On problems of determining the source function in a parabolic equation

This work is to examine the existence and uniqueness of solutions to problems of determining the source function in the heat equation when the sought-for function depends on all independent variables of the equation. We have found sufficient conditions for unique solvability of these problems. We examine the well-posedness of problems of source identification in a parabolic equation under various overdetermining conditions. The problem of determining the source function in a parabolic equation has been considered by many authors (see, e.g. [1-4,6,8-12, 14, 15]). They generally assume that the sought-for function is independent of one or more variables of the equation. In this work, we do not impose this constraint: the unknown source is a function of all independent variables. We obtain sufficient conditions for unique solvability of such problems and give examples demonstrating that the problems may have more than one solution when the conditions are violated. 1. FORMULATION OF PROBLEMS, DEFINITIONS AND RESULTS Problem 1. Find a pear of functions *(*,*), F(t,x) = f ( t ) g ( x ) satisfying ut(t, x) = uxx(t, z) + f(t)g(x) , (*, x) e QT (1.1) u(0,z) = u°(x), ze[0,z0] (1.2) t**(t,0) = ux(t,xQ) = 0, *e[0,T] (1.3) <7» = u\x}, z€[0,z 0 ] (1.4) u(t, Xl) = (t) , t € [Ο, Τ] , Ο < ζι < ζο (1.5) in the domain QT = [t, x\ Ο < t < T, 0 < x < XQ}. We assume that conditions (1.2)-(1.5) are consistent. Definition 1.1. A pair of functions u <E W/(QT) n C(0, T; W2(0, x0)) Π C(0, χ0; «&0, T)) , F G satisfying (1.1)-(1.5) is called a classical solution of Problem 1. * Mathematics Dept, Krasnoyarsk State University, Krasnoyarsk 660062, Russia