A quasilinear chemotaxis–haptotaxis model: The roles of nonlinear diffusion and logistic source

Abstract

In this paper, we study the following quasilinear chemotaxis–haptotaxis system urn:x-wiley:mma:media:mma4126:mma4126-math-0001 in a bounded smooth domain under zero‐flux boundary conditions, where the nonlinearities D,S1, and S2 are supposed to generalize the prototypes urn:x-wiley:mma:media:mma4126:mma4126-math-0003 with , and f∈C1([0,+∞) × [0,+∞)) satisfies urn:x-wiley:mma:media:mma4126:mma4126-math-0005 with r > 0 and b > 0. If the nonnegative initial data u0(x)∈W1,∞(Ω),v0(x)∈W1,∞(Ω), and for some α∈(0,1), it is proved that For n = 1, if and then (⋆) has a unique nonnegative classical solution, which is globally bounded. For n = 2, if and then (⋆) has a unique nonnegative classical solution, which is globally bounded. For n≥3, if and then (⋆) has a unique nonnegative classical solution, which is globally bounded. Copyright © 2016 John Wiley & Sons, Ltd.