Singular perturbation of the wave equation

Abstract

Although Dirichlet conditions for the wave equation is a classical case of a non-well-posed problem in partial differential equations, there has been recent interest in studying those conditions under which the problem is well-posed. This question is of applied interest (V. Barcilon, Mathematika 15 (1968), 93) as well as theoretical interest (A. I. Abdul-Latif and J. B. Diaz, Appi. Anal. 1 (1971), 1). In this paper the onset of nonuniqueness is studied by considering a higher order equation with a small parameter ε in the limit ε→0. The method is quite similar to the method of small viscosity used to study the onset of shock waves