Unstable patterns in reaction-diffusion model of early carcinogenesis (Q390809)

The authors consider the following system for \(x\in [0,1]\) and \(t>0\) NEWLINE\[NEWLINE\begin{aligned} u_t&=\left(\frac{av}{u+v}-d_c\right)u,\\ v_t&=-d_bv+u^2w-dv,\\ w_t&=\frac{1}{\gamma}w_{xx}-d_gw-u^2w+dv+k_0, \end{aligned}NEWLINE\]NEWLINE incorporated with homogeneous Neumann boundary condition and nonnegative initial value condition. They show the global existence of nonnegative solutions, the asymptotic stability of the trivial equilibrium \((0,0,\frac{k_0}{d_g})\). However, this model exhibits diffusion-driven instability (Turing type instability). All the Turing type patterns, i.e. regular stationary solutions, including continuous and discontinuous, are proved to be unstable in the Lyapunov sense.