TY - JOUR
T1 - Rigorous numerical inclusion of the blow-up time for the Fujita-type equation
AU - Mizuguchi, Makoto
AU - Sekine, Kouta
AU - Hashimoto, Kouji
AU - Nakao, Mitsuhiro T.
AU - Oishi, Shin’ichi
N1 - Funding Information:
We appreciate editors in this journal and anonymous reviewers’ useful comments for improving quality of this paper. This work was supported by CREST, JST Grant no. JPMJCR14D4, JSPS KAKENHI no. 18K13462, JSPS KAKENHI no. 18K03434, JSPS KAKENHI no. 18K03440, and JSPS KAKENHI no. 21H00998.
Publisher Copyright:
© 2022, The Author(s).
PY - 2022
Y1 - 2022
N2 - 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].
AB - 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].
KW - Blow-up time
KW - Computer-assisted proof
KW - Fujita-type equation
KW - Numerical verification method
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U2 - 10.1007/s13160-022-00545-8
DO - 10.1007/s13160-022-00545-8
M3 - Article
AN - SCOPUS:85141419813
JO - Japan Journal of Industrial and Applied Mathematics
JF - Japan Journal of Industrial and Applied Mathematics
SN - 0916-7005
ER -