## 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

### MSC:

 35R30 Inverse problems for PDEs 35K05 Heat equation

### Keywords:

stability; heat sources; uniqueness; spherical averages
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