## 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
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### 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
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