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

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 L-2 (Omega) 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 L-2 (Omega) 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) = 192/5 x(x - 1)(x(2) - x - 1) in a one-dimensional domain Omega and demonstrate its efficiency in predicting L-2 (Omega) blow-up time. The existing theory cannot prove that the solution of the equation blows up in L-2 (Omega). However, our proposed method shows that the solution is the L-2 (Omega) blow-up solution and the L-2 (Omega) blow-up time is in the interval (0.3068, 0.317713].