Global Classical Solutions, Stability of Constant Equilibria, and Spreading Speeds in Attraction-Repulsion Chemotaxis Systems with Logistic Source on RN

In this paper, we consider the following chemotaxis systems of parabolic- ellipticelliptic type on RN, ut = u -.1.( u. v1) +.2.( u. v2) + u( a - bu), x. RN, t > 0, 0 = ( -.1 I) v1 + mu 1u, x. RN, t > 0, 0 = ( -.2 I) v2 + mu 2u, x. RN, t > 0, u( center dot, 0) = u0, x. RN where.i = 0,.i > 0, mu i > 0 ( i = 1, 2) and a > 0, b > 0 are constant real numbers, and N is a positive integer. First, under some conditions on the parameters.i, mu i,.i, a, b and N, we prove the global existence and boundedness of classical solutions ( u( x, t; u0), v1( x, t; u0), v2( x, t; u0)) for nonnegative, bounded, and uniformly continuous initial functions u0( x). Next, we explore the asymptotic stability of the constant equilibrium ( a b, mu 1.1 a b, mu 2.2 a b) and prove under some further assumption on the parameters that, for every strictly positive initial u0( x), lim t.8 u( center dot, t; u0) - a b 8 + .1v1( center dot, t; u0) - a b mu 1 8 + .2v2( center dot, t; u0) - a b mu 2 8 = 0. Finally, we investigate the spreading properties of the global solutions with compactly supported initial functions. We show that under some conditions on the parameters, there are two positive numbers 0 < c *- (.1, mu 1,.1,.2, mu 2,.2) = c *+ (.1, mu 1,.1,.2, mu 2,.2) such that for every nonnegative initial function u0( x) with nonempty and compact support, we have lim t.8 sup | x|= ct | u( x, t; u0) - a b |+ sup | x|= ct |.1v1( x, t; u0) - a b mu 1|+ sup | x|= ct |.2v2( x, t; u0) - a b mu 2| = 0 whenever 0 = c < c *- (.1, mu 1,.1,.2, mu 2,.2), and lim t.8 sup | x|= ct | u( x, t; u0)| + sup | x|= ct | v1( x, t; u0)| + sup | x|= ct | v2( x, t; u0)| = 0 whenever c > c *+ (.1, mu 1,.1,.2, mu 2,.2). Furthermore we show that lim (.1,.2).( 0,0) c *- (.1, mu 1,.1,.2, mu 2,.2) = lim (.1,.2).( 0,0) c *+ (.1, mu 1,.1,.2, mu 2,.2) = 2 root a.