Carleman Estimates for Singular Parabolic Equations with Interior Degeneracy and Non-Smooth Coefficients

Adv. Nonlinear Anal. 2017; 6 (1):61–84

Research Article

Genni Fragnelli and Dimitri Mugnai* Carleman estimates for singular parabolic equations with interior degeneracy and non-smooth coefficients

DOI: 10.1515/anona-2015-0163 Received November 18, 2015; accepted January 14, 2016

Abstract: We establish Carleman estimates for singular/degenerate parabolic Dirichlet problems with degeneracy and singularity occurring in the interior of the spatial domain. Our results are completely new, since this situation is not covered by previous contributions for degeneracy and singularity on the boundary. In addition, we consider non-smooth coefficients, thus preventing the use of standard calculations in this framework.

Keywords: Carleman estimates, singular/degenerate equations, non-smooth coefficients, Hardy–Poincaré inequality, Caccioppoli inequality, observability inequality, null controllability

MSC 2010: 35Q93, 93B05, 93B07, 34H15, 35A23, 35B99

1 Introduction

Controllability issues for parabolic problems have been a mainstream topic in recent years, and several de- velopments have been pursued: starting from the heat equation in bounded and unbounded domain, related contributions have been found for more general situations. A common strategy in showing controllability results is to prove that certain global Carleman estimates hold true for the operator which is the adjoint of the given one. In this paper we focus on a class of singular parabolic operators with interior degeneracy of the form

λ ut − (a(x)ux)x − u, (1.1) b(x) associated to Dirichlet boundary conditions and with (t, x) ∈ QT := (0, T) × (0, 1), T > 0 being a fixed num- ber. Here a and b degenerate at the same interior point x0 ∈ (0, 1), and λ ∈ ℝ satisfies suitable assumptions (see condition (2.4) below). The fact that both a and b degenerate at x0 is just for the sake of simplicity and shortness. All the stated results are still valid if they degenerate at different points. The prototypes we have K K in mind are a(x) = |x − x0| 1 and b(x) = |x − x0| 2 for some K1, K2 > 0. The main goal is to establish global Carleman estimates for operators of the form given in (1.1). Such estimates for uniformly parabolic operators without degeneracies or singularities have been largely developed (see, e.g., [32]). Recently, these estimates have been also studied for operators which are not uniformly parabolic. Indeed, as pointed out by several authors, many problems coming from Physics (see [36]), Biology (see [21]) and Mathematical Finance (see [35]) are described by degenerate parabolic equations.

Genni Fragnelli: Dipartimento di Matematica, Università di Bari "Aldo Moro", Via E. Orabona 4, 70125 Bari, Italy, e-mail: [email protected] *Corresponding author: Dimitri Mugnai: Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy, e-mail: [email protected]. http://orcid.org/0000-0001-8908-5220 62 Ë G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations

In particular, new Carleman estimates (and consequently null controllability properties) were established in [1], and also in [15, 40], for the operator

ut − (aux)x + c(t, x)u, (t, x) ∈ QT ,

1 where a(0) = a(1) = 0, a ∈ C (0, 1) and c ∈ L∞(QT) (see also [12, 13, 25] for problems in non-divergence form). An interesting situation is the case of parabolic operators with singular inverse-square potentials. First results in this direction were obtained in [46] for the non-degenerate singular potentials with heat-like operator 1 ut − ∆u − λ 2 u, (t, x) ∈ (0, T) × Ω, (1.2) |x| N with associated Dirichlet boundary conditions in a bounded domain Ω ⊂ ℝ containing the singularity x = 0 in the interior (see also [47] for the wave and Schrödinger equations and [16] for boundary singularity). Sim- ilar operators of the form 1 u ∆u λ u, t, x 0, T Ω, t − − K ( ) ∈ ( ) × |x| 2 arise, for instance, in quantum mechanics (see, e.g., [4, 19]), or in combustion problems (see, e.g., [6, 10, 20, 33]), and is known to generate interesting phenomena. For example, in [4] and in [5] it was proved that, for all values of λ, global positive solutions exist if K2 < 2, whereas instantaneous and complete blow-up occurs if K2 > 2. In the critical case, i.e., K2 = 2, the value of the parameter λ determines the behavior of the equation. If λ ≤ 1/4 (which is the optimal constant of the Hardy inequality, see [9]) global positive solutions exist, while, if λ > 1/4, instantaneous and complete blow-up occurs (for other comments on this argument we refer to [45]). We recall that in [46], Carleman estimates were established for (1.2) under the condition λ ≤ 1/4. On the contrary, if λ > 1/4, in [22] it was proved that null controllability fails. We remark that the non-degenerate problems studied in [4, 16, 22, 45–47] cover the multidimensional case, while here we treat the case N = 1, like Vancostenoble [45], who studied the operator that couples a degenerate diffusion coefficient with a singular potential. In particular, for K1 ∈ [0, 2) and K2 ≤ 2 − K1, she established Carleman estimates for the operator 1 K1 ut x ux x λ u, t, x QT , − ( ) − xK2 ( ) ∈ unifying the results of [14] and [46] in the purely degenerate operator and in the purely singular one, respectively. This result was then extended in [24] and in [23] to the operators 1 ut a x ux x λ u, t, x QT , (1.3) − ( ( ) ) − xK2 ( ) ∈

K for a ∼ x 1 , K1 ∈ [0, 2) and K2 ≤ 2 − K1. Here, as before, the function a degenerates at the boundary of the space domain, and Dirichlet boundary conditions are in force. We remark the fact that all the papers cited so far, with the exception of [22], consider a singular/degenerate operator with degeneracy or singularity appearing at the boundary of the domain. For example, in (1.3) as a one can also consider the double power function

k κ a(x) = x (1 − x) , x ∈ [0, 1], where k and κ are positive constants. To the best of our knowledge, [8, 29, 30] are the first papers dealing with Carleman estimates (and, consequently, null controllability) for operators (in divergence and in non- divergence form with Dirichlet or Neumann boundary conditions) with mere degeneracy at the interior of the space domain (for related systems of degenerate equations we refer to [7]). We also recall [28] and [27] for other type of control problems associated to parabolic operators with interior degeneracy in divergence and non-divergence form, respectively. G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations Ë 63

We emphasize the fact that an interior degeneracy does not imply a simple adaptation of previous results and of the techniques used for boundary degeneracy. Indeed, imposing homogeneous Dirichlet boundary conditions, in the latter case one knows a priori that any function in the reference functional space vanishes exactly at the degeneracy point. Now, since the degeneracy point is in the interior of the spatial domain, such information is not valid anymore, and we cannot take advantage of this fact. For this reason, the present paper is devoted to study the operator defined in (1.1), that couples agen- eral degenerate diffusion coefficient with a general singular potential with degeneracy and singularity atthe interior of the space domain. In particular, under suitable conditions on all the parameters of the operator, we establish Carleman estimates and, as a consequence, null controllability for the associated generalized heat problem. Clearly, this result generalizes the one obtained in [29, 30]. In fact, if λ = 0 (that is, if we consider the purely degenerate case), we recover the main contributions therein. See also [26] for the problem in non-divergence form for both Dirichlet and Neumann boundary conditions. We also remark the fact that, though we have in mind prototypes as power functions for the degeneracy and the singularity, we don not limit our investigation to these functions, which are analytic out of their zero. Indeed, in this paper, pure powers singularities and degeneracies are considered only as a by-product of our main results, which are valid for non-smooth general coefficients. This is quite a new view-point when dealing with Carleman estimates, since in this framework it is natural to assume that all the coefficients in force are quite regular. However, though this strategy has been successful for years, it is clear that also more irregular coefficients can be considered and appear in a natural way (for instance, see [34, 37]). Nevertheless, itwillbe clear from the proof that Carleman estimates do hold without particular conditions also in the non-smooth setting, while for observability (and thus controllability) another technical condition is needed; however, such a condition is trivially true for the prototypes. For this reason, for the first time to our best knowledge, in [30] non-smooth degenerate coefficients were treated. Continuing in this direction, here we consider operators which contain both degenerate and singular coefficients, as in [23, 24, 45], but with low regularity. The classical approach to study singular operators in dimension 1 relies in the validity of the Hardy– Poincaré inequality

1 1 u2 dx 4 uὔ 2 dx, (1.4) X x2 ≤ X( ) 0 0 1 which is valid for every u ∈ H (0, 1) with u(0) = 0. Similar inequalities are the starting point to prove well- posedness of the associated problems in the Sobolev spaces under consideration. In our situation, we prove an inequality related to (1.4), but with a degeneracy coefficient in the gradient term. Such an estimate isvalid in a suitable Hilbert space H we shall introduce below, and it states the existence of C > 0 such that for all u ∈ H, we have 1 1 u2 dx C a uὔ 2 dx. X b ≤ X ( ) 0 0 This inequality, which is related to another weighted Hardy–Poincaré inequality (see Proposition 2.7), is the key step for the well-posedness of (1.5). Once this is done, global Carleman estimates follow, provided that an ad hoc choice of the weight functions is made (see Theorem 3.3). The introduction of the space H (which may coincide with the usual Sobolev space in some cases) is another feature of this paper, which is completely new with respect to all the previous approaches. Includ- 2 ing the integrability of u /b in the definition of H has the advantage of obtaining immediately some useful functional properties, that in general could be hard to show in the usual Sobolev spaces. Indeed, solutions were already found in suitable function spaces for the “critical” and “supercritical” cases (when λ equals or exceeds the best constant in the classical Hardy–Poincaré inequality) in [47] and [48] for purely singular problems. However, as already done in the purely degenerate case (see [1, 7, 8, 12–14, 23–25, 29–31]), a weighted Sobolev space must be used. For this reason, we believe that it is natural to unify these approaches in the singular/degenerate, as we do. 64 Ë G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations

Now, let us consider the evolution problem λ 6. ut − (aux)x − u = h(t, x)χω(x), (t, x) ∈ QT , 6 b(x) 6 (1.5) > u(t, 0) = u(t, 1) = 0, t ∈ (0, T), 6 6 u 0, x u0 x , x 0, 1 , F ( ) = ( ) ∈ ( ) 2 2 where u0 ∈ L (0, 1), the control h ∈ L (QT) acts on a non-empty interval ω ⊂ (0, 1) and χω denotes the char- acteristic function of ω. 2 As usual, we say that problem (1.5) is null controllable if there exists h ∈ L (QT) such that u(T, x) ≡ 0 for x ∈ [0, 1]. A common strategy to show that (1.5) is null controllable is to prove Carleman estimates for any solution v of the adjoint problem of (1.5) λ . v av v 0, t, x Q , 6 t + ( x)x + b x = ( ) ∈ T 6 ( ) > v(t, 0) = v(t, 1) = 0, t ∈ (0, T), 6 6 v T, x vT x , F ( ) = ( ) and then deduce an observability inequality of the form 1 T 2 2 X v (0, x) dx ≤ CT X X v (t, x) dx dt, (1.6) 0 0 ω where CT > 0 is a universal constant. In the non-degenerate case this has been obtained by a well-established procedure using Carleman and Caccioppoli inequalities. In our singular/degenerate non-smooth situation, we need a new suitable Caccioppoli inequality (see Proposition 4.6), as well as global Carleman estimates in the non-smooth non-degenerate and non-singular case (see Proposition 4.8), which will be used far away from x0 within a localization procedure via cut-off functions. Once these tools are established, we will beable to prove an observability inequality like (1.6), and then controllability results for (1.5). However, we cannot do that in all cases, since we have to exclude that both the degeneracy and the singularity are strong, see condition (SSD) below. Finally, we remark that our studies with non-smooth coefficients are particularly useful. In fact, though null controllability results could be obtained also in other ways, for example by a localization technique (at least when x0 ∈ ω), in [30] it is shown that with non-smooth coefficients, even when λ = 0, this is not always the case. For this, our approach with observability inequalities is very general and permits to cover more involved situations. The paper is organized as follows. In Section 2, we study the well-posedness of problem (1.5), giving some general tools that we shall use several times. In Section 3, we provide one of the main results of this paper, i.e., Carleman estimates for the adjoint problem to (1.5). In Section 4, we apply the previous Carleman estimates to prove an observability inequality, which, together with a Caccioppoli type inequality, lets us derive new null controllability results for the associated singular/degenerate problem, also when the degeneracy and the singularity points are inside the control region. A final comment on the notation: by c or C we shall denote universal positive constants, which are allowed to vary from line to line.

2 Well-posedness

The ways in which a and b degenerate at x0 can be quite different, and for this reason we distinguish four different types of degeneracy. In particular, we consider the following cases.

Hypothesis 2.1 (Doubly weakly degenerate case (WWD)). There exists x0 ∈ (0, 1) such that a(x0) = b(x0) = 1,1 0, a, b > 0 on [0, 1]\{x0}, a, b ∈ W (0, 1), and there exist K1, K2 ∈ (0, 1) such that (x − x0)aὔ ≤ K1a and (x − x0)bὔ ≤ K2b a.e. in [0, 1]. G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations Ë 65

Hypothesis 2.2 (Weakly-strongly degenerate case (WSD)). There exists x0 ∈ (0, 1) such that a(x0) = b(x0) = 1,1 1, 0, a, b > 0 on [0, 1]\{x0}, a ∈ W (0, 1), b ∈ W ∞(0, 1), and there exist K1 ∈ (0, 1) and K2 ≥ 1 such that (x − x0)aὔ ≤ K1a and (x − x0)bὔ ≤ K2b a.e. in [0, 1].

Hypothesis 2.3 (Strongly-weakly degenerate case (SWD)). There exists x0 ∈ (0, 1) such that a(x0) = b(x0) = 1, 1,1 0, a, b > 0 on [0, 1]\{x0}, a ∈ W ∞(0, 1), b ∈ W (0, 1), and there exist K1 ≥ 1 and K2 ∈ (0, 1) such that (x − x0)aὔ ≤ K1a and (x − x0)bὔ ≤ K2b a.e. in [0, 1].

Hypothesis 2.4 (Doubly strongly degenerate case (SSD)). There exists x0 ∈ (0, 1) such that a(x0) = b(x0) = 1, 0, a, b > 0 on [0, 1]\{x0}, a, b ∈ W ∞(0, 1), and there exist K1, K2 ≥ 1 such that (x − x0)aὔ ≤ K1a and (x − x0)bὔ ≤ K2b a.e. in [0, 1]. K K Typical examples for the previous degeneracies and singularities are a(x) = |x − x0| 1 and b(x) = |x − x0| 2 , with 0 < K1, K2 < 2.

Remark 2.5. The restriction Ki < 2 is related to the controllability issue. Indeed, it is clear from the proof of Theorem 2.22 that such a condition is useless, for example, when λ < 0. On the other hand, concerning K controllability, we will not consider the case Ki ≥ 2, since if a(x) = |x − x0| 1 , K1 ≥ 2 and λ = 0, by a standard change of variables (see [30]), problem (1.5) may be transformed in a non-degenerate heat equation on an unbounded domain, while the control remains distributed in a bounded domain. This situation is now well- understood, and the lack of null controllability was proved by Micu and Zuazua in [41].

We will use the following result several times; we state it for a, but an analogous one holds for b replacing K1 with K2.

Lemma 2.6 ([29, Lemma 2.1]). Assume that there exists x0 ∈ (0, 1) such that a(x0) = 0, a > 0 on [0, 1]\{x0}, and either of the following holds: 1,1 ∙ a ∈ W (0, 1) and there exist K1 ∈ (0, 1) such that (x − x0)aὔ ≤ K1a a.e. in [0, 1], 1, ∙ a ∈ W ∞(0, 1) and there exist K1 ∈ [1, 2) such that (x − x0)aὔ ≤ K1a a.e. in [0, 1]. Then, the following hold: γ |x−x0| (i) For all γ ≥ K1 the map x Ü→ a is non-increasing on the left of x = x0 and non-decreasing on the right of x x0, so that = γ |x − x0| lim = 0 for all γ > K1. x→x0 a 1 1 (ii) If K1 < 1, then a ∈ L (0, 1). 1 1 1 1 (iii) If K1 1, 2 , then L 0, 1 and L 0, 1 . ∈ [ ) $a ∈ ( ) a ̸∈ ( ) For the well-posedness of the problem, we start by introducing the following weighted Hilbert spaces, which are suitable to study all situations, namely the (WWD), (SSD), (WSD) and (SWD) cases:

1 1,1 ὔ 2 Ha(0, 1) := u ∈ W0 (0, 1) : $au ∈ L (0, 1) and 1 1 u 2 Ha,b(0, 1) := u ∈ Ha(0, 1) : ∈ L (0, 1), $b endowed with the inner products 1 1

u, v 1 : auὔvὔdx uv dx, ⟨ ⟩Ha(0,1) = X + X 0 0 and 1 1 1 uv ὔ ὔ u, v H1 0,1 : au v dx uv dx dx, ⟨ ⟩ a,b( ) = X + X + X b 0 0 0 respectively. 1 ὔ 2 ὔ ὔ Note that, if u ∈ Ha(0, 1), then au ∈ L (0, 1), since |au | ≤ (max[0,1] $a)$a|u |. We recall the following weighted Hardy–Poincaré inequality, see [29, Proposition 2.6]. 66 Ë G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations

Proposition 2.7. Assume that p ∈ C([0, 1]), p > 0 on [0, 1]\{x0}, p(x0) = 0, and there exists q > 1 such that the following condition is satisfied: p(x) (C1) The function x q is non-increasing on the left of x x0 and non-decreasing on the right of x x0. Ü→ |x−x0| = = Then, there exists a constant CHP > 0 such that for any function w, locally absolutely continuous on the set [0, x0) ∪ (x0, 1] and satisfying

1 2 w(0) = w(1) = 0 with X p(x)|wὔ(x)| dx < +∞, 0 the following inequality holds:

1 1 p x ( ) 2 ὔ 2 X 2 w (x) dx ≤ CHP X p(x)|w (x)| dx. (2.1) x x0 0 ( − ) 0

Remark 2.8. Actually, such a proposition was proved in [29] also requiring q < 2. However, as it is clear from the proof, the result is true without such an upper bound on q, that in [29] was used for other estimates.

Moreover, we will also need other types of Hardy inequalities. Let us start with the following crucial one.

Lemma 2.9. If K1 + K2 ≤ 2 and K2 < 1, then there exists a constant C > 0 such that

1 1 u2 dx C a uὔ 2 dx (2.2) X b ≤ X ( ) 0 0

1 for every u ∈ Ha(0, 1). 2 (x−x0) Proof. We set p(x) := b , so that p satisfies condition (C1) of Proposition 2.7 with q = 2 − K2 > 1, by 1 Lemma 2.6. Thus, taken u ∈ Ha(0, 1), by Proposition 2.7, we get

1 1 1 u2 p x ( ) 2 ὔ 2 X dx = X 2 u dx ≤ CHP X p(x)|u (x)| dx. b x x0 0 0 ( − ) 0

Now, by Lemma 2.6, x x K1 x x K2 2 K1 K2 ( − 0) ( − 0) p(x) = (x − x0) − − a(x) ≤ ca(x) a(x) b(x) for some c > 0, and the claim follows.

Remark 2.10. A similar proof shows that, when K1 + 2K2 ≤ 2 and K2 < 1/2, then

1 1 u2 dx C a uὔ 2 dx X b2 ≤ X ( ) 0 0

1 for every u ∈ Ha(0, 1). 1 1 Lemma 2.9 implies that Ha(0, 1) = Ha,b(0, 1) when K1 + K2 ≤ 2 and K2 < 1. However, inequality (2.2) holds in other cases, see Proposition 2.14 below. In order to prove such a proposition, we need a preliminary result.

1 Lemma 2.11. If K2 ≥ 1, then u(x0) = 0 for every u ∈ Ha,b(0, 1). 1,1 L Proof. Since u ∈ W0 (0, 1), there exists limx→x0 u(x) = L ∈ ℝ. If L ̸= 0, then |u(x)| ≥ 2 in a neighborhood of x0, that is u x 2 L2 | ( )| L1 0, 1 , b ≥ 4b ̸∈ ( ) by Lemma 2.6, and thus L = 0. G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations Ë 67

We also need the following result, whose proof, with the aid of Lemma 2.11, is a simple adaptation of the one given in [31, Lemma 3.2].

Lemma 2.12. If K2 ≥ 1, then

1 1 Hc (0, 1) := u ∈ H0(0, 1) : supp u ⊂ (0, 1)\{x0}

1 is dense in Ha,b(0, 1). In the spirit of [18, Lemma 5.3.1], we are now ready for the following “classical” Hardy inequality in the space 1 α 2−α Ha,b(0, 1) for a(x) = |x − x0| and b(x) = |x − x0| . However, note that our inequality is more interesting than the classical one, since we admit a singularity inside the interval.

Lemma 2.13. For every α ∈ ℝ, the inequality 1 1 1 α 2 u2 ( − ) α ὔ 2 X 2 α dx ≤ X |x − x0| (u ) dx 4 x x0 − 0 | − | 0

1 holds true for every u H α 2−α 0, 1 . ∈ |x−x0| ,|x−x0| ( ) Proof. The case α = 1 is trivial. So, take β = (1 − α)/2 ̸= 0 and ε ∈ (0, 1 − x0).

First case: β < 0 (α > 1). In this case we have

1 1 2 α 2 α β β ὔ 1 X (x − x0) (uὔ) dx = X (x − x0) (x − x0) (x − x0)− u + β(x − x0)− u dx x0+ε x0+ε 1 1 2 α 2 2 α β 1 β ὔ ≥ β X (x − x0) − u dx + 2β X (x − x0) + − u(x − x0)− u dx x0+ε x0+ε 1 1 2 α−2 2 −β 2ᐈ = β X (x − x0) u dx + β(x − x0) u ᐈ (since α + β − 1 = −β) ᐈx0+ε x0+ε 1 2 α 2 2 ≥ β X (x − x0) − u dx. x0+ε

Letting ε → 0+, we get that 1 1 α 2 2 α 2 2 X(x − x0) (uὔ) dx ≥ β X(x − x0) − u dx. (2.3) x0 x0

Second case: β > 0. In this situation we have 2 − α > 1. Thus, in view of Lemma 2.12 with K2 = 2 − α, we 1 1 1 will prove (2.3) first if u Hc 0, 1 and then, by density, if u H α 2−α 0, 1 . Thus, take u Hc 0, 1 ; ∈ ( ) ∈ |x−x0| ,|x−x0| ( ) ∈ ( ) proceeding as above, we get

1 1 1 α ὔ 2 2 α−2 2 −α 2ᐈ X (x − x0) (u ) dx ≥ β X (x − x0) u dx + β(x − x0) u ᐈ ᐈx0+ε x0+ε x0+ε 1 2 α 2 2 ≥ β X (x − x0) − u dx, x0+ε since u(x0 + ε) = 0 for ε small enough. Passing to the limit as ε → 0+, and using Lemma 2.12, we get that 1 (2.3) holds true for every u H α 2−α 0, 1 . ∈ |x−x0| ,|x−x0| ( ) Operating in a symmetric way on the left of x0, we get the conclusion. As a corollary of the previous result, we get the following improvement of Lemma 2.9. 68 Ë G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations

Proposition 2.14. If one among Hypotheses 2.1–2.3 holds with K1 + K2 ≤ 2, then we have that (2.2) holds for 1 every u ∈ Ha,b(0, 1). 1 Proof. By Lemma 2.6 and Lemma 2.13 with α = 2 − K2, we immediately get that for every u ∈ Ha,b(0, 1), 1 1 1 1 1 u2 u2 dx c dx c x x 2−K2 uὔ 2 dx c x x K1 uὔ 2 dx c a uὔ 2 dx. X ≤ X K ≤ X| − 0| ( ) ≤ X| − 0| ( ) ≤ X ( ) b x x0 2 0 0 | − | 0 0 0

Remark 2.15. It is well known that when K1 = K2 = 1, an inequality of the form (2.2) does not hold (see [42]). Being such an inequality fundamental for the observability inequality (see Lemma 4.9), it is no surprise if with our techniques we cannot handle this case in Section 4.

The fundamental space in which we will work is clearly the one where the Hardy–Poincaré-type inequality (2.2) holds. In view of Proposition, it is clear that such a space is

1 H := Ha,b(0, 1). 2 Remark 2.16. Under the assumptions of Proposition 2.14, the standard norm ‖ ⋅ ‖H is equivalent to 1 u 2 : a uὔ 2 dx ‖ ‖∘ = X ( ) 0 for all u ∈ H. Indeed, for all u ∈ H, we have 1 1 1 u2 u2 dx b dx c a uὔ 2 dx, X = X b ≤ X ( ) 0 0 0 and this is enough to conclude the assertion. Moreover, when λ < 0, an equivalent norm is given by 1 1 u2 u 2 : a uὔ 2 dx λ dx. ‖ ‖∼ = X ( ) − X b 0 0 This is particularly useful if Hypothesis 2.4 holds (see the proof of Theorem 2.22).

First, let us call C∗ the best constant of (2.2) in H. From now on, we make the following assumptions on a, b and λ.

Hypothesis 2.17. Either of the following holds: (i) One among Hypotheses 2.1–2.3 holds true with K1 + K2 ≤ 2, and we assume that 1 λ 0, . (2.4) ∈ C∗ (ii) Hypotheses 2.1, 2.2, 2.3 or 2.4 hold with λ < 0.

Observe that the assumption λ ̸= 0 is not restrictive, since the case λ = 0 was already considered in [29] and in [30]. Using the previous lemmas one can prove the next inequality.

Proposition 2.18. Assume Hypothesis 2.17. Then, there exists Λ ∈ (0, 1] such that for all u ∈ H, 1 1 1 u2 a uὔ 2 dx λ dx Λ a uὔ 2 dx. X ( ) − X b ≥ X ( ) 0 0 0 1 Proof. If λ < 0, the result is obvious taking Λ = 1. Now, assume that λ ∈ (0, C∗ ). Then, 1 1 1 1 1 u2 a uὔ 2 dx λ dx a uὔ 2 dx λC∗ a uὔ 2 dx Λ a uὔ 2 dx. X ( ) − X b ≥ X ( ) − X ( ) ≥ X ( ) 0 0 0 0 0 G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations Ë 69

We recall the following definition.

2 2 Definition 2.19. Let u0 ∈ L (0, 1) and h ∈ L (QT). A function u is said to be a (weak) solution of (1.5) if 2 1 u ∈ L (0, T; H) ∩ H ([0, T]; H∗), and it satisfies (1.5) in the sense of H∗-valued distributions.

2 Note that, by [43, Lemma 11.4], any solution belongs to C([0, T]; L (0, 1)). Finally, we introduce the Hilbert space

2 1 ὔ 1 2 Ha,b(0, 1) := u ∈ Ha(0, 1) : au ∈ H (0, 1) and Au ∈ L (0, 1), where λ Au : auὔ ὔ u with D A H2 0, 1 . = ( ) + b ( ) = a,b( ) Remark 2.20. Observe that if u D A , then u and u L2 0, 1 , so that u H1 0, 1 and inequality (2.2) ∈ ( ) b $b ∈ ( ) ∈ a,b( ) holds.

We also recall the following integration by parts with functions in the reference spaces.

Lemma 2.21 (Green’s formula, [31, Lemma 2.3]). Assume one among the Hypotheses 2.1–2.4. Then, for all 2 1 (u, v) ∈ Ha,b(0, 1) × Ha(0, 1) the following identity holds:

1 1 X(auὔ)ὔv dx = − X auὔvὔ dx. (2.5) 0 0 Observe that in the non-degenerate case, it is well known that the heat operator with an inverse-square singular potential u ut − ∆u − λ 2 v |x| gives rise to well-posed Cauchy–Dirichlet problems if and only if λ is not larger than the best Hardy inequality (see [5, 11, 48]). For this reason, it is not strange that we require an analogous condition for problem (1.5), by invoking Hypothesis 2.17. As a consequence, using the standard semigroup theory, we have that (1.5) is well-posed.

2 2 Theorem 2.22. Assume Hypothesis 2.17. For every u0 ∈ L (0, 1) and h ∈ L (QT), there exists a unique solution 2 2 of problem (1.5). In particular, the operator A : D(A) → L (0, 1) is non-positive and self-adjoint in L (0, 1) and it generates an analytic contraction semigroup of angle π/2. Moreover, let u0 ∈ D(A). Then, 1,1 2 1 2 h ∈ W (0, T; L (0, 1)) ⇒ u ∈ C (0, T; L (0, 1)) ∩ C([0, T]; D(A)), 2 1 2 h ∈ L (QT) ⇒ u ∈ H (0, T; L (0, 1)). 2 Proof. Observe that D(A) is dense in L (0, 1). The existence of the unique solution follows in a standard way by a Faedo–Galerkin procedure, see, e.g., [43, Theorem 11.3] or [39, Theorem 3.4.1 and Remark 3.4.3]. Let us prove the other facts. A is non-positive. By Proposition 2.18, Remark 2.16 and Lemma 2.21, for all u ∈ D(A), we have 1 1 1 2 λ 2 u 2 Au, u 2 auὔ ὔ u u dx a uὔ dx λ dx C u . −⟨ ⟩L (0,1) = − X( ) + b = X ( ) − X b ≥ ‖ ‖H 0 0 0

2 2 A is self-adjoint. Let T : L (0, 1) → L (0, 1) be the mapping defined in the following usual way: To each 2 h ∈ L (0, 1) associate the weak solution u = T(h) ∈ H of 1 1 uv auὔvὔ λ dx hv dx X − b = X 0 0 70 Ë G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations for every v ∈ H. Note that T is well defined by the Lax–Milgram Lemma via Proposition 2.18, which also implies that T is continuous. Now, it is easy to see that T is injective and symmetric. Thus, it is self-adjoint. 1 2 As a consequence, A = T− : D(A) → L (0, 1) is self-adjoint (for example, see [44, Proposition A.8.2]). A is m-dissipative. Being A non-positive and self-adjoint, this is a straightforward consequence of [17, Corollary 2.4.8]. Then, (A, D(A)) generates a cosine family and an analytic contractive semigroup of 2 angle π/2 on L (0, 1) (see, for instance, [3, Examples 3.14.16 and 3.7.5]). The additional regularity is a consequence of [17, Lemma 4.1.5 and Proposition 4.1.6] in the first case, and of [2, 6.2.2 and 6.2.4] in the second one.

3 Carleman estimates for singular/degenerate problems

In this section we prove one of the main result of this paper, i.e., a new Carleman estimate with boundary terms for solutions of the singular/degenerate problem λ 6. vt + (avx)x + v = h(t, x) = h, (t, x) ∈ QT , 6 b(x) 6 (3.1) > v(t, 0) = v(t, 1) = 0, t ∈ (0, T), 6 6 v T, x vT x , F ( ) = ( ) which is the adjoint of problem (1.5). On the degenerate function a we make the following assumption.

Hypothesis 3.1. Hypothesis 2.17 holds. Moreover, if K1 > 4/3, then there exists a constant θ ∈ (0, K1] such that the following condition is satisfied: a(x) (C2) The function x θ is non-increasing on the left of x x0 and non-decreasing on the right of x x0. Ü→ |x−x0| = = In addition, when K1 > 3/2 the function in (C2) is bounded below away from 0, and there exists a constant Σ > 0 such that 2θ 3 |aὔ(x)| ≤ Σ|x − x0| − for a.e. x ∈ [0, 1]. (3.2) Moreover, if λ < 0, we require that (x − x0)bὔ(x) ≥ 0 in [0, 1]. (3.3) K Remark 3.2. If a(x) = |x − x0| 1 , then condition (C2) is clearly satisfied with θ = K1. Moreover, the additional requirements for the sub-case K1 > 3/2 are technical ones and were introduced in [30] to guarantee the con- K vergence of some integrals (see [30, Appendix]). Of course, the prototype a(x) = |x − x0| 1 satisfies again such K conditions with θ = K1. Finally, (3.3) is clearly satisfied by the prototype b(x) = |x − x0| 2 .

To prove Carleman estimate, let us introduce the function φ := Θψ, where x 1 y − x0 Θ(t) := 4 and ψ(x) := c1 X dy − c2¡ (3.4) t T t a(y) [ ( − )] x0 x y−x0 with c2 sup 0,1 dy and c1 0 (for the observability inequality, c1 will be taken sufficiently large, > [ ] ∫x0 a(y) > see Lemma 4.7). Observe that Θ(t) → +∞ as t → 0+, T−, and clearly −c1c2 ≤ ψ < 0. The main result of this section is the following

Theorem 3.3. Assume Hypothesis 3.1. Then, there exist two positive constants C and s0, such that every solution v of (3.1) in 2 2 1 V := L (0, T; Ha,b(0, 1)) ∩ H (0, T; H) (3.5) satisfies, for all s ≥ s0, T 2 2 3 3 (x − x0) 2 2sφ 2 2sφ 2sφ t,x 2 x=1 sΘa vx s Θ v e dx dt C h e dx dt sc1 aΘe ( ) x x0 vx dt . X ( ) + a ≤ ¤ X + X ( − )( ) x=0¥ QT QT 0 G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations Ë 71

Remark 3.4. In [46] Vancostenoble and Zuazua proved a related Carleman inequality for the non-degenerate singular one-dimensional problem μ λ v v v h, t, x Q , 6. t + xx + x2 + β = ( ) ∈ T 6 x v t, 0 v t, 1 0, t 0, T , (3.6) 6> ( ) = ( ) = ∈ ( ) 6 v T, x v x , x 0, 1 , F ( ) = T( ) ∈ ( ) where β ∈ [0, 2). When μ = 0 and x0 = 0, such an inequality reads as follows: s v2 s v2 1 s3Θ3x2v2 Θ Θ e2sΨ dx dt h2e2sΨ dx dt, X + 2 x2 + 2 x2/3 ≤ 2 X QT QT

x2 where Ψ(x) = 2 − 1 < 0 in [0, 1]. Actually, it is proved for solutions v such that

v(t, x) = 0 for all (t, x) ∈ (0, T) × (1 − η, 1) and for some η ∈ (0, 1). (3.7) However, in [46, Remark 3.5] Vancostenoble and Zuazua say that Carleman estimates can be proved also for all solutions of (3.6) not satisfying (3.7). We think that this latter situation is much more interesting, since by the Carleman estimates, if h = 0, then v ≡ 0 even if (3.7) does not hold.

The proof of Theorem 3.3 is quite long, and several intermediate lemmas will be used. First, for s > 0, define the function sφ t,x w(t, x) := e ( )v(t, x), where v is any solution of (3.1) in V. Observe that, since v ∈ V and φ < 0, w ∈ V and satisfies e−sφ w e−sφ w a e−sφ w λ h, t, x 0, T 0, 1 , 6. ( )t + ( ( )x)x + = ( ) ∈ ( ) × ( ) 6 b w t, 0 w t, 1 0, t 0, T , (3.8) 6> ( ) = ( ) = ∈ ( ) 6 w T, x w 0, x 0, x 0, 1 . F ( ) = ( ) = ∈ ( ) As usual, we rewrite the previous problem as follows. Setting v Lv : v av λ and L w esφ L e−sφ w , = t + ( x)x + b s = ( ) then (3.8) becomes L w esφ h, . s = 6 w(t, 0) = w(t, 1) = 0, t ∈ (0, T), 6> 6 w T, x w 0, x 0, x 0, 1 . F ( ) = ( ) = ∈ ( ) Computing Ls w, one has + − Ls w = Ls w + Ls w, where w L+w : aw λ sφ w s2aφ2w, s = ( x)x + b − t + x and − Ls w := wt − 2saφx wx − s(aφx)x w. Clearly,

+ − + − + 2 − 2 2 Ls w, Ls w 2 Ls w, Ls w Ls w 2 Ls w 2 ⟨ ⟩ ≤ ⟨ ⟩ + ‖ ‖L (QT ) + ‖ ‖L (QT ) 2 sφ 2 Ls w 2 he 2 , (3.9) = ‖ ‖L (QT ) = ‖ ‖L (QT ) 2 where ⟨ ⋅ , ⋅ ⟩ denotes the scalar product in L (QT). As usual, we will separate the scalar product ⟨L+s w, L−s w⟩ in distributed terms and boundary terms. 72 Ë G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations

Lemma 3.5. The following identity holds:

+ − ⟨Ls w, Ls w⟩ = DT + BT, (3.10) where s D φ w2 dx dt 2s2 aφ φ w2 dx dt s 2a2φ aaὔφ w 2 dx dt T = 2 X tt − X x tx + X( xx + x)( x) QT QT QT aφx bὔ s3 2aφ aὔφ a φ 2w2 dx dt sλ w2 dx dt + X( xx + x) ( x) − X b2 QT QT and T 1 1 2 x=1 s 2 t=T s 2 2 t=T BT awx wt dt w φt dx a φx w dt = X x=0 − 2 X t=0 + 2 X ( ) t=0 0 0 0 T x 1 2 2 2 3 2 3 2 aφx 2 = + X−sφx(awx) + s aφt φx w − s a (φx) w − sλ w dt b x 0 0 = T 1 t=T x=1 1 2 1 2 + X−sa(aφx)x wwxx 0 dt − Xa(wx) − λ w dx, = 2 2b t 0 0 0 = i.e., DT and BT are the distributed and boundary terms, respectively.

Proof. Computing ⟨L+s w, L−s w⟩, one has that

+ − ⟨Ls w, Ls w⟩ = I1 + I2 + I3 + I4, where

2 2 I1 := X(awx)x − sφt w + s a(φx) wwt dx dt,

QT 2 2 I2 := X(awx)x − sφt w + s a(φx) w(−2saφx wx) dx dt,

QT 2 2 I3 := X(awx)x − sφt w + s a(φx) w−s(aφx)x w dx dt,

QT and w I : λ w 2saφ w s aφ w dx dt. 4 = X b t − x x − ( x)x QT

By several integrations by parts in space and in time (see [1, Lemma 3.4], [29, Lemma 3.1] or [30, Lemma 3.1]) and by observing that a aφx xx wwx dx dt 0 (by the very definition of φ), we get ∫QT ( ) = s I I I φ w2 dx dt 2s2 aφ φ w2 dx dt s 2a2φ aaὔφ w 2 dx dt 1 + 2 + 3 = 2 X tt − X x tx + X( xx + x)( x) QT QT QT T 1 3 2 2 x=1 s 2 t=T s 2aφxx aὔφx a φx w dx dt awx wt dt w φt dx + X( + ) ( ) + X x=0 − 2 X t=0 QT 0 0 1 T 2 s 2 2 t=T 2 2 2 3 2 3 2 x=1 a φx w dt sφx awx s aφt φx w s a φx w dt + 2 X ( ) t=0 + X− ( ) + − ( ) x=0 0 0 T 1 x=1 1 2 t=T sa aφx x wwx dt a wx dx. (3.11) + X− ( ) x=0 − 2 X ( ) t=0 0 0 G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations Ë 73

Next, we compute I4: 1 a aφ I λ w2 dx dt 2s φ w w dx dt s ( x)x w2 dx dt 4 = ¤ X 2b ( )t − X b x x − X b ¥ QT QT QT 1 1 2 t=T a 2 (aφx)x 2 λ w dx s φx w x dx dt s w dx dt = ¤ X 2b t=0 − X b ( ) − X b ¥ 0 QT QT 1 T x=1 1 2 t=T a 2 aφx 2 (aφx)x 2 λ w dx s φx w dt s w dx dt s w dx dt = ¤ X t=0 − X + X − X ¥ 2b b x=0 b x b 0 0 QT QT 1 T 1 aφ x=1 aφ bὔ λ w2 t=T dx s x w2 dt s x w2 dx dt . (3.12) = ¤ X t=0 − X − X 2 ¥ 2b b x=0 b 0 0 QT By adding (3.11) and (3.12), (3.10) follows immediately.

For the boundary terms in (3.10), we have the following lemma.

Lemma 3.6. The boundary terms in (3.10) reduce to

T 2 x=1 s Θ awx ψὔ dt. − X ( ) x=0 0 Proof. As in [29] or [30], using the definition of φ and the boundary conditions on w, one has that

T 1 1 2 x=1 s 2 t=T s 2 2 t=T awx wt dt w φt dx a φx w dt X x=0 − 2 X t=0 + 2 X ( ) t=0 0 0 0 T 2 2 2 3 2 3 2 x=1 sφx awx s aφt φx w s a φx w dt + X− ( ) + − ( ) x=0 0 T 1 T x=1 1 2 t=T 2 x=1 sa aφx x wwx dt a wx dx s Θ awx ψὔ dt. (3.13) + X− ( ) x=0 − 2 X ( ) t=0 = − X ( ) x=0 0 0 0

Moreover, since w ∈ V, we have that w ∈ C([0, T]; H); then w(0, x) and w(T, x) are well defined. From the boundary conditions of w, we get that 1 t T 1 2 = X w dx = 0. 2b t 0 0 = Now, consider the last boundary term

T x 1 aφx 2 = sλ X w dt. b x 0 0 = Using the definition of φ, this term becomes

T ὔ x 1 aψ 2 = λ XΘ w dt. b x 0 0 =

aψὔ 2 By the definition of ψ, the function Θ b w is bounded in (0, T). Thus, by the boundary conditions on w, one has T ὔ x 1 aψ 2 = sλ XΘ w dt = 0. b x 0 0 = 74 Ë G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations

Now, the crucial step is to prove the following estimate.

Lemma 3.7. Assume Hypothesis 3.1. Then, there exist two positive constants s0 and C such that for all s ≥ s0, the distributed terms of (3.10) satisfy the estimate s φ w2 dx dt 2s2 aφ φ w2 dx dt s 2a2φ aaὔφ w 2 dx dt 2 X tt − X x tx + X( xx + x)( x) QT QT QT T 1 aφx bὔ s3 2aφ aὔφ a φ 2w2 dx dt sλ w2 dx dt + X X( xx + x) ( x) − X b2 0 0 QT C C3 x x 2 s Θa w 2 dx dt s3 Θ3 ( − 0) w2 dx dt. ≥ 2 X ( x) + 2 X a QT QT Proof. Proceeding as in [29, Lemma 3.2] or in [30, Lemma 4.1], one can prove that, for s large enough, s φ w2 dx dt 2s2 aφ φ w2 dx dt 2 X tt − X x tx QT QT 2 2 3 2 2 + s X(2a φxx + aaὔφx)(wx) dx dt + s X(2aφxx + aὔφx)a(φx) w dx dt

QT QT 3C C3 x x 2 s Θa w 2 dx dt s3 Θ3 ( − 0) w2 dx dt, ≥ 4 X ( x) + 2 X a QT QT where C is a positive constant. Let us remark that one can assume C as large as desired, provided that s0 increases as well. Indeed, taken k > 0, from s s3 CsA C3s3A kC A k3C3 A , 1 + 2 = k 1 + k3 2 ὔ ὔ we can choose s0 = ks0 and C = kC large as needed. Now, we estimate the term aφ bὔ sλ x w2 dx dt. − X b2 QT If λ < 0, the thesis follows immediately by the previous inequality and by (3.3). Otherwise, if λ > 0, by the definition of φ and the assumption on b, one has aφ bὔ aψὔbὔ x x bὔ Θ sλ x w2 dx dt sλ Θ w2 dx dt sλc Θ ( − 0) w2 dx dt sλc K w2 dx dt. − X b2 = − X b2 = − 1 X b2 ≥ − 1 2 X b QT QT QT QT

Since w(t, ⋅ ) ∈ H for every t ∈ [0, 1], for w ∈ V, by (2.2) we get Θ w2 dx dt C∗ Θa w 2 dx dt. X b ≤ X ( x) QT QT Hence, aφx bὔ sλ w2 dx dt sλc K C∗ Θa w 2 dx dt, − X b2 ≥ − 1 2 X ( x) QT QT and we can assume, in view of what remarked above, that this last quantity is greater than C s Θa w 2 dx dt. − 4 X ( x) QT

Summing up, the distributed terms of L+wL−w dx dt can be estimated as ∫QT s s C C3 x x 2 D s Θa w 2 dx dt s3 Θ3 ( − 0) w2 dx dt T ≥ 2 X ( x) + 2 X a QT QT for s large enough and C > 0. G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations Ë 75

From Lemma 3.5, Lemma 3.6 and Lemma 3.7, we deduce immediately that there exist two positive constants C and s0 such that for all s ≥ s0, T 2 2 3 3 (x − x0) 2 2 2 x=1 L+wL−w dx dt Cs Θa wx dx dt Cs Θ w dx dt s Θa w ψὔ dt. (3.14) X s s ≥ X ( ) + X a − X x x=0 QT QT QT 0

Thus, a straightforward consequence of (3.9) and of (3.14) is the next result.

Lemma 3.8. Assume Hypothesis 3.1. Then, there exist two positive constants C and s0 such that for all s ≥ s0,

T 2 2 3 3 (x − x0) 2 2 2sφ t,x 2 2 x=1 s Θa wx dx dt s Θ w dx dt C h e ( ) dx dt s Θa wx ψὔ dt . (3.15) X ( ) + X a ≤ ¤ X + X ( ) x=0 ¥ QT QT QT 0

sφ sφ sφ Recalling the definition of w, we have v = e− w and vx = −sΘψὔe− w + e− wx. Thus, substituting in (3.15), Theorem 3.3 follows.

4 Observability results and application to null controllability

In this section we shall apply the just established Carleman inequalities to observability and controllability issues. For this, we assume that the control set ω satisfies the following assumption.

Hypothesis 4.1. The subset ω is such that either of the following holds: (i) it is an interval which contains the degeneracy point, i.e.,

ω = (α, β) ⊂ (0, 1) and x0 ∈ ω, (4.1)

(ii) it is an interval lying on one side of the degeneracy point, i.e.,

ω = (α, β) ⊂ (0, 1) and x0 ̸∈ ω̄ . (4.2)

On the coefficients a and b we essentially start with the assumptions made so far, with the exception of Hypothesis 2.4, and we add another technical one. We summarize all of them in the following.

Hypothesis 4.2. ∙ Assume one among Hypotheses 2.1–2.3 with K1 + K2 ≤ 2 and λ < 1/C∗. ∙ If λ < 0, (3.3) holds. ∙ If K1 > 4/3, condition (C2) holds, and if K1 > 3/2, (3.2) is satisfied. ∞ 1,∞ ∙ If Hypothesis 2.1 or 2.2 holds, there exist two functions g ∈ Lloc([0, 1]\{x0}), h ∈ Wloc ([0, 1]\{x0}) and two strictly positive constants g0, h0 such that g(x) ≥ g0 for a.e. x in [0, 1] and

B aὔ(x) − ¤ X g(t) dt + h0¥ + xa(x)g(x) = h(x, B) (4.3) 2 a x x ( ) x

for a.e. x, B ∈ [0, 1] with x < B < x0 or x0 < x < B.

Remark 4.3. Since we require identity (4.3) far from x0, once a is given, it is easy to find g, h, g0 and h0 with α the desired properties. For example, if a(x) := |x − x0| , α ∈ (0, 1), we can take g(x) ≡ g0 = h0 = 1 and

α 1 α h x, B x x 2 − sign x x B 1 x x x ( ) = | − 0| 2 ( − 0)( + − ) + | − 0| for all x and B ∈ [0, 1] with x < B < x0 or x0 < x < B. Clearly,

∞ 1,∞ ∞ g ∈ Lloc([0, 1]\{x0}) and h ∈ Wloc ([0, 1]\{x0}; L (0, 1)). 76 Ë G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations

Now, we associate to problem (1.5) the homogeneous adjoint problem λ 6. vt + (avx)x + v = 0, (t, x) ∈ QT , 6 b(x) 6 (4.4) > v(t, 0) = v(t, 1) = 0, t ∈ (0, T), 6 6 v T, x vT x , F ( ) = ( ) 2 where T > 0 is given and vT(x) ∈ L (0, 1). By the Carleman estimate in Theorem 3.3, we will deduce the following observability inequality for all the degenerate cases.

Proposition 4.4. Assume Hypotheses 4.1 and 4.2. Then, there exists a positive constant CT such that every 2 2 solution v ∈ C([0, T]; L (0, 1)) ∩ L (0, T; H) of (4.4) satisfies 1 T 2 2 X v (0, x) dx ≤ CT X X v (t, x) dx dt. (4.5) 0 0 ω Using the observability inequality (4.5) and a standard technique (e.g., see [38, Section 7.4]), one can prove the following null controllability result for the linear degenerate problem (1.5).

2 2 Theorem 4.5. Assume Hypotheses 4.1 and 4.2. Then, given u0 ∈ L (0, 1), there exists h ∈ L (QT) such that the solution u of (1.5) satisfies u(T, x) = 0 for every x ∈ [0, 1]. Moreover, 1 2 2 X h dx dt ≤ C X u0(x) dx

QT 0 for some positive constant C.

4.1 Proof of Proposition 4.4

In this subsection we will prove, as a consequence of the Carleman estimate proved in Section 3, the observability inequality (4.5). For this purpose, we will give some preliminary results. As a first step, consider the adjoint problem v Av 0, t, x Q , . t + = ( ) ∈ T 6 v(t, 0) = v(t, 1) = 0, t ∈ (0, T), (4.6) 6> 6 v T, x v x D A2 , F ( ) = T( ) ∈ ( ) where 2 D(A ) = u ∈ D(A) : Au ∈ D(A) and u Au : au λ . = ( x)x + b 2 Observe that D(A ) is densely defined in D(A) for the graph norm (see, for example, [9, Lemma 7.2]) and 2 hence in L (0, 1). As in [12, 13, 25, 29], define the following class of functions:

W := v is a solution of (4.6). Obviously (see, for example, [9, Theorem 7.5]),

1 2 W ⊂ C ([0, T]; Ha,b(0, 1)) ⊂ V ⊂ U, where V is defined in (3.5) and

2 2 U := C([0, T]; L (0, 1)) ∩ L (0, T; H). (4.7) We start with the following proposition. G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations Ë 77

Proposition 4.6 (Caccioppoli’s inequality). Assume Hypothesis 2.17. Let ωὔ and ω be two open subintervals of (0, 1) such that ωὔ ⊂⊂ ω ⊂ (0, 1) and x0 ̸∈ ω̄ ὔ. Let φ(t, x) = Θ(t)Υ(x), where Θ is defined in (3.4) and 1 Υ ∈ C([0, 1], (−∞, 0)) ∩ C ([0, 1]\{x0}, (−∞, 0)) is such that c |Υx| ≤ in [0, 1]\{x0} (4.8) $a for some c > 0. Then, there exist two positive constants C and s0 such that every solution v ∈ W of the adjoint problem (4.6) satisfies T T 2 2sφ 2 X X(vx) e dx dt ≤ C X X v dx dt (4.9) 0 ωὔ 0 ω for all s ≥ s0.

Of course, our prototype for Υ is the function ψ defined in (3.4), since 2 x x0 1 1 ψὔ x c | − | c , | ( )| = 1{ a x ≤ ( ) xa(x) xa(x) by Lemma 2.6.

Proof of Proposition 4.6. The proof follows the one of [29, Proposition 4.2], but it is different due to the pres- ence of the singular term. Let us consider a smooth function ξ : [0, 1] → ℝ such that 0 ≤ ξ(x) ≤ 1 for all x ∈ [0, 1], ξ(x) = 1, x ∈ ωὔ, ξ(x) = 0, x ∈ [0, 1]\ ω. Since v solves (4.6), we have T 1 d 0 ξ 2e2sφ v2 dx dt = X dt ¤ X ¥ 0 0 2 2sφ 2 2 2sφ = X 2sξ φt e v + 2ξ e vvt dx dt

QT v 2 ξ 2sφ e2sφ v2 dx dt 2 ξ 2e2sφ v λ av dx dt = X t + X − b − ( x)x QT QT v2 2 ξ 2sφ e2sφ v2 dx dt 2λ ξ 2e2sφ dx dt 2 ξ 2e2sφ v av dx dt. (4.10) = X t − X b + X( )x x QT QT QT If λ < 0, then, differentiating the last term in (4.10), weget v2 2 ξ 2e2sφ a v 2 dx dt 2λ ξ 2e2sφ dx dt 2 ξ 2sφ e2sφ v2 dx dt 2 ξ 2e2sφ avv dx dt X ( x) = X b − X t − X( )x x QT QT QT QT 2 2sφ 2 2 2sφ ≤ −2 X ξ sφt e v dx dt − 2 X(ξ e )x avvx dx dt,

QT QT and then one can proceed as in the proof of [29, Proposition 4.2], obtaining the claim. sφ Otherwise, if λ > 0 and ε > 0 is fixed, then by the Cauchy–Schwarz inequality, for w = ξe v, we have 1 1 v2 ξ 2e2sφ dx C∗ a w 2 dx X b ≤ X ( x) 0 0 1 1 sφ 2 2 2 2sφ 2 ≤ Cε X a[(ξe )x] v dx + ε X ξ e a(vx) dx for some Cε > 0. 0 0 78 Ë G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations

Moreover,

sφ 2 2sφ 2 2 sφ [(ξe )x] ≤ Cχωe + s (φx) e 1 Cχ 1 ≤ ω + a

2sφ 2 2 2sφ for some positive constant C. Indeed, e < 1, while s (φx) e can be estimated with

c 2 c 2 (Υx) ≤ , (− max Υ) a by (4.8), for some constants c > 0. Thus, v2 2λ ξ 2e2sφ dx dt 2λC a ξesφ 2v2 dx dt 2λε ξ 2e2sφ a v 2 dx dt X b ≤ ε X [( )x] + X ( x) QT QT QT T 2 2 2sφ 2 ≤ C X X v dx dt + 2λε X ξ e a(vx) dx dt (4.11)

0 ω QT for a positive constant C depending on ε. Hence, differentiating the last term in (4.10) and using (4.11), we get

v2 2 ξ 2e2sφ a v 2 dx dt 2λ ξ 2e2sφ dx dt 2 ξ 2sφ e2sφ v2 dx dt 2 ξ 2e2sφ avv dx dt X ( x) = X b − X t − X( )x x QT QT QT QT T 2 2 2sφ 2 ≤ C X X v dxdt + 2λε X ξ e a(vx) dx dt

0 ω QT 2 2sφ 2 2 2sφ − 2 X ξ sφt e v dx dt − 2 X(ξ e )x avvx dx dt.

QT QT

Thus, applying again the Cauchy–Schwarz inequality, we get

T 2 2sφ 2 2 2 2sφ 2 2 2sφ (2 − 2λε) X ξ e a(vx) dx dt ≤ C X X v dx dt − 2 X ξ sφt e v dx dt − 2 X(ξ e )x avvx dx dt

QT 0 ω QT QT T T 2 2 2sφ 2 ≤ C X X v dx dt − 2 X X ξ sφt e v dx dt 0 ω 0 ω T T ξ 2e2sφ 2ε aξesφ v 2 dx dt D a ( )x v 2 dx dt + X X$ x + ε X X$ ξesφ 0 ω 0 ω T T 2 2 2sφ 2 = C X X v dx dt − 2 X X ξ sφt e v dx dt 0 ω 0 ω T T ξ 2e2sφ 2 2ε ξ 2e2sφ a v 2 dx dt D [( )x] av2 dx dt + X X ( x) + ε X X ξ 2e2sφ 0 ω 0 ω for some Dε > 0. Hence, T 2 2sφ 2 2(1 − ε − λε) X X ξ e a(vx) dx dt 0 ω T T T ξ 2e2sφ 2 C v2 dx dt 2 ξ 2sφ e2sφ v2 dx dt D [( )x] av2 dx dt. ≤ X X − X X t + ε X X ξ 2e2sφ 0 ω 0 ω 0 ω G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations Ë 79

Since x0 ̸∈ ω̄ ὔ, we have T T 2sφ 2 2 2sφ 2 2 1 ε λε inf a x e vx dx dt 2 1 ε λε ξ e a vx dx dt ( − − ) ωὔ ( ) X X ( ) ≤ ( − − ) X X ( ) 0 ωὔ 0 ω̄ ὔ T 2 2sφ 2 ≤ 2(1 − ε − λε) X X ξ e a(vx) dx dt 0 ω T T 2 2 2sφ 2 ≤ C X X v dx dt − 2 X X ξ sφt e v dx dt 0 ω 0 ω T ξ 2e2sφ 2 D [( )x] av2 dx dt. + ε X X ξ 2e2sφ 0 ω Finally, we show that there exists a positive constant C (still depending on ε) such that

T T T ξ 2e2sφ 2 2 ξ 2sφ e2sφ v2 dx dt D [( )x] av2 dx dt C v2 dx dt, − X X t + ε X X ξ 2e2sφ ≤ X X 0 ω 0 ω 0 ω so that the claim will follow. Indeed,

2 2sφ 2 [(ξ e )x] ξ 2e2sφ can be estimated by 2sφ 2 2 2sφ Ce + s (φx) e χω , 1 and proceeding as before, we get the claim, choosing ε small enough, namely ε < (1 + λ)− . We shall also use the following lemma.

Lemma 4.7. Assume Hypotheses 4.1 and 4.2. Then, there exist two positive constants C and s0 such that every solution v ∈ W of (4.6) satisfies, for all s ≥ s0, T x x 2 sΘa v 2 s3Θ3 ( − 0) v2 e2sφ dx dt C v2 dx dt. X ( x) + a ≤ X X QT 0 ω

Here Θ and φ are as in (3.4) with c1 sufficiently large. Using the following non-degenerate classical Carleman estimate, one has that the proof of the previous lemma is a simple adaptation of the proof of [30, Lemma 5.1 and 5.2], to which we refer, also to explain why c1 must be large. Proposition 4.8 (Non-degenerate non-singular Carleman estimate). Let z be the solution of

z 2 zt + (azx)x + λ = h ∈ L ((0, T) × (A, B)), . b (4.12) > z t, A z t, B 0, t 0, T , F ( ) = ( ) = ∈ ( ) 80 Ë G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations where b ∈ C([A, B]) is such that b ≥ b0 > 0 in [A, B] and a satisfies either of the following: 1,1 1 1, (a1) a ∈ W (A, B), a ≥ a0 > 0 in (A, B) and there exist two functions g ∈ L (A, B), h ∈ W ∞(A, B) and two strictly positive constants g0, h0 such that g(x) ≥ g0 for a.e. x in [A, B] and B aὔ(x) − ¤ X g(t) dt + h0¥ + xa(x)g(x) = h(x) for a.e. x ∈ [A, B]. 2 a x x ( ) x 1, (a2) a ∈ W ∞(A, B) and a ≥ a0 > 0 in (A, B). Then, for all λ ∈ ℝ, there exist three positive constants C, r and s0 such that for any s > s0, T B T B 2 3 3 2 2sΦ 2 2sΦ X XsΘ(zx) + s Θ z e dx dt ≤ C¤ X X h e dx dt − BT¥, (4.13) 0 A 0 A where T B x B . = 6sr a3/2e2sΦ Θ g τ dτ h z 2 dt if (a ) holds, 6 X ¤ X ( ) + 0¥( x) ¡ 1 6 x A B 6 0 x = T = T 6> x=B 6 2sΦ rζ 2 6sr ae Θe vx dt if (a2) holds. 6 X ( ) ¡ x=A F 0 Here the function Φ is defined as Φ(t, x) := Θ(t)ρ(x), where Θ is as in (3.4), x B x 6. 1 h0 6−r X X g(s) ds dt + X dt¡ − c if (a1) holds, ρ(x) := xa(t) xa(t) (4.14) 6> A t A 6 rζ x e ( ) c if (a2) holds, F − and B 1 ζ x d dt. ( ) = X a t x ( )

d ὔ ∞ c Here = ‖a ‖L (A,B) and > 0 is chosen, in the second case, in such a way that max[A,B] ρ < 0. z Proof. Rewrite the equation of (4.12) as zt + (azx)x = h̄, where h̄ := h − λ b . Then, applying [30, Theorem 3.1], there exist two positive constants C and s0 > 0, such that T B T B 2 3 3 2 2sΦ 2 2sΦ X XsΘ(zx) + s Θ z e dx dt ≤ C¤ X X h̄ e dx dt − BT¥ (4.15) 0 A 0 A T B for all s s . Using the definition of h, the term e2sΦ h2 dx dt can be estimated in the following way: ≥ 0 ̄ ∫0 ∫A ̄ T B T B T B z2 h̄2e2sΦ dx dt 2 h2e2sΦ dx dt 2λ2 e2sΦ dx dt. (4.16) X X ≤ X X + X X b2 0 A 0 A 0 A sΦ 2 Applying the classical Poincaré inequality to w(t, x) := e z(t, x) and observing that 0 < inf Θ ≤ Θ ≤ cΘ , one has T B T B T B 2 2 2 2 z 2sΦ 2 w λ 2 2λ e dx dt 2λ dx dt 2 C wx dx dt X X b2 = X X b2 ≤ b2 X X( ) 0 A 0 A 0 0 A T B 2 2 2 2 2sΦ ≤ C X Xs Θ z + (zx) e dx dt 0 A T B T B s s3 Θ z 2e2sΦ dx dt Θ3z2e2sΦ dx dt ≤ X X 2 ( x) + X X 2 0 A 0 A G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations Ë 81 for s large enough. Using this last inequality in (4.16), we have

T B T B T B T B s s3 h̄2e2sΦ dx dt 2 e2sΦ h2 dx dt Θ z 2e2sΦ dx dt Θ3z2e2sΦ dx dt. (4.17) X X ≤ X X + X X 2 ( x) + X X 2 0 A 0 A 0 A 0 A Using this inequality in (4.15), (4.13) follows immediately.

In order to prove Proposition 4.4, the last result that we need is the following.

Lemma 4.9. Assume Hypotheses 4.1 and 4.2. Then, there exists a positive constant CT such that every solution v ∈ W of (4.6) satisfies 1 T 2 2 X v (0, x) dx ≤ CT X X v (t, x) dx dt. 0 0 ω

Proof. Multiplying the equation of (4.6) by vt and integrating by parts over (0, 1), one has 1 v 0 v av λ v dx = X t + ( x)x + b t 0 1 vv v2 av v λ t dx = X t + ( x)x t + b 0 1 1 1 2 2 x=1 λ d v v dx avx vt avx vtx dx dx = X t + x=0 − X + 2 dt X b 0 0 0 1 1 1 1 d λ d v2 v2 dx a v 2 dx = X t − 2 dt X ( x) + 2 dt X b 0 0 0 1 1 1 d λ d v2 a v 2 dx dx. ≥ −2 dt X ( x) + 2 dt X b 0 0 Thus, the function 1 1 v2 t a v 2 dx λ dx Ü→ X ( x) − X b 0 0 is non-decreasing for all t ∈ [0, T]. In particular, 1 1 1 1 1 v2 0, x v2 t, x a v 2 0, x dx λ ( ) dx a v 2 t, x dx λ ( ) dx 1 λ C∗ a v 2 t, x dx, X ( x) ( ) − X b x ≤ X ( x) ( ) − X b x ≤ ( + | | ) X ( x) ( ) 0 0 ( ) 0 0 ( ) 0 by Proposition 2.14. Integrating the previous inequality over [T/4, 3T/4], Θ being bounded therein, we find 1 1 3T 4 1 2 / 2 v (0, x) 2 2 X a(x)(vx) (0, x) dx − λ X dx ≤ (1 + |λ|C∗) X X a(vx) dx dt b(x) T 0 0 T/4 0 3T/4 1 2 2sφ ≤ CT X X sΘa(vx) e dx dt. (4.18) T/4 0 Hence, from the previous inequality and Lemma 4.7, if λ < 0, then 1 1 1 T v2 0, x a v 2 0, x dx a v 2 0, x dx λ ( ) dx C v2dxdt X ( x) ( ) ≤ X ( x) ( ) − X b x ≤ X X 0 0 0 ( ) 0 ω for some positive constant C > 0. 82 Ë G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations

If λ > 0, using again Lemma 4.7 and (4.18), one has

1 1 T v2 0, x a v 2 0, x dx λ ( ) dx C v2 dx dt. (4.19) X ( x) ( ) − X b x ≤ X X 0 0 ( ) 0 ω

Hence, by (2.2) and (4.19), we have

1 1 T 1 T v2 0, x a v 2 0, x dx λ ( ) dx C v2 dx dt λC∗ a v 2 0, x dx C v2 dx dt. X ( x) ( ) ≤ X b x + X X ≤ X ( x) ( ) + X X 0 0 ( ) 0 ω 0 0 ω

Thus, 1 T 2 2 (1 − λC∗) X a(vx) (0, x) dx ≤ C X X v dx dt 0 0 ω for a positive constant C. In every case, there exists C > 0 such that

1 T 2 2 X a(vx) (0, x) dx ≤ C X X v dx dt. (4.20) 0 0 ω

The Hardy–Poincaré inequality (see Proposition 2.7) and (4.20) imply that

1 1 1 3 a / 2 p 2 X 2 v (0, x) dx ≤ X 2 v (0, x) dx x x0 x x0 0 ( − ) 0 ( − ) 1 2 ≤ CHP X p(vx) (0, x) dx 0 1 2 ≤ cCHP X a(vx) (0, x) dx 0 T 2 ≤ C X X v dx dt 0 ω

4 1/3 4/3 1/3 for a positive constant C. Here p(x) = (a(x)|x − x0| ) if K1 > 4/3, while p(x) = |x − x0| max[0,1] a otherwise, and c, C are obtained by Lemma 2.6. Again by Lemma 2.6, we have

1 3 1 3 1 3 a(x) / a(1) / a(0) / 2 ≥ C3 := min¦ 2 , 2 § > 0. (x − x0) (1 − x0) x0 Hence, 1 T 2 2 C3 X v(0, x) dx ≤ C X X v dx dt 0 0 ω and the claim follows.

Proof of Proposition 4.4. It follows by a density argument as for the proof of [29, Proposition 4.1].

Funding: The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), and supported by the GNAMPA project 2014 Systems with irregular operators. Dimitri Mugnai was also supported by the M.I.U.R. project Vari- ational and perturbative aspects of nonlinear differential problems. G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations Ë 83

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