Quasi-variational structure approach to systems with degenerate diffusions

Abstract

We consider the following system consisting of one strongly nonlinear partial differential inclusion, one linear PDE, and one ODE, which describes a tumor invasion phenomenon with a haptotaxis effect and was originally proposed by Chaplain and Anderson in 2003: $$ \begin{equation\*} \begin{cases} u\_t-\nabla \cdot (d\_u (v) \nabla \beta (v{};u)-u \nabla \lambda (v))+\beta (v{};u) \ni 0,\ v\_t=-avw,\ w\_t=d\_w \Delta w -bw+cu. \end{cases} \end{equation\*} $$ This system has two interesting characteristics. One is that the diffusion coefficient $d\_u$ for the unknown function $u$ in the partial differential inclusion depends on the function $v$, which is also unknown in this system. The other one is that the diffusion flux $\nabla \beta (v{};u)$ of $u$ also depends on $v$. Moreover, we are especially interested in the case that $\beta (v{};u)$ is nonsmooth and degenerate under suitable assumptions. These facts make it difficult for us to treat this system mathematically. In order to overcome these mathematical difficulties, we apply the theory of evolution inclusions on the real Hilbert space $V^\*$, the dual space of $V$, with a quasi-variational structure for the inner products, which was established by the author in 2019, and show the existence of time global solutions to the initial-boundary value problem for this system.