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 $${\mathrm{\partial }}_{t}U=U\{f(U)-dV\},{\mathrm{\partial }}_{t}V={\mathrm{\partial }}_x\{f(U){\mathrm{\partial }}_{x}V\}+rVf(V),$$ where $f(u) = 1-u$andtheparameters$d,r$arepositive\mathrm{.}Denotingby$($\mathcal{U}$,$\mathcal{V}$)$thetravelingwaveprofileandby$($\mathcal{U}$\pm,$\mathcal{V}$\pm)$itsasymptoticstatesat$\pm\infty$, we investigate existence in the regimes i) $d>1$ (homogeneous invasion) : $({\mathcal{U}}_{-},{\mathcal{V}}_{-})=(0,1)$, $({\mathcal{U}}_{+},{\mathcal{V}}_{+})=(1,0)$; ii) $d<1$ (heterogeneous invasion) : $($\mathcal{U}$-,$\mathcal{V}$-) = (1-d,1)$,$($\mathcal{U}$+,$\mathcal{V}$+) = (1,0)$. 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 $c\to 0$.