We consider the system(0.1){ut=(d(x)u)xx−(d(x)uwx)x,wt=−ug(w), which arises as a simple model for haptotactic migration in heterogeneous environments, such as typically occurring in the invasive dynamics of glioma. A particular focus is on situations when the diffusion herein is degenerate in the sense that the zero set of d is not empty.It is shown that if such possibly present degeneracies are sufficiently mild in the sense that(0.2)∫Ω1d<∞, then under appropriate assumptions on the initial data a corresponding initial-boundary value problem for (0.1), posed under no-flux boundary conditions in a bounded open interval Ω⊂R, possesses at least one globally defined generalized solution.Moreover, despite such degeneracies the myopic diffusion mechanism in (0.1) is seen to asymptotically determine the solution behavior in the sense that for some constant μ∞>0, the obtained solution satisfies(0.3)u(⋅,t)⇀μ∞din L1(Ω)andw(⋅,t)→0in L∞(Ω)as t→∞, and that hence in the degenerate case the solution component u stabilizes toward a state involving infinite densities, which is in good accordance with experimentally observed phenomena of cell aggregation.Finally, under slightly stronger hypotheses inter alia requiring that 1d belong to LlogL(Ω), a substantial effect of diffusion is shown to appear already immediately by proving that for a.e. t>0, the quantity ln(du(⋅,t)) is bounded in Ω. In degenerate situations, this particularly implies that the blow-up phenomena expressed in (0.3) in fact occur instantaneously.