An inverse problem for the heat equation

The authors consider the problem of the determination of a unknown source f=f(x,t) in the heat conduction equation from overspecified data. For certain types of functions f they demonstrate the uniqueness of the unknown source and derive a priori estimates of the continuous dependence upon the data. As an example of the results they state one of their theorems. Let chi _(a,b)(x) denote the characteristic function of the interval 0<a<or=x<or=b and let u=u(x,t) and f=f(t) satisfy u_t-U_xx=f(t) chi _(a,b)(x), - infinity <x< infinity , 0<t< infinity ; u(x,0)=0, - infinity <x< infinity ; u(0,t)=g(t), 0<t< infinity ; f(0)=0, and f' and f" bounded. Then for fixed T>0, there exists a constant M>0 such that mod f(t) mod <or=M(log(//g//^-1, 0<or=t<or=T, for the L₂ norm //g//=( integral ₀^infinity (g(t))²dt)¹2/<1.