Propagation fronts in a simplified model of tumor growth with degenerate cross-dependent self-diffusivity

Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby–Gawlinski model ∂tU = U {f(U) − dV } , ∂tV = ∂x {f(U) ∂xV } + rV f(V ) , where f(u) = 1 − u and the parameters d, r are positive. Denoting by (U, V) the traveling wave profile and by (U±, V±) its asymptotic states at ±∞, we investigate existence in the regimes d > 1 : (U−, V−)=(0, 1) and (U+, V+)=(1, 0), d < 1 : (U−, V−)=(1 − d, 1) and (U+, V+)=(1, 0), which are called, respectively, homogeneous invasion and heterogeneous invasion. In both cases, we prove that a propagating front exists whenever the speed parameter c is strictly positive. We also derive an accurate approximation of the front profile in the singular limit c → 0.