Fredholm Solvability of Periodic Neumann Problem for a Linear Telegraph Equation

We investigate the linear telegraph equation $$ u_{tt}-u_{xx}+2μu_t=f(x,t) $$ 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 $μ$ becomes variable and discontinuous or an additional zero-order term appears. We also show that the solutions to the problem are smoothing.

💡 Research Summary

The paper studies the linear telegraph equation
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