Solution regularity and smooth dependence for abstract equations and applications to hyperbolic PDEs
Abstract
First we present a generalized implicit function theorem for abstract equations of the type F (λ , u) = 0. We suppose that u_{0} is a solution for λ = 0 and that F (λ , ṡ) is smooth for all λ, but, mainly, we do not suppose that F (ṡ , u) is smooth for all u. We state conditions such that for all λ ≈ 0 there exists exactly one solution u ≈u_{0}, that u is smooth in a certain abstract sense, and that the datatosolution map λ ↦ u is smooth. Then we apply this to timeperiodic solutions of firstorder hyperbolic systems
 Publication:

Journal of Differential Equations
 Pub Date:
 December 2015
 DOI:
 10.1016/j.jde.2015.07.029
 arXiv:
 arXiv:1411.5562
 Bibcode:
 2015JDE...259.6287K
 Keywords:

 Generalized implicit function theorem;
 Nonlinear firstorder and secondorder hyperbolic PDEs;
 Boundary value problems;
 Timeperiodic solutions;
 Mathematics  Analysis of PDEs
 EPrint:
 48 pages