Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis

Phase field models recently gained a lot of interest in the context of tumour growth models. Typically Darcy-type flow models are coupled to Cahn–Hilliard equations. However, often Stokes or Brinkman flows are more appropriate flow models. We introduce and mathematically analyse a new Cahn–Hilliard–Brinkman model for tumour growth allowing for chemotaxis. Outflow boundary conditions are considered in order not to influence tumour growth by artificial boundary conditions. Existence of global-in-time weak solutions is shown in a very general setting.     

Do some chemotaxis-growth models possess Lyapunov functionals?

In spirit of a result by W. Alt from 1980 we give some sufficient criteria that guarantee the existence of Lyapunov functionals for parabolic cross-diffusion models including chemotaxis-growth models with non-diffusive chemotactic signals (resp. with non-diffusive memory).     

Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source

This paper deals with a mathematical model of cancer invasion of tissue. The model consists of a system of reaction-diffusion-taxis partial differential equations describing interactions between cancer cells, matrix degrading enzymes, and the host tissue. In two space dimensions, we prove global existence and uniqueness of classical solutions to this model for any μ>0 (where μ is the logistic growth rate of cancer cells). The crucial point of proof is to raise the regularity estimate of a solution from L¹(Ω) to L³(Ω×(0,T)) (where Ω⊂R² is some bounded domain and T>0 is some constant). This paper develops new estimate techniques and improves greatly our previous results [Y. Tao, M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity 21 (2008) 2221-2238] in 2 dimensions.     

Asymptotic Behavior of Solution to Some Models Involving Two Species All with Chemotaxis

This paper is concerned with the asymptotic behavior of solution to the following model involving two species all with chemotaxis：{αp/αt=Dp△（p△lnp/w）,αp/αt=Dq△（q△lnq/w）,αw/αt=βp-δw,p△ln（p/w）.^-n=q△ln（q/w）.^-n=0.We prove that the solution exists globally asβ ≥ 0. Asβ 〈 0, whether the solution exists globally or not depends on the initial data. By function transformation and compari- son, the asymptotical behavior of the solution is studied.     

Global existence of weak solutions for the 3D chemotaxis-Navier–Stokes equations

In this paper, we consider the Cauchy problem for the three dimensional chemotaxis-Navier–Stokes equations. By exploring the new a priori estimates, we prove the global existence of weak solutions for the 3D chemotaxis-Navier–Stokes equations.