Existence of solutions for linear hyperbolic initial-boundary value problems in a rectangle

Abstract

In a recent article, we achieved the well-posedness of linear hyperbolic initial and boundary value problems (IBVP) in a rectangle via semigroup method, and we found that there are only two elementary modes called hyperbolic and elliptic modes in the system. It seems that, there is only one set of boundary conditions for the hyperbolic mode, while there are infinitely many sets of boundary conditions for the elliptic mode, which can lead to well-posedness. In this article, we continue to consider linear hyperbolic IBVP in a rectangle in the constant coefficients case and we show that there are also infinitely many sets of boundary conditions for hyperbolic mode which will lead to the existence of a solution. We also have uniqueness in some special cases. The boundary conditions satisfy the reflection conditions introduced in Section 3, which turn out to be equivalent to the strictly dissipative conditions.

Keywords:

• first-order hyperbolic PDE
• initial-boundary value problem
• existence and uniqueness
• reflection boundary conditions

AMS Subject Classifications:

• 35L02
• 35L50
• 35F16
• 34A12

Acknowledgements

The author is indebted to his advisor, Professor Roger Temam, for providing insightful comments and invaluable advice.

Notes

This work was partially supported by the National Science Foundation [grant number NSF DMS-1206438] and by the Research Fund of Indiana University.