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

摘要

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)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L^2(varOmega )$$end{document} 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)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L^2(varOmega )$$end{document} 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)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$u(0,x)=frac{192}{5}x(x-1)(x^2-x-1)$$end{document} in a one-dimensional domain Ωdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$varOmega$$end{document} and demonstrate its efficiency in predicting L2(Ω)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L^2(varOmega )$$end{document} blow-up time. The existing theory cannot prove that the solution of the equation blows up in L2(Ω)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L^2(varOmega )$$end{document}. However, our proposed method shows that the solution is the L2(Ω)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L^2(varOmega )$$end{document} blow-up solution and the L2(Ω)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L^2(varOmega )$$end{document} blow-up time is in the interval (0.3068, 0.317713].