On backward uniqueness for parabolic equations when Osgood continuity of the coefficients fails at one point

We prove uniqueness for backward parabolic equations whose coefficients are Osgood continuous in time for $t>0$ but not at $t=0$.


Introduction
Let us consider the following backward parabolic operator We assume that all coefficients are defined in [0, T ] × R n x , measurable and bounded; (a jk (t, x)) jk is a real symmetric matrix for all (t, x) ∈ [0, T ] × R n x and there exists λ 0 ∈ (0, 1] such that n j,k=1 a jk (t, x)ξ j ξ k ≥ λ 0 |ξ| 2 for all (t, x) ∈ [0, T ] × R n x and ξ ∈ R n ξ . Given a functional space H we say that the operator L has the Huniqueness property if, whenever u ∈ H, Lu = 0 in [0, T ] ×R n x and u(0, x) = 0 in R n x , then u = 0 in [0, T ] × R n x . Our choice for H is the space of functions H = H 1 ((0, T ), L 2 (R n x )) ∩ L 2 ((0, T ), H 2 (R n x )).
This choice is natural, since it follows from elliptic regularity results that the domain of the operator − n j,k=1 ∂ x j (a jk (t, x)∂ x k ) in L 2 (R n ) is H 2 (R n ) for all t ∈ [0, T ].
In our previous papers [5,6] we investigated the problem of finding the minimal regularity assumptions on the coefficients a jk ensuring the Huniqueness property to L. Namely, we proved the H-uniqueness property for the operator L when the coefficients a jk are Lipschitz continuous in x and the regularity in t is given in terms of a modulus of continuity µ, i. e. Suitable counterexamples show that Osgood condition is sharp for backward uniqueness in parabolic equations: given any non-Osgood modulus of continuity µ, it is possible to construct a backward parabolic equation, whose coefficients are C ∞ in x and µ-continuous in t, for which the Huniqueness property does not hold. In the mentioned counterexamples the coefficients are in fact C ∞ in t for t = 0, and Osgood continuity fails only at t = 0.
In this paper we show that if the loss of Osgood continuity is properly controlled as t → 0, then we can recover the H-uniqueness property for L. Our hypothesis reads as follows: given a modulus of continuity µ satisfying the Osgood condition, we assume that the coefficients a jk are Hölder continuous with respect to t on [0, T ], and for all t ∈]0, T ] where 0 < β < 1. The cofficients a jk are assumed to be globally Lipschitz continuous in x. Under such hypothesis we prove that the H-uniqueness property holds for L. As in our previous papers [5,6], the uniqueness result is consequence of a Carleman estimate with a weight function shaped on the modulus of continuity µ. The weight function is obtained as solution of a specific second order ordinary differential equation. In the previous results cited above, the corresponding o.d.e. is autonomous. Here, on the contrary, the time dependent control (1) yields to a non-autonomous o.d.e.. Also, the "Osgood singularity" of a jk at t = 0 introduces a number of new technical difficulties which are not present in the fully Osgood-regular situation considered before. The result is sharp in the following sense: we exhibit a counterexample in which the coefficients a jk are Hölder continuous with respect to t on [0, T ], for all t ∈]0, T ] and for all ǫ > 0 and the operator L does not have the H-uniqueness property. The borderline case ǫ = 0 in (2) is considered in paper [7]. In such a situation only a very particular uniqueness result holds and the problem remains essentially open.

Main result
We start with the definition of modulus of continuity. • for all s ∈ [0, 1], µ(s) ≥ s; • on (0, 1], the function s → µ(s) s is decreasing; • the limit lim s→0 + µ(s) s exists; Let µ be a modulus of continuity and let ϕ : We introduce the notion of Osgood modulus of continuity.
where all the coefficients are supposed to be complex valued, defined in [0, T ] × R n , measurable and bounded. Let (a j,k (t, x)) j,k be a real symmetric matrix and suppose there exists λ 0 ∈ (0, 1) such that n j,k=1 for all (t, x) ∈ [0, T ] × R n and for all ξ ∈ R n . Under this condition L is a backward parabolic operator. Let H be the space of functions such that Let µ be a modulus of continuity satisfying the Osgood condition. Suppose that there exist α ∈ (0, 1) and C > 0 such that, i) for all j, k = 1, . . . , n, ii) for all j, k = 1, . . . , n and for all t ∈ (0, T ],
We remark that φ ′ (1) = 1 and φ ′ is decreasing in [1, +∞[, so that φ is a concave function. We remark also that φ −1 : We define i. e. ψ γ is a solution to the differential equation ).
In particular, for t ∈ (0, . We can now state the Carleman estimate.
Theorem 2. In the previous hypotheses there exist γ 0 > 0, C > 0 such that The way of obtaining the H-uniqueness from the inequality (14) is a standard procedure, the details of which can be found in [5,Par. 3.4].

Littlewood-Paley decomposition
We will use the so called Littlewood-Paley theory. We refer to [2], [3], [9] and [1] for the details. Let ψ ∈ C ∞ ([0, +∞[, R) such that ψ is non-increasing and We set, for ξ ∈ R n , Given a tempered distribution u, the dyadic blocks are defined by where we have denoted by F −1 the inverse of the Fourier transform. We introduce also the operator We recall some well known facts on Littlewood-Paley demposition.
Moreover there exists C > 1, depending only on n and s, such that, for all u ∈ H s , Moreover there exists C > 0, depending only on n, such that, for all a ∈ Lip and for all k ∈ N, where a Lip = a L ∞ + ∇a L ∞ . We recall some known facts on modified Bony's paraproduct.
Then a − T m a maps L 2 into H 1 and there exists C ′ > 0 depending only on n, m, such that, for all u ∈ L 2 , A similar result remains valid for valued functions when a is replaced by a positive definite matrix (a j,k ) j,k .
We end this subsection with a property which will needed in the proof of the Carleman estimate.

Approximated Carleman estimate
We set v(t, x) = e 1 γ Φγ (γ(T −t)) u(t, x). The Carleman estimate (14) becomes: there exist γ 0 > 0, C > 0 such that for all γ > γ 0 and for all v ∈ C ∞ 0 (R n+1 ) such that Supp u ⊆ [0, T 2 ] × R n x . First of all, using Proposition 4, we fix a value for m in such a way that Next we consider Proposition 3 and in particular from (19) we deduce that (23) will be a consequence of since the difference between (23) and (25) is absorbed by the right side part of (25) with possibly a different value of C and γ 0 . With a similar argument, using (16) and (22), (25) will be deduced from where we have denoted by v h the dyadic block ∆ h v. We fix our attention on each of the terms We have Let consider the last term in (27). We define, for ε ∈ (0, T 2 ], . With a strightforward computation, form (7) and (8), we obtain and for all j, k = . . . , n and for all (t, x) ∈ [0, T 2 ] × R n x . We deduce Now, T m a j,k − T m a j,k,ε = T m a j,k −a j,k,ε and, from (18) and (28), where C depends only on n, m and a j,k L ∞ and N > 0 can be chosen arbitrarily. Similarly

From (18) and (29) we have
and, from (21), where C depends only on n, m and a j,k Lip and N > 0 can be chosen arbitrarily. As a conclusion, from (27), we finally obtain

End of the proof
We start considering (30) for h = 0. We fix ε = 1 2 . Recalling (13) we have Choosing a suitable γ 0 , we have that, for all γ > γ 0 , We consider (30) for h ≥ 1. We fix ε = 2 −2h . We have From (24) it is possible to deduce that n j,k=1 Suppose first that and then, using also (13), we obtain Then there exist γ 0 > 0 and C > 0 such that, for all γ ≥ γ 0 and for all h ≥ 1, Suppose finally that From (12), the fact that λ 0 ≤ 1 and the properties of the modulus of continuity µ Then there exist γ 0 > 0 and C > 0 such that, for all γ ≥ γ 0 and for all h ≥ 1, As a conclusion, form (31), (33) and (34), there exist γ 0 > 0 and C > 0 such that, for all γ ≥ γ 0 and for all h ∈ N, and (26) follows. The proof is complete.
The conclusion of the theorem is reached simply exchanging t with −t.