Increasing smoothness of solutions to a hyperbolic system on the plane with delay in the boundary conditions

Abstract

Under consideration is a mixed problem in the half-strip Π = {(x, t) : 0 < x < 1, t > 0} for a first order homogeneous linear hyperbolic system with delay in t in the boundary conditions. We study the behavior of the Laplace transform of a solution to this problem for the large values of the complex parameter. The boundary conditions are found under which the smoothness of a solution to the corresponding mixed problem increases with t.

Perturbations of superstable linear hyperbolic systems

“…n -norm of the function (4.21) can be estimated from above by 2d max 22) where K > 0 is a constant that depends on the coefficients a and b but does not depend on the function u l . Thus Claim 1 is proved.…”

“…It is also shown that the stabilization property is closely related to increasing smoothness of solutions to the perturbed autonomous problems. In the autonomous strictly hyperbolic case the smoothing effect is addressed in [10,21,22], while in the non-autonomous weakly hyperbolic case it is investigated in [16,17].…”