Long-time asymptotic shape of the tumor region

Characterize the limit shape of the tumor region as time tends to infinity for the tumor growth model with nutrients, including the effects of thin-finger growth and nutrient-poor regions left behind by the expanding tumor.

Background

The paper’s principal results concern finite-time singularities and space-time regularity of the Hele-Shaw free boundary. The authors emphasize that these results do not determine the eventual tumor region or its geometry as time becomes large.

They prove a lower volume-growth estimate showing that the tumor region cannot remain bounded, but leave unresolved the finer asymptotic geometry. In particular, nutrient consumption may permit growth through thin fingers while producing nutrient-poor regions that are not subsequently invaded.

References

Global problems, such as the characterization of the limit shape of the tumor patch as $t\to\infty$, remain intriguing open questions; we make some initial observations on this matter in Section~\ref{sec:tumor-applications}.

— Space-time geometry of Hele-Shaw flow  (2608.16509 - Collins et al., 17 Aug 2026) in Section 1, subsection “Tumor growth model with nutrients”; Section 8, subsection “A discussion on the size of the tumor region”