Well-posedness and finite-time extinction of a PDE-ODE spatial-network model with anisotropic diffusion

Abstract

We study a system of reaction-diffusion equations posed on a bounded domain composed of subdomains separated by a connected network with a metric graph structure. The reaction-diffusion dynamics with anisotropic diffusion on the graph edges are coupled to well-mixed ODE dynamics occurring at the vertices by junction conditions, and to similar PDE dynamics occurring on adjacent subdomains through Robin-like boundary conditions. The resulting PDE-ODE system can be used in epidemiological and ecological settings to study population movement in between cluster centers along road-like structures and into the surrounding continuum. We employ a semi-Galerkin approximation to establish the well-posedness of weak solutions to the PDE-ODE system, and examine further properties such as regularity, boundedness and finite-time extinction.

Figures (4)

• Figure 1: The domain $\Omega$ consists of three subdomains, separated by nine edges that meet at seven vertices. The external boundary is comprised of edges $e_1$, $e_2$, $e_3$, $e_4$ and $e_5$. In this example, the edges adjacent to $\Omega_1$ are $e_1$, $e_6$ and $e_9$, while the edges that share vertex $v_1$ as one of their endpoints are $e_6$, $e_7$ and $e_9$.
• Figure 2: Balance of fluxes at vertex $v_1$ from Figure \ref{['fig:domain']}.
• Figure 3: Schematics of the setting for the formal derivation of equations on edges: The region $E^\delta$ formally shrinks to the $x_1$-axis $\Gamma$ and becomes an edge separating the upper and lower half plane. The unit normals $\bm{n}$ on the dashed interfaces representing $\{x_2 = \pm \frac{\delta}{2}\}$ are oriented so that they point into $E^\delta$.
• Figure 4: Schematics of the setting for the formal derivation of equations on vertices: The shaded dashed interval $V^\delta$ formally shrinks to the origin and becomes a vertex connecting the left and right edges.

Theorems & Definitions (10)

• Remark 2.1
• Example 2.1
• Definition 3.1
• Theorem 3.1: Well-posedness of weak solutions
• Theorem 3.2: Temporal regularity under symmetric inter-edge transfer coefficients
• Remark 3.1
• Theorem 3.3: Spatial regularity in the unpopulated vertices setting
• Theorem 3.4: Finite time extinction under symmetric coefficients and unpopulated vertices
• Theorem B.1: Interior regularity, Theorem 5.2 of GKW
• Theorem B.2: Boundary regularity, Theorem 5.4 of GKW