arXiv:1507.07786v1 [math.AP] 28 Jul 2015
Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients∗ Genni Fragnelli Dipartimento di Matematica Universit`a di Bari ”Aldo Moro” Via E. Orabona 4 70125 Bari - Italy email:
[email protected] Dimitri Mugnai Dipartimento di Matematica e Informatica Universit`a di Perugia Via Vanvitelli 1, 06123 Perugia - Italy email:
[email protected]
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, Hardy–Poincar´e inequality, Caccioppoli inequality, observability inequality, null controllability 2000AMS Subject Classification: 35Q93, 93B05, 93B07, 34H15, 35A23, 35B99
1
Introduction
Controllability issues for parabolic problems have been a mainstream issue in recent years, and several developments have been pursued: starting from the heat equation in bounded and unbounded domain, related contributions have ∗ The authors are member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), and are supported by the INdAM - GNAMPA Project Systems with irregular operators
1
been found for more general situations. A common strategy in showing controllability results is to show 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 ut − (a(x)ux )x −
λ u, b(x)
(1.1)
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 (2.15) 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. However, the prototypes are a(x) = |x − x0 |K1 and b(x) = |x − x0 |K2 for some K1 , K2 > 0. The main goal we have in mind 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., Fursikov–Imanuvilovl [27]). 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 [30]), Biology (see [19]) and Mathematical Finance (see [29]) are described by degenerate parabolic equations. In particular, new Carleman estimates (and consequently null controllability properties) were established in [1], and also in [13], [33], for the operator ut − (aux )x + c(t, x)u,
(t, x) ∈ QT ,
where a(0) = a(1) = 0 and c ∈ L∞ (QT ) (see also [11], [12] or [23] for problems in non divergence form). An interesting situation is the case of parabolic operators with singular lower order terms. First results in this direction were obtained in [38] for the non degenerate heat operator with singular potentials ut − ∆u − λ
1 u, |x|2
(t, x) ∈ (0, T ) × Ω,
(1.2)
with associated Dirichlet boundary conditions in a bounded domain Ω containing 0 (see also [37] for the wave and Schr¨odinger equations and [15] for boundary degeneracy). The case K2 = 2 is the case of the so-called inverse square potential that arises for example in quantum mechanics (see, for example, [3], [17]), or in combustion problems (see, for example, [5], [9], [18], [28]). This potential is known to generate interesting phenomena. For example, in [3] and in [4] 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: 1 if λ ≤ (which is the optimal constant of the Hardy inequality) global positive 4 2
1 , instantaneous and complete blow-up occurs 4 (for other comments on this argument we refer to [36]). We recall that in [38], 1 Carleman estimates were established for (1.2) under the condition λ ≤ . On 4 1 the contrary, if λ > , in [20] it was proved that null controllability fails. 4 Recently, in [36], Vancostenoble studied the operator that couples a degenerate diffusion coefficient with a singular potential. In particular, for K1 ∈ [0, 2) and K2 ≤ 2 − K1 , the author established Carleman estimates for the operator solutions exist, while, if λ >
ut − (xK1 ux )x − λ
1 u, xK2
(t, x) ∈ QT ,
unifying the results of [14] and [38] in the purely degenerate operator and in the purely singular one, respectively. This result was then extended in [21] and in [22] to the operators ut − (a(x)ux )x − λ
1 xK2
u,
(t, x) ∈ QT ,
(1.3)
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 deal with a singular/degenerate operator with degeneracy or singularity at the boundary of the domain. For example, in (1.3) as a one can also consider the double power function a(x) = xk (1 − x)κ , x ∈ [0, 1], where k and κ are positive constants. To the best of our knowledge, [7], [24] and [25] 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 [6]). We emphasize the fact that an interior degeneracy does not imply a simple adaptation of previous results and of the techniques for boundary ones. Indeed, imposing homogeneous Dirichlet boundary conditions, in the latter case one knows a priori that any function 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, in general, and proofs 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 3
the one obtained in [24] or [25]: in fact, if λ = 0 (that is, if we consider the purely degenerate case), we recover the main contributions therein. 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 play a rˆole. For this reason, for the first time to our best knowledge, in [25] non smooth degenerate coefficients were treated. Continuing in this direction, here we consider operators which contain both degenerate and singular coefficients, as in [21], [22] and [36], but with low regularity. The classical approach to study singular operators relies in the validity of the Hardy–Poincar´e inequality Z 1 Z 1 2 u dx ≤ 4 (u′ )2 dx, (1.4) 2 0 0 x which is valid for every u ∈ H 1 (0, 1) with u(0) = 0 (or of analogous ones for operators acting in higher dimensions). 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 Z 1 Z 1 2 u a(u′ )2 dx. dx ≤ C b 0 0 This inequality, which is related to another weighted Hardy-Poincar´e inequality (see Proposition 2.1), 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 3.1). 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. Indeed, including the validity of a Hardy–Poincar´e inequality with double weight 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. The choice of suitable function spaces where posing the problem was already done for the “critical” and “superctitical” cases (when λ equals or exceeds the best constant in the classical Hardy–Poincar´e inequality) in [37]and [39] for purely singular problems. However, as already done in the purely degenerate case ([1, 6, 7, 11, 12, 14, 21, 22, 23, 24, 25, 26]), a weighted Sobolev space must be used. 4
For this reason, we believe that it is natural to unify these approaches in the singular/degenerate, as we do. As it is well known, Carleman estimates are a crucial step in proving null controllability properties for the corresponding evolution problem λ ut − (aux )x − b(x) u = h(t, x)χω (x), (t, x) ∈ QT , (1.5) u(t, 0) = u(t, 1) = 0, t ∈ (0, T ), u(0, x) = u0 (x), x ∈ (0, 1),
i.e. in showing that there exists h ∈ L2 (QT ) such that u(T, x) ≡ 0 for x ∈ [0, 1]. Here, u0 ∈ L2 (0, 1), the control h ∈ L2 (QT ) acts on a non empty interval ω ⊂ (0, 1) and χω is the characteristic function of ω. In order to obtain such a result, the usual strategy is to use Carleman estimates to prove an observability inequality of the form Z
0
1
v 2 (0, x)dx ≤ CT
Z
0
T
Z
v 2 (t, x)dxdt
(1.6)
ω
for any solution v of the adjoint problem of (1.5) λ v = 0, (t, x) ∈ QT , v + (avx )x + t b(x) v(t, 0) = v(t, 1) = 0, t ∈ (0, T ), v(T, x) = vT (x),
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 4.2), as well as global Carleman estimates in the non smooth non degenerate and non singular case (see Proposition 4.3), which will be used far away from x0 within a localization procedure via cut–off functions. Once these tools are established, we are able to prove an observability inequality like (1.6), and then controllability results for (1.5). 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 [25] 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 in the following way: 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,
5
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. Acknowledgments. The authors are very grateful to both the anonymous referees for the careful reading of the paper and for their comments and remarks, which lead to a much better organization of the paper.
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. Double weakly degenerate case (WWD): there exists x0 ∈ (0, 1) such that a(x0 ) = b(x0 ) = 0, a, b > 0 on [0, 1] \ {x0 }, a, b ∈ W 1,1 (0, 1) and there exists K1 , K2 ∈ (0, 1) such that (x−x0 )a′ ≤ K1 a and (x−x0 )b′ ≤ K2 b a.e. in [0, 1]. Hypothesis 2.2. Double strongly degenerate case (SSD): there exists x0 ∈ (0, 1) such that a(x0 ) = b(x0 ) = 0, a, b > 0 on [0, 1]\{x0 }, a, b ∈ W 1,∞ (0, 1) and there exist K1 , K2 ∈ [1, 2) such that (x − x0 )a′ ≤ K1 a and (x − x0 )b′ ≤ K2 b a.e. in [0, 1]. Hypothesis 2.3. Weakly-strongly degenerate case (WSD): there exists x0 ∈ (0, 1) such that a(x0 ) = b(x0 ) = 0, a, b > 0 on [0, 1] \ {x0 }, a ∈ W 1,1 (0, 1), b ∈ W 1,∞ (0, 1) and there exist K1 ∈ (0, 1), K2 ∈ [1, 2) such that (x − x0 )a′ ≤ K1 a and (x − x0 )b′ ≤ K2 b a.e. in [0, 1]. Hypothesis 2.4. Strongly-weakly degenerate case (SWD): there exists x0 ∈ (0, 1) such that a(x0 ) = b(x0 ) = 0, a, b > 0 on [0, 1] \ {x0 }, a ∈ W 1,∞ (0, 1), b ∈ W 1,1 (0, 1), and there exist K1 ∈ [1, 2), K2 ∈ (0, 1) such that (x − x0 )a′ ≤ K1 a and (x − x0 )b′ ≤ K2 b a.e. in [0, 1]. 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. Remark 1. We do not consider the cases Ki ≥ 2, since if a(x) = xKi and Ki ≥ 2, by a standard change of variables 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, even in the case λ = 0 where the lack of null controllability was proved by Micu and Zuazua in [34]. We will use the following result several times; we state it for a, but an analogous one holds for b replacing K1 with K2 : 6
Lemma 2.1 (Lemma 2.1, [24]). Assume that there exists x0 ∈ (0, 1) such that a(x0 ) = 0, a > 0 on [0, 1] \ {x0 }, and • a ∈ W 1,1 (0, 1) and there exist K1 ∈ (0, 1) such that (x − x0 )a′ ≤ K1 a a.e. in [0, 1], or • a ∈ W 1,∞ (0, 1) and there exist K1 ∈ [1, 2) such that (x − x0 )a′ ≤ K1 a a.e. in [0, 1]. 1. Then for all γ ≥ K1 the map |x − x0 |γ is non increasing on the left of x = x0 a and non decreasing on the right of x = x0 ,
x 7→
so that lim
x→x0
2. If K1 < 1, then
|x − x0 |γ = 0 for all γ > K1 . a
1 ∈ L1 (0, 1). a
1 1 3. If K1 ∈ [1, 2), then √ ∈ L1 (0, 1) and 6∈ L1 (0, 1). a a For the well–posedness of the problem, we start introducing the following weighted Hilbert spaces, which are suitable to study all situations, namely the (WWD), (SSD), (WSD) and (SWD) cases: o n √ Ha1 (0, 1) := u ∈ W01,1 (0, 1) : au′ ∈ L2 (0, 1)
and
1 Ha,b (0, 1) :=
u u ∈ Ha1 (0, 1) : √ ∈ L2 (0, 1) , b
endowed with the inner products hu, viHa1 (0,1) := and 1 (0,1) = hu, viHa,b
Z
Z
1
au′ v ′ dx +
0
Z
1
uv dx,
0
1
au′ v ′ dx +
0
Z
0
1
uv dx +
Z
0
1
uv dx, b
respectively. √ √ Note that, if u ∈ Ha1 (0, 1), then au′ ∈ L2 (0, 1), since |au′ | ≤ (max a) a|u′ |. [0,1]
We recall the following weighted Hardy–Poincar´e inequality, see [24, Proposition 2.6]:
7
Proposition 2.1. Assume that p ∈ C([0, 1]), p > 0 on [0, 1] \ {x0 }, p(x0 ) = 0 and there exists q > 1 such that the function x 7→
p(x) is non increasing on the left of x = x0 |x − x0 |q and non decreasing on the right of x = x0 .
(2.7)
Then, there exists a constant CHP > 0 such that for any function w, locally absolutely continuous on [0, x0 ) ∪ (x0 , 1] and satisfying Z 1 w(0) = w(1) = 0 and p(x)|w′ (x)|2 dx < +∞ , 0
the following inequality holds: Z 1 Z 1 p(x) 2 w (x) dx ≤ CHP p(x)|w′ (x)|2 dx. 2 0 0 (x − x0 )
(2.8)
Remark 2. Actually, such a proposition was proved in [24] 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 [24] was used for other estimates. We start with the following crucial Lemma 2.2. If K1 + K2 ≤ 2 and K2 < 1, then there exists a constant C > 0 such that Z 1 Z 1 2 u dx ≤ C a(u′ )2 dx (2.9) b 0 0 for every u ∈ Ha1 (0, 1).
(x − x0 )2 , so that p satisfies (2.7) with q = 2 − K2 > 1 b by Lemma 2.1.1. Thus, taken u ∈ Ha1 (0, 1), by Proposition 2.1, we get Z 1 2 Z 1 Z 1 u p(x) 2 u dx ≤ CHP dx = p(x)|u′ (x)|2 dx. 2 b 0 0 (x − x0 ) 0 Proof. We set p(x) :=
Now, by Lemma 2.1, p(x) = (x − x0 )2−K1 −K2 a(x)
(x − x0 )K1 (x − x0 )K2 ≤ ca(x) a(x) b(x)
for some c > 0, and the claim follows. Remark 3. A similar proof shows that, when K1 + 2K2 ≤ 2 and K2 < 1/2, then Z 1 Z 1 2 u a(u′ )2 dx dx ≤ C 2 0 0 b for every u ∈ Ha1 (0, 1).
8
1 Lemma 2.3. If K2 ≥ 1, then u(x0 ) = 0 for every u ∈ Ha,b (0, 1).
Proof. Since u ∈ W01,1 (0, 1), there exists limx→x0 u(x) = L ∈ R. If L 6= 0, then |u(x)| ≥ L2 in a neighborhood of x0 , that is |u(x)|2 L2 ≥ 6∈ L1 (0, 1) b 4b by Lemma 2.1, and thus L = 0. We also need the following result, whose proof, with the aid of Lemma 2.3, is a simple adaptation of the one given in [26, Lemma 3.2]. Lemma 2.4. If K2 ≥ 1, then n o Hc1 (0, 1) := u ∈ H01 (0, 1) such that supp u ⊂ (0, 1) \ {x0 }
1 is dense in Ha,b (0, 1).
Lemma 2.5. If K1 + K2 ≤ 2, K1 < 1 and K2 ≥ 1, then (2.9) holds for every 1 u ∈ Ha,b (0, 1). Proof of Lemma 2.5. Take u ∈ Hc1 (0, 1). Then, for every D > 0 2 Z 1 D |x − x0 |K1 /2 u′ − 0≤ (x − x )u dx 0 |x − x0 |2−K1 /2 0 Z 1 Z 1 D2 ′ 2 K1 2 |x − x0 | (u ) + = |x − x0 |K1 −2 (x − x0 )uu′ dx. u dx − 2D |x − x0 |2−K1 0 0 (2.10) Now, take x < x0 . Then, Z x Z x K1 −2 ′ 2 |t − x0 | (t − x0 )uu dt = |t − x0 |K1 −2 (t − x0 )(u2 )′ dt 0 0 Z x = |x − x0 |K1 −2 (x − x0 )u2 (x) − (K1 − 1) |t − x0 |K1 −2 u2 dt. 0
Similarly, if y ∈ (x0 , 1), we have Z 1 2 |t − x0 |K1 −2 (t − x0 )uu′ dt y
= −|y − x0 |K1 −2 (y − x0 )u2 (y) − (K1 − 1) x− 0
Now, letting x → and y → Z 1 converges to , since
x+ 0,
Z
y
1
|t − x0 |K1 −2 u2 dt.
the sum of the two previous integrals
0
lim |x − x0 |K1 −2 (x − x0 )u2 (x) = lim+ |y − x0 |K1 −2 (y − x0 )u2 (y) = 0,
x→x− 0
y→x0
9
for u ∈ Hc1 (0, 1). In conclusion, we have obtained 0≤
Z
1
0
D2 + D(K1 − 1) 2 |x − x0 |K1 (u′ )2 + dx. u |x − x0 |2−K1
Choosing D ∈ (0, 1 − K1 ), we get that there exists c > 0 such that Z
0
1
u2 dx ≤ c |x − x0 |2−K1
Z
1
0
|x − x0 |K1 (u′ )2 dx
1 for every u ∈ Hc1 (0, 1), and by Lemma 2.4, for every u ∈ Ha,b (0, 1). Finally, by Lemma 2.1.1, we can estimate the right-hand-side of the previous inequality Z 1
a(u′ )2 . Moreover, since 2 − K1 ≥ K2 , we can estimate the left-hand0 Z 1 2 u . side from below with c b 0 with c
In the (SSD) case K1 = K2 = 1 we need an additional assumption, which is obviously satisfied from the prototypes a(x) = b(x) = |x − x0 |, namely there exists K > 0 such that (a′ )2 ≤ K
a a.e. in [0, 1]. b
(2.11)
Lemma 2.6. Let K1 = K2 = 1 and assume (2.11). Then, (2.9) holds for all 1 u ∈ Ha,b (0, 1). 1 Proof. By Lemma 2.3 we know that, taken u ∈ Ha,b (0, 1), then u(x0 ) = 0. Fix ε ∈ 0, min{x0 , 1 − x0 } and write
Z
0
1
u2 dx = b
Z
x0 −ε
+
0
Z
x0
+
x0 −ε
Z
x0 +ε
x0
+
Z
1
x0 +ε
u2 dx. b
Now, by the Poincar´e inequality applied to functions in [0, x0 − ε] vanishing at 0, we get Z x0 −ε 2 Z x0 −ε Z x0 −ε u 1 1 dx ≤ u2 dx ≤ (u′ )2 dx b min b 0 min b 0 0 [0,x0 −ε] [0,x0 −ε] (2.12) Z 1 Z x0 −ε 1 ′ 2 ′ 2 a(u ) dx a(u ) dx ≤ C ≤ min b min a 0 0 [0,x0 −ε]
[0,x0 −ε]
for some C > 0 independent of u. A similar estimate holds for
Z
1
x0 +ε
Moreover, by Lemma 2.1, there exists C = C(a, b) > 0 such that √ Z x0 √ 2 Z x0 Z x0 ( au) ( au)2 u2 dx. dx = dx ≤ C 2 ab x0 −ε x0 −ε |x − x0 | x0 −ε b 10
u2 dx. b
(2.13)
√ √ a′ au, we have w ∈ L2 (0, 1) and w′ = au′ + √ u; hence, by 2 a the Cauchy Schwarz inequality, fixed δ < 0, there exists Cδ > 0 such that δ (a′ )2 2 (w′ )2 ≤ u + Cδ a(u′ )2 ∈ L2 (x0 − ε, x0 ), so that w ∈ H 1 (x0 − ε, x0 ). 4 a Being w(x0 ) = 0, the classical Hardy–Poincar´e inequality and (2.11) imply that Z x0 Z Z x0 Z x0 δC x0 (a′ )2 2 w2 ′ 2 (w ) dx ≤ dx ≤ C a(u′ )2 dx u dx + CC δ 2 4 x0 −ε a x0 −ε x0 −ε x0 −ε |x − x0 | Z Z 1 δC x0 u2 ≤ dx + CCδ a(u′ )2 dx. 4 x0 −ε b 0 (2.14) From (2.13) and (2.14), choosing δ small enough, we immediately get Setting w =
Z
x0
x0 −ε
u2 dx ≤ C b
Z
1
a(u′ )2 dx
0
for some C > 0. By (2.12), and operating in a similar way in [x0 , 1], the claim follows. In view of Lemmas 2.2, 2.5 and 2.6, we introduce the space 1 Ha (0, 1) (if K1 + K2 ≤ 2 and K2 < 1, H := if K1 + K2 ≤ 2, K1 < 1 and K2 ≥ 1, or 1 Ha,b (0, 1) K1 = K2 = 1 and (2.11) holds,
where the Hardy–Poincar´e–type inequality (2.9) holds. From now on, we make the following assumptions on a, b and λ:
Hypothesis 2.5. 1. One among Hypothesis 2.1, 2.2 or 2.3 holds true with K1 + K2 ≤ 2, or Hypothesis 2.4 holds with K1 = K2 = 1 and (2.11); 2. setting C ∗ the best constant of (2.9) in H, we assume that λ 6= 0 and λ
4 , then there exists 3
a constant θ ∈ (0, K1 ] such that ( is non increasing on the left of x = x0 , a(x) x 7→ θ |x − x0 | is non decreasing on the right of x = x0 .
(3.18)
3 In addition, when K1 > the function in (3.18) is bounded below away from 0 2 and there exists a constant Σ > 0 such that |a′ (x)| ≤ Σ|x − x0 |2θ−3 for a.e. x ∈ [0, 1].
(3.19)
Remark 6. If a(x) = |x − x0 |K1 , then (3.18) is clearly satisfied with θ = K1 . 3 are technical Moreover, the additional requirements for the sub-case K1 > 2 ones and are introduced in [25] to guarantee the convergence of some integrals (see [25, Appendix]). Of course, the prototype a(x) = |x − x0 |K1 satisfies again such conditions with θ = K1 .
14
To prove Carleman estimate, let us introduce the function ϕ := Θψ, where Z x y − x0 1 , (3.20) and ψ(x) := c dy − c Θ(t) := 1 2 [t(T − t)]4 x0 a(y) Z x y − x0 where c2 > sup dy and c1 > 0 (for the observability inequality c1 will [0,1] x0 a(y) be taken sufficiently large, see Lemma 4.1). Observe that Θ(t) → +∞ as t → 0+ , T − , and clearly −c1 c2 ≤ ψ < 0. The main result of this section is the following Theorem 3.1. Assume Hypothesis 3.1. Then, there exist two positive constants C and s0 , such that every solution v of (3.17) in 2 (3.21) (0, 1) ∩ H 1 0, T ; H V := L2 0, T ; Ha,b
satisfies, for all s ≥ s0 , Z 2 2 2 3 3 (x − x0 ) v e2sϕ dxdt sΘa(vx ) + s Θ a QT ! Z Th Z ix=1 aΘe2sϕ(t,x) (x − x0 )(vx )2 dt ≤C h2 e2sϕ dxdt + sc1 . QT
x=0
0
Remark 7. In [38] the authors prove a related Carleman inequality for the non degenerate singular 1-D system λ µ vt + vxx + x2 + xβ v = h (t, x) ∈ QT , (3.22) v(t, 0) = v(t, 1) = 0 t ∈ (0, T ), v(T, x) = vT (x) x ∈ (0, 1),
where β ∈ [0, 2). In our case, i.e. when µ = 0 and x0 = 0, such an inequality reads as follows: Z Z s v2 1 s v2 3 3 2 2 s Θ x v + Θ 2 + Θ 2/3 e2sΨ dxdt ≤ h2 e2sΨ dxdt, 2 x 2 x 2 QT QT x2 − 1 < 0 in [0, 1]. Actually, such an inequality is proved for where Ψ(x) = 2 solutions v such that v(t, x) = 0 for all (t, x) ∈ (0, T ) × (1 − η, 1) for some η ∈ (0, 1).
(3.23)
However, in [38, Remark 3.5] the authors say that Carleman estimates can be proved also for all solutions of (3.22) not satisfying (3.23). We think that this latter situation is much more interesting, since by the Carleman estimates, if h = 0, then v ≡ 0 even if (3.23) does not hold. 15
The proof of Theorem 3.1 is quite long, and several intermediate lemmas will be used. First, for s > 0, define the function w(t, x) := esϕ(t,x) v(t, x), where v is any solution of (3.17) in V; observe that, since v ∈ V and ϕ < 0, then w ∈ V and satisfies e−sϕ w −sϕ w)t + (a(e−sϕ w)x )x + λ = h, (t, x) ∈ (0, T ) × (0, 1), (e b (3.24) w(t, 0) = w(t, 1) = 0, t ∈ (0, T ), w(T, x) = w(0, x) = 0, x ∈ (0, 1).
As usual, we re–write the previous problem as follows: setting Lv := vt + (avx )x + λ
v b
and Ls w = esϕ L(e−sϕ w),
then (3.24) becomes sϕ Ls w = e h, w(t, 0) = w(t, 1) = 0, t ∈ (0, T ), w(T, x) = w(0, x) = 0, x ∈ (0, 1).
Computing Ls w, one has
− Ls w = L+ s w + Ls w,
where L+ s w := (awx )x + λ
w − sϕt w + s2 aϕ2x w, b
and L− s w := wt − 2saϕx wx − s(aϕx )x w. Of course, − 2 − + − + 2 2hL+ s w, Ls wi ≤ 2hLs w, Ls wi + kLs wkL2 (QT ) + kLs wkL2 (QT )
= kLs wk2L2 (QT ) = khesϕ k2L2 (QT ) ,
(3.25)
where h·, ·i denotes the scalar product in L2 (QT ). As usual, we will separate − the scalar product hL+ s w, Ls wi in distributed terms and boundary terms.
16
Lemma 3.1. The following identity holds: − hL+ s w, Ls wi Z Z s 2 2 2 = ϕtt w dxdt − 2s aϕx ϕtx w dxdt 2 QT QT Z 2 ′ 2 (2a ϕxx + aa ϕx )(wx ) dxdt +s {D.T.} QT Z (2aϕxx + a′ ϕx )a(ϕx )2 w2 dxdt + s3 QT Z ′ aϕx b 2 w dxdt − sλ 2 b QT Z T Z 1 Z s s2 1 x=1 2 t=T 2 2 t=T + [awx wt ]x=0 dt − [w ϕt ]t=0 dx + [a(ϕx ) w ]t=0 dt 2 0 2 0 0 Z T aϕ x 2 x=1 2 2 2 3 2 3 2 + [−sϕx (awx ) + s aϕt ϕx w − s a (ϕx ) w − sλ w ]x=0 dt {B.T.} b 0 Z T Z 1h i t=T 1 1 2 x=1 2 + [−sa(aϕx )x wwx ]x=0 dt − dx. a(wx ) − λ w 2 2b t=0 0 0 (3.26)
− Proof. Computing hL+ s w, Ls wi, one has that
− hL+ s w, Ls wi = I1 + I2 + I3 + I4 ,
where I1 :=
Z
QT
I2 :=
Z
QT
I3 :=
Z
QT
and
(awx )x − sϕt w + s2 a(ϕx )2 w wt dxdt,
(awx )x − sϕt w + s2 a(ϕx )2 w (−2saϕx wx )dxdt,
(awx )x − sϕt w + s2 a(ϕx )2 w (−s(aϕx )x w)dxdt,
I4 := λ
Z
QT
w wt − 2saϕx wx − s(aϕx )x w dxdt. b
By several integrations by parts in space and in timeR (see [1, Lemma 3.4], [24, Lemma 3.1] or [25, Lemma 3.1]), and observing that QT a(aϕx )xx wwx dxdt = 0
17
(by the very definition of ϕ), we get I1 + I2 + I3 Z Z s = aϕx ϕtx w2 dxdt ϕtt w2 dxdt − 2s2 2 QT QT Z (2a2 ϕxx + aa′ ϕx )(wx )2 dxdt +s QT Z (2aϕxx + a′ ϕx )a(ϕx )2 w2 dxdt + s3
(3.27)
QT
+
Z
T
0
+
Z
T
0
+
Z
0
T
[awx wt ]x=1 x=0 dt −
s 2
1
Z
0
[w2 ϕt ]t=T t=0 dx +
2
s 2
Z
1
0
[a(ϕx )2 w2 ]t=T t=0 dt
[−sϕx (awx )2 + s2 aϕt ϕx w2 − s3 a2 (ϕx )3 w2 ]x=1 x=0 dt [−sa(aϕx )x wwx ]x=1 x=0 dt −
1 2
Z
1
0
it=T h a(wx )2 dx. t=0
Next, we compute I4 : Z Z 1 2 a I4 = λ (w )t dxdt − 2s ϕx wx wdxdt 2b QT QT b Z (aϕx )x 2 −s w dxdt b QT Z 1 Z Z a (aϕx )x 2 1 2 t=T =λ [w ]t=0 dx − s ϕx (w2 )x dxdt − s w dxdt b QT b QT 0 2b Z Th Z 1 i x=1 a 1 2 t=T dt =λ [w ]t=0 dx − s ϕx w2 b x=0 0 0 2b Z Z (aϕx )x 2 aϕx 2 w dxdt w dxdt − s +s b x b QT QT ! Z Z Th Z 1 aϕx b′ 2 aϕx 2 ix=1 1 2 t=T dt − s w dxdt . =λ [w ]t=0 dx − s w b b2 x=0 QT 0 0 2b (3.28) Adding (3.27)-(3.28), (3.26) follows immediately. For the boundary terms in (3.26), we have: Lemma 3.2. The boundary terms in (3.26) reduce to −s
Z
0
T
x=1 Θ(awx )2 ψ ′ x=0 dt.
Proof. As in [24] or [25], using the definition of ϕ and the boundary conditions
18
on w, one has that Z
T
[awx wt ]x=1 x=0 dt −
0
+
Z
T
0
+
Z
0
s 2
Z
1 0
[w2 ϕt ]t=T t=0 dx +
s2 2
Z
1
0
[a(ϕx )2 w2 ]t=T t=0 dt
[−sϕx (awx )2 + s2 aϕt ϕx w2 − s3 a2 (ϕx )3 w2 ]x=1 x=0 dt
T
1 − 2
[−sa(aϕx )x wwx ]x=1 x=0 dt
Z
0
1
Z it=T h 2 a(wx ) dx = −s t=0
0
T
x=1 Θ(awx )2 ψ ′ x=0 dt.
(3.29) Moreover, since w ∈ V, w ∈ C [0, T ]; H ; thus w(0, x), w(T, x) are well defined, and using the boundary conditions of w, we get that Z
1
0
1 2 w 2b
t=T
dx = 0.
t=0
T h aϕx 2 ix=1 Now, consider the last boundary term sλ dt. Using the w b x=0 0 Z T x=1 aψ ′ 2 Θ definition of ϕ, this term becomes sλ w dt. By definition of ψ b 0 x=0 aψ ′ is bounded on [0, 1]. Thus, by the and using Hypothesis 2.5.2, the function b boundary conditions on w, one has
Z
sλ
Z
0
T
aψ ′ 2 w Θ b
x=1
dt = 0.
x=0
Now, the crucial step is to prove the following estimate: Lemma 3.3. Assume Hypothesis 3.1. Then there exist two positive constants s0 and C such that for all s ≥ s0 the distributed terms of (3.26) satisfy the estimate Z Z s 2 2 ϕtt w dxdt − 2s aϕx ϕtx w2 dxdt 2 QT QT Z (2a2 ϕxx + aa′ ϕx )(wx )2 dxdt +s + s3
QT T
Z
0
C ≥ s 2
Z
1
0
Z
QT
(2aϕxx + a′ ϕx )a(ϕx )2 w2 dxdt − sλ
C3 3 Θa(wx )2 dxdt + s 2
Z
QT
Z
QT
aϕx b′ 2 w dxdt b2
(x − x0 )2 2 Θ3 w dxdt. a
Proof. Proceeding as in [24, Lemma 3.2] or in [25, Lemma 4.1], one can prove
19
that, for s large enough, Z Z s aϕx ϕtx w2 dxdt ϕtt w2 dxdt − 2s2 2 QT QT Z 2 ′ (2a ϕxx + aa ϕx )(wx )2 dxdt +s Q ZT 3 (2aϕxx + a′ ϕx )a(ϕx )2 w2 dxdt +s QT
3C s ≥ 4
C3 3 Θa(wx ) dxdt + s 2
Z
2
QT
Z
Θ3
QT
(x − x0 )2 2 w dxdt, a
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 CsA1 + C 3 s3 A2 = kC A1 + k 3 C 3 3 A2 , k k we can choose s′0 = ks0 and C ′ = kCZ large as needed. aϕx b′ 2 Now, we estimate the term −sλ w dxdt. Using the definition of b2 QT ϕ and the assumption on b, one has Z Z aψ ′ b′ 2 aϕx b′ 2 Θ w dxdt = −sλ w dxdt −sλ b2 b2 QT QT Z (x − x0 )b′ 2 = −sλc1 Θ w dxdt b2 QT Z Θ 2 w dxdt. ≥ −sλc1 K2 QT b Since w(t, ·) ∈ H for every t ∈ [0, 1], for w ∈ V, we get Z Z Θ 2 ∗ w dxdt ≤ C Θa(wx )2 dxdt. QT b QT Hence, −sλ
Z
QT
aϕx b′ 2 w dxdt ≥ −sλc1 K2 C ∗ b2
Z
Θa(wx )2 dxdt,
QT
and we can assume, in view of what remarked above, that this last quantity is smaller than Z C Θa(wx )2 dxdt. −s 4 QT R − Summing up, the distributed terms of QT L+ s wLs wdxdt can be estimated as {D.T.} ≥
C s 2
Z
Θa(wx )2 dxdt +
QT
for s large enough and C > 0. 20
C3 3 s 2
Z
QT
Θ3
(x − x0 )2 2 w dxdt, a
From Lemma 3.1, Lemma 3.2 and Lemma 3.3, we deduce immediately that there exist two positive constants C and s0 , such that for all s ≥ s0 , Z Z + − Θa(wx )2 dxdt Ls wLs wdxdt ≥ Cs QT
QT
+ Cs3
Z
QT
Θ3
(x − x0 )2 2 w dxdt − s a
Z
T
0
2 2 ′ x=1 Θa wx ψ x=0 dt.
(3.30) Thus, a straightforward consequence of (3.25) and of (3.30) is the next result. Lemma 3.4. Assume Hypothesis 3.1. Then, there exist two positive constants C and s0 , such that for all s ≥ s0 , Z Z (x − x0 )2 2 Θ3 Θa(wx )2 dxdt + s3 w dxdt s a QT QT ! (3.31) Z T Z 2 2 ′ x=1 2 2sϕ(t,x) Θa wx ψ x=0 dt. . ≤C h e dxdt + s 0
QT
e
Recalling the definition of w, we have v = e−sϕ w and vx = −sΘψ ′ e−sϕ w + wx . Thus, substituting in (3.31), Theorem 3.1 follows.
−sϕ
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 (i) it is an interval which contains the degeneracy point: ω = (α, β) ⊂ (0, 1) is such that x0 ∈ ω,
(4.32)
or (ii) it is an interval lying on one side of the degeneracy point: ω = (α, β) ⊂ (0, 1) is such that x0 6∈ ω ¯.
(4.33)
On the function a we make the following assumptions: Hypothesis 4.2. Hypothesis 3.1 is satisfied. Moreover, if Hypothesis 2.1 or 1,∞ 2.3 holds, then there exist two functions g ∈ L∞ loc ([0, 1] \ {x0 }), h ∈ Wloc ([0, 1] \
21
{x0 }) and two strictly positive constants g0 , h0 such that g(x) ≥ g0 for a.e. x in [0, 1] and ! Z B p a′ (x) − p g(t)dt + h0 + a(x)g(x) = h(x, B) for a.e. x, B ∈ [0, 1] 2 a(x) x (4.34) with x < B < x0 or x0 < x < B. Remark 8. Since we require identity (4.34) far from x0 , once a is given, it is easy to find g, h, g0 and h0 with the desired properties. For example, if a(x) := |x − x0 h|α , α ∈ (0, 1), we can take g(x) ≡ gi0 = h0 = 1 and h(x, B) = |x − α α x0 | 2 −1 sign(x − x0 )(B + 1 − x) + |x − x0 | , for all x and B ∈ [0, 1], with 2 1,∞ x < B < x0 or x0 < x < B. Clearly, g ∈ L∞ loc ([0, 1] \ {x0 }) and h ∈ Wloc ([0, 1] \ ∞ {x0 }; L (0, 1)). Now, we associate to problem (1.5) the homogeneous adjoint problem λ v = 0, (t, x) ∈ QT , v + (avx )x + t b(x) (4.35) v(t, 0) = v(t, 1) = 0, t ∈ (0, T ), v(T, x) = vT (x),
where T > 0 is given and vT (x) ∈ L2 (0, 1). By the Carleman estimate in Theorem 3.1, we will deduce the following observability inequality for all the degenerate cases:
Proposition 4.1. Assume Hypotheses 4.1 and 4.2. Then there exists a positive constant CT such that every solution v ∈ C([0, T ]; L2 (0, 1)) ∩ L2 (0, T ; H) of (4.35) satisfies Z TZ Z 1 v 2 (t, x)dxdt. (4.36) v 2 (0, x)dx ≤ CT 0
0
ω
Using the observability inequality (4.36) and a standard technique (e.g., see [31, Section 7.4]), one can prove the null controllability result for the linear degenerate problem (1.5): Theorem 4.1. Assume Hypotheses 4.1 and 4.2. Then, given u0 ∈ L2 (0, 1), there exists h ∈ L2 (QT ) such that the solution u of (1.5) satisfies u(T, x) = 0 for every x ∈ [0, 1]. Moreover Z
QT
2
h dxdt ≤ C
for some positive constant C.
22
Z
0
1
u20 (x)dx,
4.1
Proof of Proposition 4.1
In this subsection we will prove, as a consequence of the Carleman estimate proved in Section 3, the observability inequality (4.36). For this purpose, we will give some preliminary results. As a first step, consider the adjoint problem vt + Av = 0, (t, x) ∈ QT , (4.37) v(t, 0) = v(t, 1) = 0, t ∈ (0, T ), v(T, x) = v (x) ∈ D(A2 ), T
where
n o D(A2 ) = u ∈ D(A) : Au ∈ D(A)
u and Au := (aux )x + λ . Observe that D(A2 ) is densely defined in D(A) for b the graph norm (see, for example, [8, Lemma 7.2]) and hence in L2 (0, 1). As in [11], [12], [23] or [24], define the following class of functions: n o W := v is a solution of (4.37) . Obviously (see, for example, [8, Theorem 7.5])
2 (0, 1) ⊂ V ⊂ U, W ⊂ C 1 [0, T ] ; Ha,b
where, V is defined in (3.21) and
U := C([0, T ]; L2 (0, 1)) ∩ L2 (0, T ; H).
(4.38)
We start with Proposition 4.2 (Caccioppoli’s inequality). Let ω ′ and ω be two open subintervals of (0, 1) such that ω ′ ⊂⊂ ω ⊂ (0, 1) and x0 6∈ ω ¯ ′ . Let ϕ(t, x) = Θ(t)Υ(x), where Θ is defined in (3.20) and Υ ∈ C([0, 1], (−∞, 0)) ∩ C 1 ([0, 1] \ {x0 }, (−∞, 0)) is such that
c |Υx | ≤ √ in [0, 1] \ {x0 } a
(4.39)
for some c > 0. Then, there exist two positive constants C and s0 such that every solution v ∈ W of the adjoint problem (4.37) satisfies Z
0
T
Z
ω′
(vx )2 e2sϕ dxdt ≤ C
for all s ≥ s0 .
23
Z
0
T
Z
v 2 dxdt, ω
(4.40)
Of course, our prototype for Υ is the function ψ defined in (3.20), since s 1 |x − x0 |2 1 p |ψ ′ (x)| = c1 ≤ cp a(x) a(x) a(x)
by Lemma 2.1.
Proof. The proof follows the one of [24, Proposition 4.2], but it is different for the presence 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 ∈ ω′, ξ(x) = 0, x ∈ [ 0, 1] \ ω.
Since v solves (4.37), we have Z 1 Z Z T d 2 2sϕ 2 2sξ 2 ϕt e2sϕ v 2 + 2ξ 2 e2sϕ vvt dxdt ξ e v dx dt = 0= QT 0 0 dt Z Z v ξ 2 e2sϕ v −λ − (avx )x dxdt ξ 2 sϕt e2sϕ v 2 dxdt + 2 =2 b QT QT Z Z Z 2 v (ξ 2 e2sϕ v)x avx dxdt. ξ 2 e2sϕ dxdt + 2 ξ 2 sϕt e2sϕ v 2 dxdt − 2λ =2 b QT QT QT (4.41) If λ ≤ 0, then, differentiating the last term in (4.41), we get Z Z Z 2 2 2sϕ v 2 2sϕ 2 ξ 2 sϕt e2sϕ v 2 dxdt dxdt − 2 ξ e ξ e a(vx ) dxdt = 2λ 2 b QT QT QT Z (ξ 2 e2sϕ )x avvx dxdt −2 QT Z Z 2 2sϕ 2 (ξ 2 e2sϕ )x avvx dxdt, ξ sϕt e v dxdt − 2 ≤ −2 QT
QT
and then one can proceed as for the proof of [24, Proposition 4.2], obtaining the claim. Otherwise, if λ > 0, fixed ε > 0, by the Cauchy–Schwarz inequality, we have Z 1 Z 1 v2 ξ 2 e2sϕ dx ≤ C ∗ a(wx )2 dx b 0 0 Z 1 Z 1 sϕ 2 2 ≤ Cε a[(ξe )x ] v dx + ε ξ 2 e2sϕ a(vx )2 dx 0
0
for some Cε > 0. Moreover, sϕ
2
[(ξe )x ] ≤ Cχω (e
2sϕ
1 + s (ϕx ) e ) ≤ Cχω 1 + a 2
24
2 sϕ
for some positive constant C. Indeed, e2sϕ < 1, while s2 (ϕx )2 e2sϕ can be estimated with c c (Υx )2 ≤ 2 (− max Υ) a by (4.39), for some constants c > 0. Thus Z Z v2 ξ 2 e2sϕ dxdt ≤ 2λCε a[(ξesϕ )x ]2 v 2 dxdt 2λ b QT QT Z ξ 2 e2sϕ a(vx )2 dxdt + 2λε ≤C
Z
0
QT T Z
v 2 dxdt + 2λε
ω
Z
(4.42)
ξ 2 e2sϕ a(vx )2 dxdt,
QT
for a positive constant C depending on ε. Hence, differentiating the last term in (4.41) and using (4.42), we get Z Z Z 2 2 2sϕ v 2 2sϕ 2 ξ e ξ e a(vx ) dxdt = 2λ 2 ξ 2 sϕt e2sϕ v 2 dxdt dxdt − 2 b QT QT QT Z (ξ 2 e2sϕ )x avvx dxdt −2 QT T
Z
≤C Z −2
0
QT
Z
Z
ξ 2 e2sϕ a(vx )2 dxdt ω QT Z 2 2sϕ 2 (ξ 2 e2sϕ )x avvx dxdt. ξ sϕt e v dxdt − 2 v 2 dxdt + 2λε
QT
Thus, applying again the Cauchy-Schwarz inequality, we get Z
2 2sϕ
2
ξ e a(vx ) dxdt ≤ C (2 − 2λε) QT Z (ξ 2 e2sϕ )x avvx dxdt −2 QT T
≤C
Z
0
+ Dε =C
Z
Z
+ Dε
ω
Z
0
v 2 dxdt − 2
Z
0
T
Z
0
ω
2
v dxdt − 2
ξ 2 sϕt e2sϕ v 2 dxdt + 2ε ω
Z
ω
T
Z
ω
0 2
Z
T
0
ω
0
[(ξ 2 e2sϕ )x ] av 2 dxdt ξ 2 e2sϕ
25
ξ 2 sϕt e2sϕ v 2 dxdt
QT
Z √ (ξ 2 e2sϕ )x 2 v dxdt a ξesϕ ω Z Z TZ Z v 2 dxdt − 2 ξ 2 sϕt e2sϕ v 2 dxdt + 2ε
T
0 T
0
Z
Z TZ
Z
ω
T
Z
ω
√
aξesϕ vx
2
dxdt
ξ 2 e2sϕ a(vx )2 dxdt
for some Dε > 0. Hence, Z
2(1 − ε − λε) −2
Z
T
0
Z
T
Z
0
ξ 2 e2sϕ a(vx )2 dxdt ≤ C
ω
ξ 2 sϕt e2sϕ v 2 dxdt + Dε
ω
Z
T
0
Z
ω
Z
T 0
Z
v 2 dxdt
ω
[(ξ 2 e2sϕ )x ]2 2 av dxdt. ξ 2 e2sϕ
Since x0 6∈ ω ¯ ′ , then 2(1 − ε − λε) inf′ a(x) ω
≤ 2(1 − ε − λε) ≤ 2(1 − ε − λε) ≤C
Z
T 0
Z
ω
Z
Z
T 0
Z
T 0
Z
e2sϕ (vx )2 dxdt
ω′
Z
ξ 2 e2sϕ a(vx )2 dxdt
ω ¯′
T 0
Z
ξ 2 e2sϕ a(vx )2 dxdt
ω
v 2 dxdt − 2
Z
0
T
Z
ξ 2 sϕt e2sϕ v 2 dxdt + Dε
ω
Z
T
0
Z
ω
[(ξ 2 e2sϕ )x ]2 2 av dxdt. ξ 2 e2sϕ
Finally, we show that there exists a positive constant C (still depending on ε) such that −2
Z
T
0
Z
≤C
ξ 2 sϕt e2sϕ v 2 dxdt + Dε
ω
Z
0
Z
T
0
T
Z
Z
ω
[(ξ 2 e2sϕ )x ]2 2 av dxdt ξ 2 e2sϕ
v 2 dxdt,
ω
so that the claim will follow. Indeed, |sϕt e2sϕ | ≤ c
1 1/4 s0 (− max Υ)1/4
,
˙ ≤ cΘ5/4 and |Θ| |sϕt e2sϕ | ≤ cs(−Υ)Θ5/4 e2sϕ ≤
c 5/4 s(−Υ)
for some constants c > 0 which may vary at every step. [(ξ 2 e2sϕ )x ]2 can be estimated by On the other hand, ξ 2 e2sϕ C e2sϕ + s2 (ϕx )2 e2sϕ χω ,
and proceeding as before, we get the claim, choosing ε small enough, namely ε < (1 + λ)−1 . We shall also use the following 26
Lemma 4.1. Assume Hypotheses 4.1 and 4.2. Then there exist two positive constants C and s0 such that every solution v ∈ W of (4.37) satisfies, for all s ≥ s0 , Z Z TZ (x − x0 )2 2 2sϕ sΘa(vx )2 + s3 Θ3 v 2 dxdt. v e dxdt ≤ C a QT 0 ω Here Θ and ϕ are as in (3.20) 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 [25, Lemma 5.1 and 5.2], to which we refer, also to explain why c1 must be large. Proposition 4.3 (Nondegenerate nonsingular Carleman estimate). Let z be the solution of ( z zt + (azx )x + λ = h ∈ L2 (0, T ) × (A, B) , b (4.43) z(t, A) = z(t, B) = 0, t ∈ (0, T ), where b ∈ C [A, B] is such that b ≥ b0 > 0 in [A, B] and a satisfies (a1 ) a ∈ W 1,1 (A, B), a ≥ a0 > 0 in (A, B) and there exist two functions g ∈ L1 (A, B), h ∈ W 1,∞ (A, B) and two strictly positive constants g0 , h0 such that g(x) ≥ g0 for a.e. x in [A, B] and ! Z B p a′ (x) − p g(t)dt + h0 + a(x)g(x) = h(x) for a.e. x ∈ [A, B]; 2 a(x) x or
(a2 ) a ∈ W 1,∞ (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 ! Z TZ B Z TZ B 2sΦ 2 3 3 2 2 2sΦ sΘ(zx ) + s Θ z e dxdt ≤ C h e dxdt − (B.T.) , 0
A
0
A
(4.44)
where
(B.T.) =
Z sr
T
a
0
Z sr
0
"
T
3/2 2sΦ
e
Θ
Z
B
g(τ )dτ + h0 x
ae2sΦ Θerζ (vx )2
x=B x=A
dt,
!
2
(zx )
#x=B
dt,
if (a1 ) holds,
x=A
if (a2 ) holds.
Here the function Φ is defined as Φ(t, x) := Θ(t)ρ(x), where Θ is as in (3.20), # "Z Z B Z x x h0 1 −r p p dt − c, if (a1 ) holds, g(s)dsdt + ρ(x) := a(t) t a(t) A A rζ(x) e − c, if (a2 ) holds (4.45) 27
and ζ(x) = d
Z
B
x
1 dt. a(t)
Here d = ka′ kL∞ (A,B) and c > 0 is chosen in the second case in such a way that max ρ < 0. [A,B]
z Proof. Rewrite the equation of (4.43) as zt + (azx )x = ¯h, where ¯h := h − λ . b Then, applying [25, Theorem 3.1], there exist two positive constants C and s0 > 0, such that ! Z TZ B Z TZ B ¯ 2 e2sΦ dxdt − (B.T.) , sΘ(zx )2 + s3 Θ3 z 2 e2sΦ dxdt ≤ C h 0
A
0
A
(4.46)
¯ the term for all s ≥ s0 . Using the definition of h, estimated in the following way: Z TZ B Z TZ ¯ 2 e2sΦ dxdt ≤ 2 h
B
Z
T
Z
0
B
¯ 2 dxdt can be e2sΦ h
A
T
B
z 2 2sΦ e dxdt. 2 0 A 0 A 0 A b (4.47) Applying the classical Poincar´e inequality to w(t, x) := esΦ z(t, x) and observing that 0 < inf Θ ≤ Θ ≤ cΘ2 , one has Z TZ B 2 Z TZ B Z TZ B 2 z 2sΦ λ2 w 2 2λ2 e dxdt = 2λ dxdt ≤ 2 C (wx )2 dxdt 2 2 2 b b b 0 A 0 A 0 A 0 Z TZ B ≤C (s2 Θ2 z 2 + (zx )2 )e2sΦ dxdt 0
≤
Z
0
T
Z
h2 e2sΦ dxdt + 2λ2
A B
A
Z
s Θ(zx )2 e2sΦ dxdt + 2
Z
0
Z
T
Z
B
A
s3 3 2 2sΦ Θ z e dxdt, 2
for s large enough. Using this last inequality in (4.47), we have Z TZ B Z TZ B Z TZ B s 2sΦ 2 2 2sΦ ¯ e h dxdt + h e dxdt ≤ 2 Θ(zx )2 e2sΦ dxdt 0 A 2 0 A 0 A Z TZ B 3 s 3 2 2sΦ Θ z e dxdt. + 0 A 2 (4.48) Using this inequality in (4.46), (4.44) follows immediately. In order to prove Proposition 4.1, the last result that we need is the following: Lemma 4.2. Assume Hypotheses 4.1 and 4.2. Then there exists a positive constant CT such that every solution v ∈ W of (4.37) satisfies Z 1 Z TZ v 2 (0, x)dx ≤ CT v 2 (t, x)dxdt. 0
0
28
ω
Proof. Multiplying the equation of (4.37) by vt and integrating by parts over (0, 1), one has Z 1 Z 1 Z 1 v vvt x=1 2 0= vt + (avx )x + λ vt + (avx )x vt + λ vt2 dx + [avx vt ]x=0 vt dx = dx = b b 0 0 0 Z 1 Z Z 1 Z 1 2 Z 1 d d 1 λ v λ d 1 v2 dx = dx vt2 dx − a(vx )2 + − avx vtx dx + 2 dt 0 b 2 dt 0 2 dt 0 b 0 0 Z Z λ d 1 v2 1 d 1 dx. a(vx )2 dx + ≥− 2 dt 0 2 dt 0 b Thus, the function t 7→
Z
0
1
2
a(vx ) dx − λ
Z
1
0
v2 dx b
is non decreasing for all t ∈ [0, T ]. In particular, Z 1 Z 1 2 Z 1 Z 1 2 v (t, x) v (0, x) dx ≤ a(vx )2 (t, x)dx − λ dx a(vx )2 (0, x)dx − λ b(x) b(x) 0 0 0 0 Z 1 ∗ ≤ (1 + |λ|C ) a(vx )2 (t, x)dx. 0
T 3T , Θ being bounded therein, we , 4 4 Z 1 Z 1 2 v (0, x) dx a(x)(vx )2 (0, x)dx − λ b(x) 0 0 Z 1 Z 3T 4 2 ∗ ≤ (1 + |λ|C ) a(vx )2 dxdt (4.49) T T 0 4 Z 3T Z 1 4 ≤ CT sΘa(vx )2 e2sϕ dxdt.
Integrating the previous inequality over find
T 4
0
Hence, from the previous inequality and Lemma 4.1, if λ ≤ 0 Z TZ Z 1 Z 1 Z 1 2 v (0, x) dx ≤ C v 2 dxdt a(vx )2 (0, x)dx ≤ a(vx )2 (0, x)dx − λ b(x) 0 ω 0 0 0 for some positive constant C > 0. If λ > 0, using again Lemma 4.1 and (4.49), one has Z 1 Z 1 2 Z TZ v (0, x) 2 a(vx ) (0, x)dx − λ dx ≤ C v 2 dxdt. b(x) 0 0 0 ω Hence, by (2.9) and (4.50), we have Z TZ Z 1 2 Z 1 v (0, x) dx + C v 2 dxdt a(vx )2 (0, x)dx ≤ λ b(x) 0 ω 0 0 Z 1 Z TZ ∗ 2 ≤ λC a(vx ) (0, x)dx + C v 2 dxdt. 0
0
29
ω
(4.50)
Thus (1 − λC ∗ )
Z
0
1
a(vx )2 (0, x)dx ≤ C
Z
T
0
Z
v 2 dxdt,
ω
for a positive constant C. In every case, there exists C > 0 such that Z
1
0
T
Z
a(vx )2 (0, x)dx ≤ C
Z
0
v 2 dxdt.
(4.51)
ω
The Hardy- Poincar´e inequality (see Proposition 2.1) and (4.51) imply that Z
0
1
a (x − x0 )2
1/3
1
p v 2 (0, x)dx 2 0 (x − x0 ) Z 1 ≤ CHP p(vx )2 (0, x)dx
2
v (0, x)dx ≤
Z
0
≤ cCHP ≤C
Z
T
0
Z
1
a(vx )2 (0, x)dx
0
Z
v 2 dxdt,
ω
4 , while 3 4/3 p(x) = |x − x0 | max a otherwise, and c, C are obtained by Lemma 2.1. for a positive constant C. Here p(x) = (a(x)|x − x0 |4 )1/3 if K1 > [0,1]
Again by Lemma 2.1, we have
a(x) (x − x0 )2
1/3
≥ C3 := min
Hence C3
Z
0
and the claim follows.
1
(
1/3 1/3 ) a(0) > 0. , x20
a(1) (1 − x0 )2
2
v(0, x) dx ≤ C
Z
0
T
Z
v 2 dxdt
ω
Proof of Proposition 4.1. It follows by a density argument as for the proof of [24, Proposition 4.1].
References [1] F. Alabau-Boussouira, P. Cannarsa, G Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Eqs 6 (2006), 161-204. [2] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics 96 (2001), Birkh¨ auser Verlag, Basel. 30
[3] P. Baras, J. Goldstein, Remarks on the inverse square potential in quantum mechanics, Differential Equations, North-Holland Math. Stud. 92 (1984), 31–35. [4] P. Baras, J. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), 121–139. [5] J. Bebernes, D. Eberly, Mathematical problems from combustion theory, Math. Sci., 83 (1989), Springer-Verlag, New York. [6] I. Boutaayamou, G. Fragnelli, L. Maniar, Lipschitz stability for linear cascade parabolic systems with interior degeneracy, Electron. J. Diff. Equ. 2014 (2014), 1–26. [7] I. Boutaayamou, G. Fragnelli, L. Maniar, Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions, J. Anal. Math., in press. [8] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science+Business Media, LLC 2011. [9] H. Brezis, J.L. Vazquez, Blow-up solutions of some nonlinear elliptic equations, Rev. Mat. Complut. 10 (1997), 443–469. [10] X. Cabr´e, Y. Martel, Existence versus explosion instantan´ee pour des ´equations de la chaleur lin´eaires avec potentiel singulier, C. R. Math. Acad. Sci. Paris, 329 (1999), 973 978. [11] P. Cannarsa, G. Fragnelli, D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ. 8 (2008), 583–616. [12] P. Cannarsa, G. Fragnelli, D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media 2 (2007), 693– 713.accre [13] P. Cannarsa, P. Martinez, J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim. 47 (2008), 1–19. [14] P. Cannarsa, P. Martinez, J. Vancostenoble, Null controllability of the degenerate heat equations, Adv. Differential Equations 10 (2005), 153–190. [15] C. Cazacu, C. Cazacu, Controllability of the heat equation with an inversesquare potential localized on the boundary, SIAM J. Control Optim. 52 (2014), 2055–2089. [16] T. Cazenave, A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford 1998.
31
[17] A.S. De Castro, Bound states of the Dirac equation for a class of effective quadratic plus inversely quadratic potentials, Ann. Phys., 311 (2004), 170– 181. [18] J.W. Dold, V.A. Galaktionov, A.A. Lacey, J.L. Vazquez, Rate of approach to a singular steady state in quasilinear reaction-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 26 (1998), 663–687. [19] C.L. Epstein, R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology, Ann. of Math. Stud. 2013. [20] S. Ervedoza, Null Controllability for a singular heat equation: Carleman estimates and Hardy inequalities, Comm. Partial Differential Equations 33 (2008), 1996-2019. [21] M. Fotouhi, L. Salimi, Controllability results for a class of one dimensional degenerate/singular parabolic equations, Commun. Pure Appl. Anal. 12 (2013), 1415–1430. [22] M. Fotouhi, L. Salimi, Null controllability of degenerate/singular parabolic equations, J. Dyn. Control Syst. 18 (2012), 573–602. [23] G. Fragnelli, Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates, Discrete Contin. Dyn. Syst. Ser. S 6, (2013), 687–701. [24] G. Fragnelli, D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Advances in Nonlinear Analysis 2, (2013), 339–378. [25] G. Fragnelli, D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, to appear in Mem. Amer. Math. Soc. [26] G. Fragnelli, G. Ruiz Goldstein, J.A. Goldstein, S. Romanelli, Generators with interior degeneracy on spaces of L2 type, Electron. J. Differential Equations 2012 (2012), 1–30. [27] A. V. Fursikov, O. Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes Series, Research Institute of Mathematics, Global Analysis Research Center, Seoul National University 34, 1996. [28] V. Galaktionov, J.L. Vazquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), 1–67. [29] P. Hagan, D. Woodward, Equivalent Black volatilities, Appl. Math. Finance 6 (1999), 147-159.
32
[30] N.I. Karachalios, N.B. Zographopoulos, On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence, Calc. Var. Partial Differential Equations 25 (2006), 361–393. [31] J. Le Rousseau, G. Lebeau, On carleman estimates for elliptic and parabolic operators. applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var. 18 (2012), 712–747. [32] Lions Magenes, Non-homogeneous boundary value problems and applications. Vol. I. Grundlehren Math. Wiss. 181. Springer-Verlag, New YorkHeidelberg, 1972. [33] P. Martinez, J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Eq. 6 (2006), 325–362. [34] S. Micu, E. Zuazua, On the lack of null controllability of the heat equation on the half-line, Trans. Amer. Math. Soc. 353 (2001), 1635–1659. [35] M.E. Taylor, Partial Differential Equations I. Basic theory. Second edition. Applied Mathematical Sciences 115. Springer, New York, 2011. [36] J. Vancostenoble, Improved Hardy-Poincar´e inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), 761–790. [37] J. Vancostenoble, E. Zuazua, Hardy inequalities, observability, and control for the wave and Schr¨ odinger equations with singular potentials, SIAM J. Math. Anal. 41 (2009), 1508–1532. [38] J. Vancostenoble, E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal. 254 (2008), 1864–1902. [39] J.L. V´ azquez, E. Zuazua,The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal. 173 (2000), 103–153.
33
