Blow-up results of a time fractional heat equation with a nonlinear Neumann boundary condition

Hameed H. G., SELÇUK B., Rasheed M. A.

Tamkang Journal of Mathematics, vol.57, no.1, pp.51-62, 2026 (ESCI, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 57 Issue: 1
  • Publication Date: 2026
  • Doi Number: 10.5556/j.tkjm.57.2026.5910
  • Journal Name: Tamkang Journal of Mathematics
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, MathSciNet, zbMATH
  • Page Numbers: pp.51-62
  • Keywords: blow up, Fractional heat equation, maximum principles, nonlinear boundary condition
  • Ankara University Affiliated: Yes

Abstract

The study of blow-up phenomena in fractional diffusion equations is of great interest due to its numerous applications and the fact that these types of problems are encountered in several areas of science and engineering. This article is concerned with the blow-up solutions of a one-dimensional time-fractional heat equation, where the time derivative is defined in the sense of the Caputo fractional formula, subject to a nonlinear Neumann boundary condition of a power-type function. Firstly, global existence and blow-up are studied. Under a restricted condition on the nonlinear boundary term, it is proved that every positive solution blows up in finite time; otherwise, positive solutions are continued globally. Secondly, we prove that the blow-up phenomenon can occur only on the boundary.