A Mixed Problem for a Hyperbolic System on the Plane with Delay in the Boundary Conditions

Abstract

We examine well-posedness of the boundary value problem in a half-strip for a first-order linear hyperbolic system with delay (lumped and distributed) in the boundary conditions. In the case of the negative real parts of the eigenvalues of the corresponding spectral problem we prove a time uniform estimate for a solution to the homogeneous problem which enables us to justify the linearization principle for analysis of stability of stationary solutions to the nonlinear problem.

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

“…As shown in [9], the set Λ(L A ) lies on the left of some straight line parallel to the imaginary axis; so we introduce the number…”

“…This expansion is constructed in [9] for k = 1, but it is not enough for us. Nevertheless, we will use the results and notation from the article.…”

“…Wellposedness of a mixed problem was considered for a linear hyperbolic system on the plane with delay (lumped and distributed) in the boundary conditions in [9]. The problems of this type arise in the mathematical modeling of counterflow chemical reactors with recycle in the case when some substances after the exit from a reactor returns to the entrance of a reactor with some delay in time necessary for transportation.…”

“…Proportionality coefficients usually indicate which part of the substance returns. In the case of negative real parts of the eigenvalues of the corresponding spectral problem, a uniform estimate with respect to time is established in [9], which permits us to justify the linearization principle for the analysis of stability of stationary solutions to the nonlinear problem.…”

“…To state the existence theorem of a classical solution to the problem in question, we define the agreement conditions for the initial data [9]. We say that the initial data U (x, t) meet the agreement conditions of zero order whenever…”