UNIQUENESS OF DETERMINATION OF THE UNKNOWN SOURCE TERM IN SOME MULTIDIMENSIONAL PARABOLIC EQUATIONS

The purpose of this paper is to identify the unknown source term in a multidimensional parabolic equation by means of a one-point interior measurement of the solution at x0 ∈ Ω, i.e. u(x0, .) on [0, T ]; or a one-point boundary measurement, i.e. u(x0, .) on [0, T ] with x0 ∈ ∂Ω. 2000 Mathematics Subject Classification : 35K30, 35K55.


Introduction
Let Ω be an open bounded domain of R n (n ≥ 1) with ∂Ω ∈ C 2+α boundary, α ∈]0, 1[.Let T > 0 and let g = g(x, t) be a given function defined on ∂Ω × [0, T ].We consider the following nonlinear parabolic equation: It is well known that for smooth function f , Eq. (1.1) has a unique classical solution, which we denote by u(f ), provided the data g is sufficiently regular and satisfies compatibility conditions.
In our case the nonlinear source term f is assumed to be unknown, so that additional information is needed to determine this function.
In this paper, we consider either a one-point interior measurement u(f )(x 0 , t) = θ(t), t ∈ [0, T ]; (1.2) or a one-point boundary measurement where x 0 ∈ Ω (respectively, x 0 ∈ ∂Ω) and ∂ n will denote the derivative with respect to the outward normal to ∂Ω.
More precisely, we are concerned with the unique identifiability of the unknown source term; i.e. with the injectivity of the mapping which leads to the uniqueness of f in the inverse problems (1.1) We also study the identifiability (or uniqueness) of the nonlinear term a in the following parabolic equation: (1.4) from the observation (1.2) or (1.3).Throughout this paper we denote by v(a) the solution of Eq. (1.4) corresponding to a.
A result in this direction has already been obtained by Choulli [3] for a one-dimensional inverse problem (1.4)−(1.2) (see also, Cannon-DuChateau [1] and DuChateau [2]).In [5], uniqueness results for the determination of the unknown f (respectively, a) were obtained by Choulli and Zeghal, in the multidimensional case, when the Dirichlet condition is replaced by a Neumann one, from a lateral overdetermination; i.e. u(f ) | ∂Ω×[0,T ] (respectively, v(a) | ∂Ω×[0,T ] ).For an extensive bibliography concerning identifiability problems, the reader is referred to the review article [11] by Nakagiri for a survey of Japanese work up to 1993, and to the survey paper by Isakov [9].Our results depend heavily on the maximum principles which are contained in the books [8], [12] or [13].
2. Some properties of u(f ) and v(a) In this section we outline some properties of the solutions of Eqs.(1.1) and (1.4).To this end, we first define In view of the identifiability problems, we assume throughout that g ∈ e G and h ∈ H.
Let µ be a positive constant and set For the parabolic equation (1.1) (respectively, Eq. (1.4)), we look for the source term f (respectively, a) in the set where δ, γ are positive constants).
It is well known (see, for instance, Ladyzhenskaja et al [10] that if f ∈ F, a ∈ A and the hypotheses on g and h are satisfied, then Eq. (1.1) (respectively, Eq. (1.4)) has a unique solution u(f Next, a simple application of the maximum principle leads to the elementary observation.
where R stands for a range of a function.
Proof.Let f ∈ F. Then the maximum principle applied successively to −u(f ), u(f ) − u(µ), together with (2.1) leads to the following Thus the range of u(f ) is contained in the domain of f .In a similar way, we obtain the second assertion.

2
We will use the following Lemma.
Then, for all s 1 > 0, there exists s 0 > 0 and T 0 , T 1 with 0 < T 0 < T 1 ≤ T such that ) be the solution of the following parabolic equation: and let ψ h be the function defined by ψ h (x, t) = ψ(x, t + h) − ψ(x, t) on Ω × [0, T 1 − h] for 0 < h < T 1 .Then ψ h is the solution of the following equation: By the maximum principle applied to −ψ h , we obtain ψ h ≥ 0 on Ω×[0, T 1 − h].Passing to the limit we deduce that ∂ t ψ ≥ 0 on Ω × [0, T 1 ] and therefore ψ(x, t) ≥ min{ψ(y, τ ); y ∈ Ω} > 0, 0 < τ ≤ t ≤ T 1 and x ∈ Ω, (the second inequality follows from an application of the maximum principle to ψ and the hypotheses on g).Since, min{ψ(y, τ ); y ∈ Ω} → 0 as τ → 0, one sees that there exists T 0 ∈]0, T 1 ] and s 0 > 0 such that s Finally, using again the maximum principle to the function ψ − u(f ), we deduce that u(f ) ≥ ψ on Ω × [0, T 1 ], which achieves the proof.(ii) For all s 1 > 0, there exists s 0 > 0 and T 0 , T 1 with 0 , where λ ∈ R is to be selected in the sequel.A straightforward calculations show that w is a solution of the following equation: where c = c(x, t) = a 0 (v(a)(x, t))∆v(a)(x, t).Since, c is bounded, we can choose λ so large that c − λ ≤ 0.Then, by a maximum principle applied to −w, we have w > 0 on Ω × (0, T ] (because ∂ t h(x, t) > 0 on ∂Ω × (0, T ]), and consequently e λt > 0 on Ω × (0, T ]. (ii) By the maximum principle applied successively to v(δ)−v(a) and v(a)− v(γ) and the fact that ∆v(δ) and ∆v(γ) are positive, we deduce where k is a positive constant.The last inequality of (2.5) follows from the regularity of v(γ) and the fact that v(γ)(x, 0) = 0. Now, the rest of the proof is similar to that of Lemma 2.1, it suffices to replace the function ψ by v(δ).

The main results
Before stating our main results, let us make some notations.Let F a (respectively, A a ) be the set of real analytic functions of F (respectively, A).
In the case of a one-point interior measurement, we have the following result.
From this result we immediately obtain uniqueness of the solution of the inverse problem (1.2).

Corollary 3.1. Under the hypotheses g ∈ e
G and h ∈ e H, the inverse problem (1.2)associated to Eq. (1.1) (respectively, Eq. (1.4)), can possess at most one solution in F a (respectively, in A a ).

A. Zeghal
Note that the condition a 1 − a 2 ∈ A c , in the previous corollary, occurs frequently in the inverse problems where the unknown is the nonlinear term appearing in the equation (see, for instance, [7], [5], [2] and [4]).
In the case of a one-point boundary measurement, we have the following result.We argue as in the proof of Theorem 3.1.Suppose that f 1 6 = f 2 .Since the minimum of the solution w of Eq. (3.1) is attained on {x 0 } × [0, T ] ⊂ ∂Ω×[0, T ] and w is not constant on Ω×[T 0 , T 1 ] (since F > 0 on Ω×[T 0 , T 1 ]), it follows from the minimum principle that which is in contradiction with (3.4).
The proof of the second item is similar.

2
The following result is immediate from the theorem above.