JournalsprimsVol. 55, No. 2pp. 235–258

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

  • Soon-Yeong Chung

    Sogang University, Seoul, Republic of Korea
  • Min-Jun Choi

    Sogang University, Seoul, Republic of Korea
  • Jea-Hyun Park

    Kunsan National University, Republic of Korea
Fujita-Type Blow-Up for Discrete Reaction–Diffusion Equations on Networks cover
Download PDF

A subscription is required to access this article.

Abstract

This paper is concerned with long-time behaviors of solutions to the reaction–diffusion equations ut=Δωu+ψ(t)uq1uu_{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 C(ψ)\mathcal{C}(\psi) in the following sense: (i) solutions blow up in finite time for qC(ψ)q\in\mathcal{C}(\psi); (ii) solutions with small initial data are exponentially decreasing for q∉C(ψ)q\not\in\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.

Cite this article

Soon-Yeong Chung, Min-Jun Choi, Jea-Hyun Park, Fujita-Type Blow-Up for Discrete Reaction–Diffusion Equations on Networks. Publ. Res. Inst. Math. Sci. 55 (2019), no. 2, pp. 235–258

DOI 10.4171/PRIMS/55-2-1