Stability of the determination of a time-dependent coefficient in parabolic equations

We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coefficient $\sigma(t)$, appearing in a Dirichlet initial-boundary value problem for the parabolic equation $\partial_tu-\Delta_x u+\sigma(t)f(x)u=0$, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation in changed to the semi-linear parabolic equation $\partial_tu-\Delta_x u=F(t,x,\sigma(t),u(x,t))$.

We prove the following theorem, where B(M ) is the ball of C 1 [0, T ] centered at 0 and with radius M > 0.

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Following the proof of Theorem 1, we prove that, under the following conditions : there exist x 1 , . . . , x p ∈ Γ such that the matrix M (t) = (f k (x l )g(x l , t)) is invertible for any t ∈ [0, T ], if σ j k ∈ B(M ), 1 k p and j = 1, 2. Here C is a constant that can depend only on data and u j = u(σ j 1 , . . . , σ j p ), j = 1, 2.
ii) We can replace the semi-linear parabolic equation in (1.4) by a semi-linear integro-dierential equation. In other words, F can be changed to Under appropriate assumptions on F 1 and F 2 , one can establish that Theorem 2 is still valid in the present case.
ii) Both in (1.1) and (1.4), the Laplace operator can be replaced by a second order elliptic operator in divergence form : ) is a vector with components in C α (Ω) and the following ellipticity condition holds Actually, the normal derivative associated to E is the boundary To our knowledge, there are only few results concerning the determination of a time-dependent coecient in an initial-boundary value problem for a parabolic equation from a single measurement. The determination of a source term of the form f (t)χ D (x), where χ D the characteristic function of the known subdomain D, was considered by J. R. Canon and S. P. Esteva. They established in [CE86-1] a logarithmic stability estimate in 1D case in a half line when the overdetermined data is the trace at the end point. A similar inverse problem problem in 3D case was studied by these authors in [CE86-2], where they obtained a Lipschitz stability estimate in weighted spaces of continuous functions. The case of a non local measurement was considered by J. R. Canon and Y. Lin in [CL88] and [CL90], where they proved existence and uniqueness for both quasilinear and semi-linear parabolic equations. The determination of a time dependent coecient in an abstract integrodierential equation was studied by the rst author in [Ch91-1]. He proved existence, uniqueness and Lipschitz stability estimate, extending earlier results by [Ch91-2], [LS87], [LS88], [PO85-1] and [PO85-2]. In [CY06], the rst author and M. Yamamoto obtained a stability result, in a restricted class, for the inverse problem of determining a source term f (x, t), appearing in a Dirichlet initial-boundary value problem for the heat equation, from Neumann boundary data. In a recent work, the rst author and M. Yamamoto [CY11] considered the inverse problem of nding a control parameter p(t) that reach a desired temperature h(t) along a curve γ(t) for a parabolic semi-linear equation with homogeneous Neumann boundary data and they established existence, uniqueness as well as Lipschitz stability. Using geometric optic solutions, the rst author [Ch09] proved uniqueness as well as stability for the inverse problem of determining a general time dependent coecient of order zero for parabolic equations from Dirichlet to Neumann map. In [E07] and [E08], G. Eskin considered the same inverse problem for hyperbolic and the Schrödinger equations with time-dependent electric and magnetic potential and he established uniqueness by gauge invariance. Recently, R. Salazar [Sa] extended the result of [E07] and obtained a stability result for compactly supported coecients.
We would like to mention that the determination of space dependent coecient f (x), in the source term σ(t)f (x), from Neumann boundary data was already considered by the rst author and M. Yamamoto [CY06]. But, it seems that our paper is the rst work where one treats the determination of a time dependent coecient, appearing in a parabolic initial-boundary value problem, from Neumann boundary data. This paper is organized as follows. In section 2 we come back to the construction of the Neumann fundamental solution by [It] and establish time-dierentiability of some potential-type functions, necessary for proving Theorems 1 and 2. Section 3 is devoted to the proof of Theorems 1 and 2.

Time-differentiability of potential-type functions
In this section, we establish time-dierentiability of some potential-type functions, needed in the proof of our stability estimates. In our analysis we follow the construction of the fundamental solution by S. Itô [It].
First of all, we recall the denition of fundamental solution associate to the heat equation plus a timedependent coecient of order zero, in the case of Neumann boundary condition. Consider the initialboundary value problem (2.1) is the solution of (2.1). We refer to [It] for the existence and uniqueness of this fundamental solution. We start with time-dierentiability of volume potential-type functions 1 .
Then, f 1 admits a derivative with respect to t and (2.2) Moreover, F given by possesses a derivative with respect to t, Recall that if ϕ = ϕ(x, t) is a continuous function then the corresponding volume potential is given by
Let then u 0 ∈ C 2 (Ω) and consider the function We show that u admits a derivative with respect to t and (2.5) We need to consider rst the case Clearly, w(x, t) is the solution of the following initial-boundary value problem Therefore, (2.5), with w in place of u, is a consequence of Theorem 9.1 of [It]. Next, let (w n 0 ) n be a sequence in C ∞ (Ω) converging to u 0 in C 2 (Ω) and v(x, t) given by Consider (w n ) n , the sequence of functions, dened by We proved that, for any n ∈ N, (2.6) From the proof of Theorem 7.1 of [It], Therefore, we can pass to the limit, as n goes to innity, in (2.6). We deduce that ∂ t w n converges to v in C(Ω × [s, t 0 ]). But, w n converges to u in C(Ω × [s, t 0 ]). Hence u admits a derivative with respect to t and hal-00673690, version 2 -28 Feb 2012 That is we proved (2.5) and consequently (2.2) holds true. Finally, we note that (2.4) is deduced easily from (2.7).
Next, we consider time-dierentiability a single layer potential-type function 2 .
is well dened, admits a derivative with respect to t and we have (Γ×(s,t0)) . (2.8) Contrary to Lemma 1, for Lemma 2 we cannot use directly the general properties of the fundamental solutions developed in [It]. We need to come back to the construction of the fundamental solution of (2.1) introduced by [It]. First, consider the heat equation ∂ t u = ∆ x u in the half space Ω 1 = {x = (x 1 , . . . , x n ); x 1 > 0} in R n with the boundary condition ∂ x1 u = 0 on Γ 1 = {x = (0, x 2 , . . . , x n ); (x 2 , . . . , x n ) ∈ R n−1 }. For any y = (y 1 , y 2 , . . . , y n ), we dene y by y = (−y 1 , y 2 , . . . , y n ). Let t denotes the Gaussian kernel and set Then, the fundamental solution U 0 (x, t; y, s) of is given by U 0 (x, t; y, s) = G 1 (x, t − s; y).
In order to construct the fundamental solution in the case of an arbitrary domain Ω, Itô introduced the following local coordinate system around each point z ∈ Γ.
Lemma 3. (Lemma 6.1 and its corollary, Chapter 6 of [It]) For every point z ∈ Γ, there exist a coordinate neighborhood W z of z and a coordinate system (x * 1 , . . . , x * n ) satisfying the following conditions: 1) the coordinate transformation between the coordinate system (x * 1 , . . . , x * n ) and the original coordinate system in W z is of class C 2 and the partial derivatives of the second order of the transformation functions are Hölder continuous ; 2) Γ ∩ W z is represented by the equation The single-layer potential corresponding to a continuous function ϕ = ϕ(x, t) is given by U (x, t; y, τ )ϕ(y, τ )dσ(y)dτ.

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Then, for any u ∈ C 1 (Ω) we have From now on, for any z ∈ Γ, we view coordinate system (x * 1 , . . . , x * n ) as a rectangular coordinate system. Moreover, using the local coordinate system of Lemma 3, for any y = (y 1 , y 2 , . . . , y n ) ∈ L(W z ), we dene y = (−y 1 , y 2 , . . . , y n ) and, without loss of generality, we assume that, for any y ∈ L(W z ), we have y ∈ L(W z ). For any interior point z of Ω, we x an arbitrary local coordinate system and a coordinate neighborhood W z contained in Ω. For any z ∈ Ω and δ > 0, we set W (z, δ) = {x : |x − z| 2 < δ} and δ z > 0 such that, for any z ∈ Ω we have W (z, δ z ) ⊂ W z .
Recall the following partition of unity lemma. Let {z 1 , . . . , z m } be the nite subset of Ω, introduced in the previous lemma. For any k ∈ {1, . . . , m}, let L k denotes the dieomorphism from W z k to L k (W z k ) dened by where (x * 1 , . . . , x * n ) is the local coordinate system of Lemma 3 dened in W z k . For any k ∈ {1, . . . , m}, the dierential operator becomes, in terms of local coordinate system x * = (x * 1 , . . . , x * n ), in L k (W z k ) × (s 0 , t 0 ). Here q k (x * , t) is Hölder continuous on L k (W z k ) × (s 0 , t 0 ) and (a ij (x * )) is the contravariant tensor of degree 2 dened by According to the construction of [It] given in Chapter 6 (see pages 42 to 45 of [It]), a ij k (x * ) is of class C 2 in L k (Ω ∩ W z k ) and it is a positive denite symmetric matrix at every point x * ∈ L k (W z k ). We set Note that, by the construction of S. Itô [It] (see page 45), for any k = 1, . . . , m, we have a ij k (L k (x)) = a ij k (L k (x)), x ∈ Ω ∩ W z k , for i = j = 1 or i, j = 2, . . . , n, (2.10) a 1j k (L k (x)) = a j1 k (L k (x)) = −a 1j k (L k (x)), x ∈ Ω ∩ W z k , for j = 1, . . . , n (2.11) and a 1j k (L k (ξ)) = a j1 k (L k (ξ)) = δ j1 , ξ ∈ Γ ∩ W z k , j = 1, . . . , n, (2.12) where δ j1 denotes the kronecker's symbol. For any k ∈ {1, . . . , m}, let G k (x, t; y) be dened, in the region Consider also H(x, t; y), dened in the region as follows As in Lemma 7.2 of [It], we dene successively: Then, following [It] (see page 53), the fundamental solution of (2.1) is given by (2.13) We are now able to prove Lemma 2 with the help of representation (2.13), the properties of H(x, t; y) and K(x, t; y, s).
According to representation (2.13), one needs to show that F 1 and F 2 admit a derivative with respect to t and (2.14) for (x, t) ∈ Γ × (0, t 0 ). We start by considering F 1 . Applying a simple substitution, we obtain H(x, s; y)f (y, t − s)dσ(y)ds.

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and (2.16), applying the substitution y = z −y √ τ −s , with z * = (z * 1 , z ) and y * = (0, y ), we obtain for 0 < s < τ < t 0 and z * ∈ L l (Ω ∩ W z k ), where, for j = 0, 1, 2, P j are polynomials and J j 0 are continuous functions, C 1 with respect to τ, s ∈ (0, t 0 ) and satisfy for some constant C l > 0. We note that ∂ v1 J j 0 ((z 1 , z ), τ ; y , s; v 1 ) is not necessarily bounded. Indeed, we show This representation and the construction of K(z, τ ; y, s) in Chapter 5 of [It] (see pages 31 to 32 for the construction in R n and page 53 for the construction in a bounded domain) lead (2.20) for 0 < s < τ < t 0 and z * ∈ L l (Ω ∩ W z k ), where, for j = 0, 1, 2, Q j are polynomials and K j are continuous functions, C 1 with respect to τ, s ∈ (0, t 0 ) and satisfy where C l > 0 is a constant. Furthermore, using representation (2.20), we have, for s < τ < t < t 0 , . . , z * n ) ∈ R n ; z * 1 > 0}. Then, applying the substitutions z = x −z √ t−τ and z 1 = z * 1 √ τ −s , we deduce, in view of the form of the functions K j , the following (2.21) for some continuous functions K 0 , K 1 and H l such that K 0 , K 1 are C 1 , with respect to s and τ , and the following estimates hold: (2.23) hal-00673690, version 2 -28 Feb 2012 for some constant C l > 0. Repeating the arguments used for (2.21) and applying some results of page 31 of [It], we obtain, for 0 < t < t 0 , (2.24) This estimate and Fubini's theorem imply Then, in view of representation (2.21), for all 0 < t < t 0 , f 1 (x , s; y , z )dy dz dz 1 dτ ds.