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 \[ \partial_t U = U\{f(U)-dV\}, \qquad \partial_t V = \partial_x \{f(U)\,\partial_x V\} + r V f(V), \] where $f(u) = 1-u$ and the parameters $d,r$ are positive. Denoting by $(\mathcal{U},\mathcal{V})$ the traveling wave profile and by $(\mathcal{U}_\pm,\mathcal{V}_\pm)$ its asymptotic states at $\pm\infty$, 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 $c \to 0$.