Carleman estimates for singular parabolic equations with interior degeneracy and non-smooth coefficients

1 Introduction

Controllability issues for parabolic problems have been a mainstream topic in recent years, and several developments 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

associated to Dirichlet boundary conditions and with (t,x)∈QT:=(0,T)×(0,1), T>0 being a fixed number. 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 in mind are a⁢(x)=|x-x0|K1 and b⁢(x)=|x-x0|K2 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. In particular, new Carleman estimates (and consequently null controllability properties) were established in [1], and also in [15, 40], for the operator

where a⁢(0)=a⁢(1)=0, a∈C1⁢(0,1) and c∈L∞⁢(QT) (see also [13, 12, 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

with associated Dirichlet boundary conditions in a bounded domain Ω⊂ℝN containing the singularity x=0 in the interior (see also [47] for the wave and Schrödinger equations and [16] for boundary singularity). Similar operators of the form

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, 46] 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

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

for a∼xK1,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

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.

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 a general degenerate diffusion coefficient with a general singular potential with degeneracy and singularity at the 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, it will be 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 [24, 23, 45], but with low regularity.

The classical approach to study singular operators in dimension 1 relies in the validity of the Hardy–Poincaré inequality

which is valid for every u∈H1⁢(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 is valid 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

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. Including the integrability of u2/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, 13, 12, 14, 24, 23, 25, 29, 30, 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.

Now, let us consider the evolution problem

where u0∈L2⁢(0,1), the control h∈L2⁢(QT) acts on a non-empty interval ω⊂(0,1) and χω denotes the characteristic function of ω.

As usual, we say that problem (1.5) is null controllable if there exists h∈L2⁢(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)

and then deduce an observability inequality of the form

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 be able 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.

Typical examples for the previous degeneracies and singularities are a⁢(x)=|x-x0|K1 and b⁢(x)=|x-x0|K2, with 0<K1,K2<2.

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

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:

endowed with the inner products

respectively.

Note that, if u∈Ha1⁢(0,1), then a⁢u′∈L2⁢(0,1), since |a⁢u′|≤(max[0,1]⁡a)⁢a⁢|u′|.

We recall the following weighted Hardy–Poincaré inequality, see [29, Proposition 2.6].

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

Lemma 2.9 implies that Ha1⁢(0,1)=Ha,b1⁢(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.

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].

In the spirit of [18, Lemma 5.3.1], we are now ready for the following “classical” Hardy inequality in the space Ha,b1⁢(0,1) for a⁢(x)=|x-x0|α and b⁢(x)=|x-x0|2-α. However, note that our inequality is more interesting than the classical one, since we admit a singularity inside the interval.

As a corollary of the previous result, we get the following improvement of Lemma 2.9.

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

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.

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.

We recall the following definition.

Note that, by [43, Lemma 11.4], any solution belongs to C⁢([0,T];L2⁢(0,1)).

Finally, we introduce the Hilbert space

where

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

Observe that in the non-degenerate case, it is well known that the heat operator with an inverse-square singular potential

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.

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

which is the adjoint of problem (1.5).

On the degenerate function a we make the following assumption.

To prove Carleman estimate, let us introduce the function φ:=Θ⁢ψ, where

with c2>sup[0,1]⁡∫x0xy-x0a⁢(y)⁢𝑑y and c1>0 (for the observability inequality, c1 will be taken sufficiently large, see Lemma 4.7). Observe that Θ⁢(t)→+∞ as t→0+,T-, and clearly -c1⁢c2≤ψ<0.