Boundedness in a quasilinear chemotaxis–haptotaxis system with logistic source

In this paper, we consider the quasilinear chemotaxis–haptotaxis system (Formula presented.) in a bounded smooth domain (Formula presented.) under zero-flux boundary conditions, where the nonlinearities (Formula presented.) and (Formula presented.) are assumed to generalize the prototypes(Formula presented.)with (Formula presented.) and (Formula presented.) fulfills(Formula presented.),where (Formula presented.) Assuming nonnegative initial data (Formula presented.) and (Formula presented.) for some (Formula presented.) we prove that (i) for (Formula presented.) if (Formula presented.) then (Formula presented.) has a unique nonnegative classical solution which is globally bounded, (ii) for (Formula presented.) if (Formula presented.) and (Formula presented.) or (Formula presented.) and (Formula presented.) then (Formula presented.) has a unique nonnegative classical solution which is globally bounded.