Concentration phenomena on \(Y\)-shaped metric graph for the Gierer-Meinhardt model with heterogeneity (Q2227849)

The author studies stationary solutions to the Gierer-Meinhardt model with heterogeneity on the \(Y\)-shaped metric graph \(\mathcal{G}=(V,E)\), \begin{align*} & -\varepsilon^2U''=-U+g_1(x) \frac{U^2}{V}, \quad x\in \mathcal{G}, \\ & -DV''=-V+g_2(x)U^2, \quad \ x\in \mathcal{G},\\ & \sum_{e\succ v}U_e'(v)=\sum_{e\succ v}V_e'(v)=0, \quad v\in V, \tag{1} \end{align*} where \(U\) and \(V\) are continuous functions on \(\mathcal{G}\), and are \(C^2\) on each edge \(e\in E\). Heterogeneous environments are provided by the positive continuous functions \(g_1(x)\) and \(g_2(x)\) on \(\mathcal{G}\), which are \(C^3\) on each \(e\in E\). Here, \(e\succ v\) means that \(e\in E\) is incident at \(v\in V\), and \(\eta_e'(v)\) denotes the differentiation at \(v\) to the outer direction concerning \(e\). Then, a criterion to \(g_1(x)\) and \(g_2(x)\) is shown for the existence of the one-peak solution.