##
**Data compatibility and conditional stability for an inverse source problem in the heat equation.**
*(English)*
Zbl 1105.35144

The author considers the problem of recovering the nonlinearity \(g\) in the following one-dimensional parabolic equation, where \(D_T=(0,1)\times (0,T)\), \(T>0\):
\[
D_tu(x,t) - D_x^2u(t,x) = a(x) g(u(t,x)),\qquad (x,t)\in D_T,
\]
subject to the initial and boundary conditions
\[
u(x,0)=0,\quad x\in [0,1], \qquad D_xu(0,t)=-h(t),\quad D_xu(1,t)=0, \quad t\in [0,T].
\]

For this purpose, function \(g\) is sought for as the one minimizing the error functional

\[ J(u)=\int_0^1 | u(T,x)-\theta(x)| ^2\,dx \]

in the class \(G\) of functions that are continuous everywhere in \({\mathbb R_+}\) and piecewise smooth there, with \(g(0)=0\), \(g(u)\geq 0\) for all \(u\in [0,M]\), \(g\neq 0\), \(M=\theta(0)\). Moreover, the data are assumed to satisfy the following properties: (i) \(a\in C([0,1])\), \(a(x)>0\) for all \(x\in (0,1)\); (ii) \(h\in C([0,1])\), \(h(0)=0\), \(h(t)>0\) for all \(t\in (0,T)\); (iii) \(\theta\) is continuous in \([0,1]\) and piecewise smooth there, with \(\theta(x)\geq 0\) and \(\theta'(x)\leq 0\) for all \(x\in (0,1)\); (iv) \(\theta'(0)=-h(T)\), \(\theta'(1)=0\).

Under such assumptions a conditional stability result – in a suitable metric space for \(g\) – is proved. However, in Theorem 3.2 the author should have explained detailly why the first variation of the functional \(J\) vanishes at a minimal point \(g_0\in S_{ad}\), since it needs not to be an interior point of the non-open set \(S_{ad}\).

The paper is supplied with two examples where numerical computations are carried out to validate the procedure.

For this purpose, function \(g\) is sought for as the one minimizing the error functional

\[ J(u)=\int_0^1 | u(T,x)-\theta(x)| ^2\,dx \]

in the class \(G\) of functions that are continuous everywhere in \({\mathbb R_+}\) and piecewise smooth there, with \(g(0)=0\), \(g(u)\geq 0\) for all \(u\in [0,M]\), \(g\neq 0\), \(M=\theta(0)\). Moreover, the data are assumed to satisfy the following properties: (i) \(a\in C([0,1])\), \(a(x)>0\) for all \(x\in (0,1)\); (ii) \(h\in C([0,1])\), \(h(0)=0\), \(h(t)>0\) for all \(t\in (0,T)\); (iii) \(\theta\) is continuous in \([0,1]\) and piecewise smooth there, with \(\theta(x)\geq 0\) and \(\theta'(x)\leq 0\) for all \(x\in (0,1)\); (iv) \(\theta'(0)=-h(T)\), \(\theta'(1)=0\).

Under such assumptions a conditional stability result – in a suitable metric space for \(g\) – is proved. However, in Theorem 3.2 the author should have explained detailly why the first variation of the functional \(J\) vanishes at a minimal point \(g_0\in S_{ad}\), since it needs not to be an interior point of the non-open set \(S_{ad}\).

The paper is supplied with two examples where numerical computations are carried out to validate the procedure.

Reviewer: Alfredo Lorenzi (Milano)

### MSC:

35R30 | Inverse problems for PDEs |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35K05 | Heat equation |

### Keywords:

unknown nonlinear source terms; nonlinear 1D parabolic equations; conditional stability; integral identity method
Full Text:
DOI

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