Uniqueness and stability of 3D heat sources

The authors discuss the uniqueness of the problem of finding a region D contained in/implied by R³ and a function U=U(x,t) such that U_t= Delta U+ chi _D(x)f(t),x in R³, t>0; U(x,0)=0, x in R³; with either U((x₁,x₂,0),t)=g(x₁,x₂,t), (x₁,x₂) in V contained in/implied by R² for some open set V, or U(p,t)=g(t) for some fixed p in R³. Let A(r)=area(rS² intersection D), where S² is the unit sphere in R³ and r is the radius of the sphere rS² centred at the origin. For the data U(0,t)=S(t) they derive an estimate. Thus spherical averages A(r) of D, which are contained in the ball centred at the origin with radius M and which are smooth enough that A(r) is Holder continuous with exponent alpha are uniquely determined by U(0,t)=g(t). Applying this result to U(x₁,x₂,t)=S(x₁,x₂,t) they see that spherical averages A((x₁,x₂,0),T), where (x₁,x₂,0) is the centre of the sphere of radius r, are uniquely determined by g(x₁,x₂,t). From this and from a known result on integral geometry it follows that D is uniquely determined.