Fredholm Solvability of Periodic Neumann Problem for a Linear Telegraph Equation

Abstract

We investigate the linear telegraph equation $$u_{tt}-u_{xx}+2\mu u_t=f(x,t)$$ with periodic Neumann boundary conditions. We prove that the operator of the problem is modeled as a Fredholm operator of index zero in the scale of Sobolev-type spaces of periodic functions. This result extends to small perturbations of the equation where $\mu$ becomes variable and discontinuous or an additional zero-order term appears. We also show that the solutions to the problem are smoothing.

12 pages

References in corpus (3)

• Fredholm Alternative for Periodic-Dirichlet Problems for Linear Hyperbolic Systems
• Fredholmness and Smooth Dependence for Linear Time-Periodic Hyperbolic System
• Smoothing Solutions to Initial-Boundary Problems for First-Order Hyperbolic Systems

Cited by in corpus (1)

• Time-Periodic Second-Order Hyperbolic Equations: Fredholmness, Regularity, and Smooth Dependence