2017/06/16 by Winkler, Michael
#35B40 #35B44 (primary) #35D30 #35K65 #92C17 (secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1706.05211
We consider the system 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 ∫Ω(1)/(d)0, the obtained solution satisfies \[ u(⋅,t)\rightharpoonup (μ_∞)/(d) in L1(Ω) and w(⋅,t) → 0 in 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 (1)/(d) belong to Llog L(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 (⋆) in fact occur instantaneously.