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.

翻译：本文提出了一种基于半群理论与计算机辅助证明的通用构造方法，用于求解半线性抛物型偏微分方程初值问题。通过有限维投影获得解的数值候选后，利用切比雪夫级数展开求解近似解附近的线性化方程，从而构造解映射算子。基于该解算子（由半群理论保证存在），我们定义了一个无穷维压缩算子，其唯一不动点及其严格界限提供了解的局部包含关系。将该技术应用于多个时间步长，可构造性地证明解在长时间区间上的存在性。作为应用实例，我们研究了三维/二维Swift-Hohenberg方程，结合显式陷阱区域构造方法证明了初值问题解全局存在且渐近收敛至非平凡平衡态。另一应用实例为二维Ohta-Kawasaki方程，该工作为处理非线性项中的导数项提供了框架。