Monotone methods for semilinear parabolic and elliptic equations on graphs

TL;DR

This paper studies the extinction and propagation of solutions to certain parabolic and elliptic equations on graphs, establishing maximum principles, stability, and providing numerical demonstrations.

Contribution

It introduces monotone methods for analyzing semilinear parabolic and elliptic equations on graphs, including maximum principles and stability analysis.

Findings

Maximum principle and upper/lower solutions established

Stability of equilibrium solutions demonstrated

Numerical experiments support theoretical results

Abstract

This paper is devoted to investigate the extinction and propagation properties of solutions to the graph Laplacian parabolic problems with Kpp type or Allen-Cahn type forcing terms on graphs. To this end, we establish the (strong) maximum principle and the upper and lower solutions method for parabolic and elliptic problems on graphs. The stability of equilibrium solutions is studied by constructing suitable upper and lower solutions. Moreover, we give an example and numerical experiments to demonstrate one of our main results.