Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type (Q2128869)

The authors study the bulk-surface Gierer-Meinhardt system \[ u_t=\varepsilon^2\Delta u-u+\frac{u^p}{v^q}+\sigma,\ \tau_sv_t=D_s\Delta v-(1+K)v+Kw+\varepsilon^{-1}\frac{u^r}{v^s} \quad \text{on }\Gamma \times (0,T) \] with \[ \tau_sw_t=D_b\Delta w-w \quad \text{in }\Omega\times (0,T), \qquad \left. D_b\frac{\partial w}{\partial \nu}\right\vert_\Gamma=Kv-Kw, \] where \(\Omega\subset \mathbb{R}^n\) is an open set with boundary \(\Gamma=\partial\Omega\). Having global-in-time well-posedness, there arises the weak convergence of the solution to that of the nonlocal surface Gierer-Meinhardt system \[ \begin{aligned} u_t= \varepsilon^2\Delta u-u+\frac{u^p}{v^q}+\sigma,\ \tau_sv_t=D_s\Delta v-(1+K)v+Kw+\varepsilon^{-1}\frac{u^r}{v^s} \quad &\text{on }\Gamma\times (0,T),\\ \tau_b\frac{dw}{dt}=-(1+K\frac{\vert \Gamma\vert}{\vert\Omega\vert})w+\frac{K}{\vert \Omega\vert}\int_\Gamma v \quad &\text{in }(0,T) \end{aligned} \] as \(D_b\rightarrow \infty\).