Asymptotic Decay for Ultrahyperbolic Operators

Abstract

This chapter discusses the asymptotic decay for ultrahyperbolic operators. The asymptotic behavior of solutions of partial differential equations concerns the rate of decay or growth of a solution as a single variable, usually designated as time, tends to infinity. An analysis of such behavior has been given for a wide variety of equations, mostly of hyperbolic and parabolic type. The chapter discusses the establishment of the maximum rate of decay when both A and B are the Laplace operator with m arbitrary and n = 2. If A is a second-order elliptic operator defined in a domain D in Rm and B is another second-order elliptic operator defined in a positive cone Γ in Rn. It presents an extension of the results to the case where A is an arbitrary self-adjoint second-order elliptic operator. Extension of the method to include the case where B is an arbitrary second-order operator in two variables is straightforward. However, the method that is developed fails for n > 2, and the validity of the results in this situation is open.