JournalsrsmupVol. 147pp. 169–235

Quasi-variational structure approach to systems with degenerate diffusions

  • Akio Ito

    Chiba, Japan
Quasi-variational structure approach to systems with degenerate diffusions cover
Download PDF

A subscription is required to access this article.

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:

{ut(du(v)β(v;u)uλ(v))+β(v;u)0,vt=avw,wt=dwΔwbw+cu.\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 dud_u for the unknown function uu in the partial differential inclusion depends on the function vv, which is also unknown in this system. The other one is that the diffusion flux β(v;u)\nabla \beta (v{};u) of uu also depends on vv. Moreover, we are especially interested in the case that β(v;u)\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 VV^*, the dual space of VV, 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.

Cite this article

Akio Ito, Quasi-variational structure approach to systems with degenerate diffusions. Rend. Sem. Mat. Univ. Padova 147 (2022), pp. 169–235

DOI 10.4171/RSMUP/96