Problems for partial differential equations are defined as improperly posed if they fail to have a global or unique solution, or if the solution does not depend continuously on the data. Ames (mathematics, U. of Alabama, Huntsville) and Straughan (mathematics, U. of Glasgow) analyze such problems in relation to continuous dependence on: the geometry, modeling backward in time, and modeling forward in time. Other non-standard problems considered include the "sideways" problem for a hyperbolic equation, the classical Cauchy problem for the Laplace equation, and linear thermoplasticity. Annotation c. by Book News, Inc., Portland, Or.
Written by two international experts in the field, this book is the first unified survey of the advances made in the last 15 years on key non-standard and improperly posed problems for partial differential equations.This reference for mathematicians, scientists, and engineers provides an overview of the methodology typically used to study improperly posed problems. It focuses on structural stability--the continuous dependence of solutions on the initial conditions and the modeling equations--and on problems for which data are only prescribed on part of the boundary.
The book addresses continuous dependence on initial-time and spatial geometry and on modeling backward and forward in time. It covers non-standard or non-characteristic problems, such as the sideways problem for a heat or hyberbolic equation and the Cauchy problem for the Laplace equation and other elliptic equations. The text also presents other relevant improperly posed problems, including the uniqueness and continuous dependence for singular equations, the spatial decay in improperly posed parabolicproblems, the uniqueness for the backward in time Navier-Stokes equations on an unbounded domain, the improperly posed problems for dusty gases, the linear thermoelasticity, and the overcoming Holder continuity and image restoration.
Key Features
* Provides the first unified survey of the advances made in the last 15 years in the field
* Includes an up-to-date compendium of the mathematical literature on these topics
