This website requires JavaScript.

A rigorous integrator and global existence for higher-dimensional semilinear parabolic PDEs via semigroup theory

Gabriel William DuchesneJean-Philippe LessardAkitoshi Takayasu
Feb 2024
0被引用
0笔记
摘要原文
In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations via semigroup theory and computer-assisted proofs. Once a numerical candidate for the solution is obtained via a finite dimensional projection, Chebyshev series expansions are used to solve the linearized equations about the approximation from which a solution map operator is constructed. Using the solution operator (which exists from semigroup theory), we define an infinite dimensional contraction operator whose unique fixed point together with its rigorous bounds provide the local inclusion of the solution. Applying this technique for multiple time steps leads to constructive proofs of existence of solutions over long time intervals. As applications, we study the 3D/2D Swift-Hohenberg, where we combine our method with explicit constructions of trapping regions to prove global existence of solutions of initial value problems converging asymptotically to nontrivial equilibria. A second application consists of the 2D Ohta-Kawasaki equation, providing a framework for handling derivatives in nonlinear terms.
展开全部
机器翻译
AI理解论文&经典十问
图表提取
参考文献
发布时间 · 被引用数 · 默认排序
被引用
发布时间 · 被引用数 · 默认排序
社区问答