Inverse Problems for Partial Differential Equations

01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information. This monograph in the Inverse and Ill-Posed Problems Series is devoted to identification problems of coefficients in equations of mathematical physics. It investigates the existence and uniqueness of the solutions for identification coefficient problems in parabolic and hyperbolic equations and equation systems of composite type. It includes a study on the problems with Cauchy data and equations in which the Fourier transform with respect to the chosen variable is supposed to occur. Differential properties of solutions for direct problems and their behaviour under great values of time are studied on the basis of solution properties for direct problems. In addition, identification problems with one or two unknown coefficients are investigated. This monograph will be of interest to specialists in mathematical physics, differential equations and inverse problems.

Table of Contents:
Auxiliary information from functional analysis and theory of differential equations; the basic notions and notations; inequalities; some concepts and theorems of functional analysis; linear partial differential equation of the first order; the maximum principle for parabolic equations of second order; the weak approximation method; examples reducing to the concept of the weak approximation method; general formulation of the weak approximation method; two theorems - an example; the linear partial differential equation; identification problems for parabolic equations with Cauchy data; the unknown source function; the unknown lowest coefficient; the unknown coefficient to the first order derivative; an unknown coefficient to the time derivative; inverse problem for a semilinear parabolic equation; equations of Burgers type; the splitting of one many-dimensional inverse problem into problems of lower dimension; the identification of the source function for a system of composite type and parabolic equation; the behaviour of the problem's solution under t-> the statement of the problem; theorems of existence and uniqueness "on the whole"; the behaviour of solution by t-> stationary problem; convergence to the stationary problem solution; unique solvability of the problem of identifying the source function for a parabolic equation; the stabilization of the solution to the inverse problem; the problem of determining the coefficient in a parabolic equation and some properties of its solution; statement of the problem; theorems of existence and uniqueness "on the whole"; properties of solution under t-> two unknown coefficients of a parabolic type equation; uniform boundary conditions; inhomogeneous conditions of over-determination; input data of the special form; some inverse boundary problems; unknown source function; nonlinear heat equation; hyperbolic equation with small parameter.