Well-posedness of Initial/Boundary Value Problems for Hyperbolic Integro-differential Systems with Nonsmooth Coefficients

In the late 1960’s, J.-L. Lions and collaborators showed that energy estimates could be used to establish existence, uniqueness, and continuous dependence on initial data for finite energy solutions of initial/boundary value problems for various linear partial differential evolution equations with nonsmooth coefficients. The second author has recently treated second order hyperbolic systems, for example linear elastodynamics, by similar methods, and extended these techniques to demonstrate continuous dependence and even differentiability (in a suitable sense) of the solution as function of the the coefficients. In the present paper, we extend Lions ’ results in a different direction, to first order symmetric hyperbolic integrodifferential systems (such as linear viscoelasticity) with bounded and measurable coefficients. We show that the initial value problem is well-posed in an appropriate space of finite-energy weak solutions, and that solutions of this class are continuous as functions of the coefficients and data. This result is sharp, in the sense that solutions are not in general locally uniformly continuous in coefficients and data. Solutions are however (Gâteaux-)differentiable as a function of the coefficients for fixed data, in case the data is smooth enough that the time derivative 1 of the solution is itself a finite-energy weak solution. The continuity result combines with the well-known domain of influence properties for hyperbolic systems with smooth coefficients to show that viscoelasticity with bounded, measureable coefficients predicts finite wave propagation speed.