Rigorous numerical inclusion of the blow-up time for the Fujita-type equation. (English) Zbl 1505.35055
Summary: 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 \(L^2 (\varOmega)\) 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 \(L^2 (\varOmega)\) 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 \(u(0,x)=\frac{192}{5}x(x-1)(x^2 -x-1)\) in a one-dimensional domain \(\varOmega\) and demonstrate its efficiency in predicting \(L^2 (\varOmega)\) blow-up time. The existing theory cannot prove that the solution of the equation blows up in \(L^2 (\varOmega)\). However, our proposed method shows that the solution is the \(L^2 (\varOmega)\) blow-up solution and the \(L^2 (\varOmega)\) blow-up time is in the interval \((0.3068, 0.317713]\).
MSC:
| 35B44 | Blow-up in context of PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35K58 | Semilinear parabolic equations |
| 35K91 | Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Software:
INTLABReferences:
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