Quasi-variational structure approach to systems with degenerate diffusions

Abstract

We consider the following system (CA) consisting of one strongly nonlinear partial differential inclusion (PDI in short) with one linear PDE and one ODE, which describes a tumor invasion phenomenon with a haptotaxis effect and was originally proposed in [1]: (CA)  ut −∇ · (du(v)∇β(v ;w)− u∇λ(v)) + β(v ;u) ∋ 0, vt = −avw, wt = dw∆w − bw + cu. This system has two interesting characteristics. One is that the diffusion coefficient du for the unknown function u in the partial differential inclusion depends on the function v, which is also unknown in this system. The other is that the diffusion flux ∇β(v ;u) of u also depends on v. Moreover, we are especially interested in the case that β(v ;u) is nonsmooth and degenerate in general 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 is established in [8], and show the existence of time global solutions to the initial-boundary value problem for this system. Mathematics Subject Classification (2010). Primary: 34G25; Secondary: 47J35, 49J40, 58E35.