2018/04/23 by Joseph W. Jerome, Jerome, Joseph W
Engineering · Mathematics · #35L05 #35L20 #47D03 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods for differential equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1804.08394
openalex publication_date 2018/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The damped nonlinear wave equation, also known as the nonlinear telegraph\nequation, is studied within the framework of semigroups and eigenfunction\napproximation. The linear semigroup assumes a central role: it is bounded on\nthe domain of its generator for all time t > 0. This permits eigenfunction\napproximation within the semigroup framework as a tool for the study of weak\nsolutions. The semigroup convolution formula, known to be rigorous on the\ngenerator domain, is extended to the interpretation of weak solution on an\narbitrary time interval. A separate approximation theory can be developed by\nusing the invariance of the semigroup on eigenspaces of the Laplacian as the\nsystem evolves. For (locally) bounded continuous L2 forcing, this permits a\nnatural derivation of a maximal solution, which can logically include a\nconstraint on the solution as well. Operator forcing allows for the\nincorporation of concurrent physical processes. A significant feature of the\nproof in the nonlinear case is verification of successive approximation without\nstandard fixed point analysis.\n