2014/11/20 by Irina Kmit, Kmit, Irina, Lutz Recke +1
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.1411.5556
openalex publication_date 2014/11/20 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
The paper concerns the general linear one-dimensional second-order hyperbolic equation ∂2tu - a2(x,t)∂2xu + a1(x,t)∂tu + a2(x,t)∂xu + a3(x,t)u=f(x,t), x∈(0,1) with periodic conditions in time and Robin boundary conditions in space. Under a non-resonance condition (formulated in terms of the coefficients a, a1, and a2) ruling out the small divisors effect, we prove the Fredholm alternative. Moreover, we show that the solutions have higher regularity if the data have higher regularity and if additional non-resonance conditions are fulfilled. Finally, we state a result about smooth dependence on the data, where perturbations of the coefficient a lead to the known loss of smoothness while perturbations of the coefficients a1, a2, and a3 do not.