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

• Published in: Nonlinear analysis: real world applications Vol. 63; p. 103387
• Main Authors: Gallay, Thierry, Mascia, Corrado
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.02.2022; Elsevier
• Subjects: Cross-dependent self-diffusivity; Degenerate diffusion; Mathematics; Reaction–diffusion systems; Singular perturbation; Traveling wave solutions; Degenerate diffusion; Reaction–diffusion systems; Cross-dependent self-diffusivity; Traveling wave solutions; Singular perturbation; 2010 Mathematics Subject Classification. 35C07 35K57 34D10 92-10 Reaction-diffusion systems cross-dependent self-diffusivity traveling wave solutions degenerate diffusion singular perturbation; traveling wave solutions; singular perturbation; cross-dependent self-diffusivity; degenerate diffusion; 2010 Mathematics Subject Classification. 35C07 35K57 34D10 92-10 Reaction-diffusion systems
• ISSN: 1468-1218; 1878-5719
• DOI: 10.1016/j.nonrwa.2021.103387

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=Uf(U)−dV,∂tV=∂xf(U)∂xV+rVf(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.