# Numerical verification for existence of a global-in-time solution to semilinear parabolic equations

@article{Mizuguchi2017NumericalVF, title={Numerical verification for existence of a global-in-time solution to semilinear parabolic equations}, author={Makoto Mizuguchi and Akitoshi Takayasu and Takayuki Kubo and Shin'ichi Oishi}, journal={J. Comput. Appl. Math.}, year={2017}, volume={315}, pages={1-16} }

## 11 Citations

### A Method of Verified Computations for Solutions to Semilinear Parabolic Equations Using Semigroup Theory

- MathematicsSIAM J. Numer. Anal.
- 2017

A recursive scheme to extend a time interval in which the validity of the solution can be verified and the existence of a global-in-time solution is demonstrated for a certain semilinear parabolic equation.

### Numerical Verification of Solutions for Nonlinear Parabolic Problems

- Mathematics, EngineeringNumerical Functional Analysis and Optimization
- 2020

A numerical verification method of solutions for nonlinear parabolic initial boundary value problems based on Nakao's projection method based on the full-discrete finite element method with constructive error estimates is presented.

### Accurate method of verified computing for solutions of semilinear heat equations

- Mathematics
- 2016

We provide an accurate verification method for solutions of heat equations with a superlinear nonlinearity. The verification method numerically proves the existence and local uniqueness of the exact…

### Rigorous numerics for nonlinear heat equations in the complex plane of time

- MathematicsNumerische Mathematik
- 2022

In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. Using a solution map operator, we…

### Pointwise a Posteriori Error Bounds for Blow-Up in the Semilinear Heat Equation

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2020

Conditional a posteriori error bounds are derived in the first order in time, implicit-explicit (IMEX), conforming finite element method in space discretization of the problem.

### Rigorous numerical computations for 1D advection equations with variable coefficients

- Mathematics, Computer ScienceJapan Journal of Industrial and Applied Mathematics
- 2019

The provided method is regarded as an efficient application of semigroup theory in a sequence space associated with the Fourier series of unknown functions as well as a foundational approach of verified numerical computations for hyperbolic PDEs.

### A Rigorous Implicit $$C^1$$ C 1 Chebyshev Integrator for Delay Equations

- Mathematics, Computer ScienceJournal of Dynamics and Differential Equations
- 2020

We present a new approach to validated numerical integration for systems of delay differential equations. We focus on the case of a single constant delay though the method generalizes to systems with…

### Numerical verification for positive solutions of Allen–Cahn equation using sub- and super-solution method

- MathematicsJournal of Advanced Simulation in Science and Engineering
- 2020

This paper describes a numerical verification method for positive solutions of the Allen–Cahn equation on the basis of the suband super-solution method. Our application range extends to…

### Rigorous FEM for 1D Burgers equation

- Mathematics
- 2020

We propose a method to integrate dissipative PDEs rigorously forward in time with the use of Finite Element Method (FEM). The technique is based on the Galerkin projection on the FEM space and…

### Numerical validation of blow-up solutions of ordinary differential equations

- MathematicsJ. Comput. Appl. Math.
- 2017

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- 2017

A recursive scheme to extend a time interval in which the validity of the solution can be verified and the existence of a global-in-time solution is demonstrated for a certain semilinear parabolic equation.

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For a semilinear parabolic initial boundary value problem we establish criterions on blow-up of the solution in finite time and give bounds for the blow-up time. We treat several applications in both…

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In this paper we give a method by computer-assistance to prove a pattern formation. As a typical model we consider two dimensional time-dependent reaction-diffusion equations with Neumann boundary…