Critical degeneracy in a nonlinear hyperbolic equation can produce atypical instability

This paper deals with the initial value problem for a semilinear wave equation on a bounded domain and solutions are required to vanish on the boundary of this domain. The essential feature of the situation considered here is that the ellipticity of the spatial part of the differential operator degenerates like the square of the distance from given point in the domain and so hyperbolicity is lost at this point. The assumptions ensure that u equivalent to 0 is a stationary solution of the problem and the object is to study the stability of this solution with respect to perturbations of the initial data. Stability, instability and asymptotic stability are all considered. The assumptions about the nonlinear terms ensure that the problem has a well-defined linearization at u equivalent to 0. There are simple cases where this linearization is asymptotically stable but u equivalent to 0 is an unstable solution of the nonlinear problem. We also establish conditions under which the stability of a stationary solution u not equivalent to 0 can be determined using our results. The quadratic degeneracy at a point treated here is typical of what is required in models for acoustic (or sonic) black holes. It also occurs in a simplified Wheeler-DeWitt model which we discuss in some detail.