×

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.

MSC:

35R30 Inverse problems for PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K05 Heat equation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Savateev, E. G., Inverse problem for the nonlinear heat equation with the final overdetermination, Math. Comput. Modell., 22, 29-43 (1995) · Zbl 0835.35157
[2] Choulli, M.; Yamamoto, M., An inverse parabolic problem with non-zero initial condition, Inverse Problems, 13, 19-27 (1997) · Zbl 0867.35112
[3] Isakov, V., Some inverse problems for the diffusion equation, Inverse Problems, 15, 3-10 (1999) · Zbl 0918.35145
[4] Tadi, M., Inverse heat conduction based on boundary measurements, Inverse Problems, 13, 1585-1605 (1997) · Zbl 0889.35123
[5] Lorenzi, A., An inverse problem for a semilinear parabolic equation, Ann. Mate, Pure Appl., 82, 145-167 (1982) · Zbl 0493.35078
[6] Pilant, M.; Rundell, W., Fixed point methods for a nonlinear parabolic inverse coefficient problem, Commun. PDE, 13, 469-493 (1988) · Zbl 0647.35083
[7] Nanda, A.; Das, P. C., Determination of the source term in the heat conduction equation, Inverse Problems, 12, 325-339 (1996) · Zbl 0851.35135
[8] Cannon, J. R.; DuChateau, P., Structural identification of an unknown source in a heat equation, Inverse Problems, 14, 535-551 (1998) · Zbl 0917.35156
[9] Gatti, S., An existence result for an inverse problem for a quasilinear parabolic equation, Inverse Problems, 14, 53-65 (1998) · Zbl 0895.35108
[10] Li, G.; Ma, Y.; Li, K., An inverse parabolic problem for nonlinear source term with nonlinear boundary conditions, J. Inverse Ill-Posed Problems, 11, 371-387 (2003) · Zbl 1048.35136
[11] Isakov, V., Inverse Problems for Partial Differential Equations (1998), Springer: Springer New York · Zbl 0908.35134
[12] Cannon, J. R., The One-dimensional Heat Equation (Encyclopedia of Math and Its Applications 23) (1984), Addison-Wesley: Addison-Wesley London
[13] DuChateau, P., Monotonicity and invertibility of coefficient to data mapping for parabolic inverse problems, SIAM J. Math. Anal., 26, 1473-1487 (1995) · Zbl 0849.35146
[14] Lions, J. L., Optimal Control of System Governed by PDE (1971), Springer: Springer Berlin · Zbl 0203.09001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.