the initial data

(u_{0}, v_{0}, w_{0})

, the corresponding initial-boundary problem possesses a unique classical solution which is global in time and bounded. In addition to coupled estimate techniques, a novel ingredient in the proof is to establish a one-sided pointwise estimate, which connects

D e l t a w

to

v

and thereby enables us to derive useful energy-type inequalities that bypass

w

. However, we note that the approach developed in this paper seems to be confined to the two-dimensional setting.

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