HYPERBOLIC EQUATIONS WITH MEMORY

Abstract

The existence of a solution to the equation governing the evolution of a displacement vector in an elastic body with non-local time and spatial memory is considered. A global weak solution to an associated initial-boundary value problem is established by constructing Galerkin approximations. Lebesgue or Sobolev spaces can be generalized for all real numbers and can be defined also on Banach spaces. They are equipped with several equivalent norms based on Fourier or Laplace transform and function expansion. These spaces help to derive suitable energy estimates