Asymptotic behavior of a three-dimensional haptotactic cross-diffusion system modeling oncolytic virotherapy

This paper deals with an initial-boundary value problem for a doubly haptotactic cross-diffusion system arising from the oncolytic virotherapy ut = u - (uv) + μu(1 - u) - uz,vt = -(u + w)v,wt = w - (wv) - w + uz,zt = Dzz - z - uz + βw, in a smoothly bounded domain ω 3 with β > 0, μ > 0 and Dz > 0. Based on a self-map argument, it is shown that under the assumption βmax{1,u0L∞(ω)} < 1 + (1 +1 minx ωu0(x))-1, this problem possesses a uniquely determined global classical solution (u,v,w,z) for certain type of small data (u0,v0,w0,z0). Moreover, (u,v,w,z) is globally bounded and exponentially stabilizes toward its spatially homogeneous equilibrium (1, 0, 0, 0) as t →∞.