Rigorous numerical inclusion of the blow-up time for the Fujita-type equation

Makoto Mizuguchi*, Kouta Sekine, Kouji Hashimoto, Mitsuhiro T. Nakao, Shin’ichi Oishi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Multiple studies have addressed the blow-up time of the Fujita-type equation. However, an explicit and sharp inclusion method that tackles this problem is still missing due to several challenging issues. In this paper, we propose a method for obtaining a computable and mathematically rigorous inclusion of the L2(Ω) blow-up time of a solution to the Fujita-type equation subject to initial and Dirichlet boundary conditions using a numerical verification method. More specifically, we develop a computer-assisted method, by using the numerically verified solution for nonlinear parabolic equations and its estimation of the energy functional, which proves that the concerned solution blows up in the L2(Ω) sense in finite time with a rigorous estimation of this time. To illustrate how our method actually works, we consider the Fujita-type equation with Dirichlet boundary conditions and the initial function u(0,x)=1925x(x-1)(x2-x-1) in a one-dimensional domain Ω and demonstrate its efficiency in predicting L2(Ω) blow-up time. The existing theory cannot prove that the solution of the equation blows up in L2(Ω). However, our proposed method shows that the solution is the L2(Ω) blow-up solution and the L2(Ω) blow-up time is in the interval (0.3068, 0.317713].

Original languageEnglish
JournalJapan Journal of Industrial and Applied Mathematics
DOIs
Publication statusAccepted/In press - 2022
Externally publishedYes

Keywords

  • Blow-up time
  • Computer-assisted proof
  • Fujita-type equation
  • Numerical verification method

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics

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