Inverse Problems for Partial Differential Equations

Yurii Ya. Belov, Siberian Federal University, Krasnoyarsk, Russia.

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.