科研之友
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摘要
In this paper, we shall study the initial-boundary value problem of a chemotaxis model with signal-dependent diffusion and sensitivity as follows @@@ {u(t) = del . (gamma(v)del u - chi(v)u del v) + alpha uF(w) + theta u - beta u(2), x is an element of Omega, t > 0, v(t) = D Delta v + u - v, x is an element of Omega, t > 0, w(t) = Delta w - uF(w), x is an element of Omega, t > 0, (*) partial derivative u/partial derivative nu = partial derivative v/partial derivative nu = partial derivative w/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), w(x, 0) = w(0)(x), x is an element of Omega, @@@ in a bounded domain Omega subset of R-2 with smooth boundary, where alpha, beta, D are positive constants, theta is an element of R and nu denotes the outward normal vector of partial derivative Omega. The functions chi(v), gamma(v) and F(v) satisfy @@@ (gamma(v), chi(v)) is an element of [C-2 [0, infinity)](2) with gamma(v) > 0, gamma'(v) < 0 and vertical bar chi(v)vertical bar+vertical bar gamma'(v)vertical bar/gamma(v) is bounded; @@@ F(w) is an element of C-1([0, infinity)), F(0) = 0, F(w) > 0 in (0, infinity) and F'(w) > 0 on [0, infinity). We first prove that the existence of globally bounded solution of system (*) based on the method of weighted energy estimates. Moreover, by constructing Lyapunov functional, we show that the solution (u, v, w) will converge to (0, 0, w*) in L-infinity with some w(*) >= 0 as time tends to infinity in the case of theta <= 0, while if theta > 0, the solution (u, v, w) will asymptotically converge to (theta/beta, theta/beta, 0) in L-infinity-norm provided D > max(0 <= v <=infinity) (theta vertical bar chi(v)vertical bar 2)(16 beta 2 gamma(v)).
