2022/10/12 by Sebastian Ohrem, Ohrem, Sebastian, Wolfgang Reichel +3
Engineering · Mathematics · #35L20 #35Q61 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 35L05 #Secondary: 35Q60 #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2210.06383
openalex publication_date 2022/10/12 · openalex created_date 2022/10/14 · openalex updated_date 2026/07/28
We consider the linear wave equation V(x) utt(x, t) - uxx(x, t) = 0 on [0, ∞)×[0, ∞) with initial conditions and a nonlinear Neumann boundary condition ux(0, t) = (f(ut(0,t)))t at x=0. This problem is an exact reduction of a nonlinear Maxwell problem in electrodynamics. In the case where f\colonℝ→ℝ is an increasing homeomorphism we study global existence, uniqueness and wellposedness of the initial value problem by the method of characteristics and fixed point methods. We also prove conservation of energy and momentum and discuss why there is no wellposedness in the case where f is a decreasing homeomorphism. Finally we show that previously known time-periodic, spatially localized solutions (breathers) of the wave equation with the nonlinear Neumann boundary condition at x=0 have enough regularity to solve the initial value problem with their own initial data.