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

Abstract

The paper concerns the general linear one-dimensional second-order hyperbolic equation $${\partial }_t^{2}u-a^2(x,t){\partial }_x^{2}u+a_1(x,t){\partial }_{t}u+a_2(x,t){\partial }_{x}u+a_3(x,t)u=f(x,t),x\in (0,1)$$ with periodic conditions in time and Robin boundary conditions in space. Under a non-resonance condition (formulated in terms of the coefficients $a$, $a_1$, and $a_2$) 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 $a_1$, $a_2$, and $a_3$ do not.

34 pages

References in corpus (2)

• Periodic Solutions to Dissipative Hyperbolic Systems. II: Hopf Bifurcation for Semilinear Problems
• Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems

Cited by in corpus (2)

• Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay
• Forced Frequency Locking for Semilinear Dissipative Hyperbolic PDEs