Fredholm Solvability of Periodic Neumann Problem for a Linear Telegraph Equation

TL;DR

This paper proves the Fredholm property and regularity of solutions for a periodic Neumann problem of a linear telegraph equation, extending results to variable coefficients and perturbations.

Contribution

It establishes the Fredholm solvability and smoothing properties of the linear telegraph equation with periodic Neumann boundary conditions, including variable and discontinuous coefficients.

Findings

The problem operator is a Fredholm operator of index zero.

Solutions exhibit smoothing effects.

Results extend to equations with variable, discontinuous coefficients, and additional zero-order terms.

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.