Weakly Hyberbolic Equations in Domains with Boundaries

Abstract

We consider weakly hyperbolic equations of the type utt(t)+a(t)Au(t)=f(t,u(t)), u(0)=u0, ut(0)=u1, u(t) in D(A), t in [0,T], for a function u:[0,T]->H, T a nonnegative real, H a separable Hilbert space, A being a non-negative, self-adjoint operator with domain D(A). The real function a is assumed to be non-negative, continuous and (piecewise) continuous differentiable, and the derivative a' will have to satisfy an integrability condition, which will admit infinitely many oscillations near the point of degeneration. For given initial data u0, u1 a global existence theorem in C([0,T],D(As)) is proved for the linear problem f=f(t). If a' does not change sign, the result can be improved, and finally a local (in time) existence theorem can be proved for nonlinearities f essentially satisfying the mapping property f(., D(As)) is subset of D(As), where s>0 describes the regularity class. In the applications, A will be a uniformly elliptic operator in a domain Omega, Omega being a bounded domain with smooth boundary in Rn, n>=2, for second-order operators then describing a weakly hyperbolic wave equation.