Existence and long-time behavior of global strong solutions to a nonlinear model of tumor growth

TL;DR

This paper proves the existence of global strong solutions for a nonlinear tumor growth model with large viscosity and small initial gradients, and analyzes their long-term behavior using level-set methods.

Contribution

It establishes the global existence of strong solutions for the tumor growth PDE system under certain conditions, advancing the understanding of regularity and long-term dynamics.

Findings

Global strong solutions exist for large viscosity and small initial gradients.

The long-time behavior of solutions is characterized by level-set analysis.

Source terms have a regularizing effect on the transport dynamics.

Abstract

In this manuscript, we study a nonlinear model of tumor growth, described by a coupled hyperbolic-elliptic system of partial differential equations. In this model, the compressible flow of tumor cells is modeled by a transport equation for the cell density, which takes into account transport via a background flow (given by a potential solving a Brinkman-type equation), and which has a source term modeling cell growth and death. In this manuscript, we show that for sufficiently large viscosity, the tumor growth system admits nontrivial global strong solutions for positive initial data having a gradient with sufficiently small norm. This illustrates the regularizing effects of the source term representing tumor cell growth and death on the resulting transport dynamics of the equation. Furthermore, we characterize the long-time behavior of global strong solutions to the tumor growth system using a level-set analysis, in which we analyze how level sets evolve as they are transported by the flow, in terms of expansion/contraction and accretion/depletion of cells. While there has been past work on global existence of weak solutions for this tumor growth system, this manuscript opens the study of well-posedness in terms of more regular strong/classical solutions, which exist globally in time.