Propagation fronts in a simplified model of tumor growth with degenerate cross-dependent self-diffusivity. (February 2022) - Gallay, Thierry; Mascia, Corrado
Abstract: 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 ∂ t U = U f ( U ) − d V, ∂ t V = ∂ x f ( U ) ∂ x V + r V 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 .