Rigorous numerical inclusion of the blow-up time for the Fujita-type equation
Rigorous numerical inclusion of the blow-up time for the Fujita-type equation
复制标题
DOI：
10.1007/s13160-022-00545-8
复制
发表时间：
2022-11
影响因子：
0.9
通讯作者：
Makoto Mizuguchi;Kouta Sekine;K. Hashimoto;M. Nakao;S. Oishi
中科院分区：
文献类型：
--
作者：
Makoto Mizuguchi;Kouta Sekine;K. Hashimoto;M. Nakao;S. Oishi
多项研究已经解决了Fujita型方程的爆破时间。然而，一个明确的和尖锐的包含方法，解决这个问题仍然是失踪，由于几个具有挑战性的问题。在本文中，我们提出了一种方法，用于获得一个可计算的和数学上严格的包含的fujita型方程的解的爆破时间的初始和Dirichlet边界条件，使用数值验证方法。更具体地说，我们发展了一种计算机辅助方法，利用数值验证的非线性抛物型方程的解及其能量泛函的估计，证明了有关的解在有限时间内爆破，并对这一时间进行了严格的估计。为了说明我们的方法实际上是如何工作的，我们考虑了Fujita型方程的Dirichlet边界条件和初始函数在一个一维区域，并证明了它在预测blow-up时间的效率。现有的理论不能证明方程的解在.然而，我们的方法表明，该解是爆破解，爆破时间在区间（0.3068，0.317713]。
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 theblow-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 thesense 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 functionin a one-dimensional domainand demonstrate its efficiency in predictingblow-up time. The existing theory cannot prove that the solution of the equation blows up in. However, our proposed method shows that the solution is theblow-up solution and theblow-up time is in the interval (0.3068, 0.317713].