Fujita-Type Blow-Up for Discrete Reaction–Diffusion Equations on Networks

Abstract

This paper is concerned with long-time behaviors of solutions to the reaction–diffusion equations $u_t={\Delta }_{\omega }u+\psi (t)|u{|}^{q-1}u$ with nontrivial and nonnegative initial data. The purpose of this paper is to introduce a critical set $\mathcal{C}(\psi )$ in the following sense: (i) solutions blow up in finite time for $q\in \mathcal{C}(\psi )$; (ii) solutions with small initial data are exponentially decreasing for $q\notin \mathcal{C}(\psi )$. In order to prove the main theorems, we first derive comparison principles for solutions of the equation above, which play an important role throughout this paper. In addition, we finally give some numerical illustrations which exploit the main results.