Energy Estimates for Symmetric Hyperbolic Integro-Differential Equations

Abstract

Publisher Summary This chapter discusses the Cauchy problem associated with the linear integro-differential equation where the partial differential equation is symmetric equation. The chapter discusses the symmetric hyperbolic differential equations by the use of a-priori L 2 estimates (so-called energy estimates). These estimates lead to the existence of unique weak solution. The estimates provide regularity results as well as show that that the solutions propagate with finite speed. The chapter proves energy estimates for which are analogous to the estimates found for symmetric hyperbolic differential equations. The chapter presents two examples associated with continuum mechanics for the materials with memory. The first example discusses the generalized linear theory with the electromagnetic theory for inhomogeneous anisotropic stable media with memory. The chapter discusses the constitutive relations proposed by Volterra.