Uniqueness and stability of 3D heat sources. (English) Zbl 0729.35144

We discuss the uniqueness of the problem of finding a region \(D\subset {\mathbb{R}}^ 3\) and a function \(U=U(x,t)\) such that \(U_ t=\Delta U+\chi_ D(x)f(t)\), \(x\in {\mathbb{R}}^ 3\), \(t>0\); \(U(x,0)=0\), \(x\in {\mathbb{R}}^ 3\); with either \(U((x_ 1,x_ 2,0),t)=g(x_ 1,x_ 2,t)\), \((x_ 1,x_ 2)\in V\subset {\mathbb{R}}^ 2\) for some open set V, or \(U(p,t)=g(t)\) for some fixed \(p\in {\mathbb{R}}^ 3\). Let \(A(r)=area(rS^ 2\cap D)\), where \(S^ 2\) is the unit sphere in \({\mathbb{R}}^ 3\) and r is the radius of the sphere \(rS^ 2\) centred at the origin. For the data \(U(0,t)=S(t)\) we derive an estimate of the form \[ \| A_ 1-A_ 2\|_ M\leq k_ 1\{1/\log (k_ 2/\| g'_ 1-g'_ 2\|_ T^{\beta /2}\}^{\alpha /2} \] where \(\| \psi \|_ I=\sup_{[0,I]}| \psi |\), and where \(M,T,k_ 1,k_ 2\) and \(\beta\) are positive constants with \(0<\alpha,\beta <1\). 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 Hölder continuous with exponent \(\alpha\), are uniquely determined by \(U(0,t)=g(t)\). Applying this result to \(U(x_ 1,x_ 2,t)=S(x_ 1,x_ 2,t)\) we see that spherical averages \(A((x_ 1,x_ 2,0),r)\), where \((x_ 1,x_ 2,0)\) is the centre of the sphere of radius r, are uniquely determined by \(g(x_ 1,x_ 2,t)\). From this and from a known result on integral geometry it follows that D is uniquely determined.
Reviewer: J.R.Cannon


35R30 Inverse problems for PDEs
35K05 Heat equation
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