Null and approximate controllability for weakly blowing up semilinear heat equations

Null and approximate controllability for weakly blowing up semilinear heat equations by Enrique FERN´ ANDEZ-CARA∗ Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla 41080 Sevilla, Spain. e-mail: [email protected] and Enrique ZUAZUA† Departamento de Matem´atica Aplicada Universidad Complutense 28040 Madrid, Spain. e-mail: [email protected] Abstract We consider the semilinear heat equation in a bounded domain of IRd, with control on a subdomain and homogeneous Dirichlet boundary conditions. We prove that the system is null-controllable at any time provided a globally defined and bounded trajectory exists and the nonlinear term f(y) is such that |f(s)|grows slower than |s|log3/2(1+|s|) as |s| → ∞. For instance, this condition is fulfilled by any function fgrowing at infinity like |s|logp(1 + |s|) with 1 < p < 3/2 (in this case, in the absence of control, blow-up occurs). We also prove that, for some functions fthat behave at infinite like |s|logp(1+|s|) with p > 2, null controllability does not hold. The problem remains open when fbehaves at infinity like |s|logp(1 + |s|), with 3/2≤p≤2. Results of the same kind are proved in the context of approximate controllability. ∗Supported by grant PB95-1242 of the DGICYT (Spain). †Supported by grant PB96-0663 of the DGES (Spain). 1 1 Introduction and main results Let Ω ⊂IRdbe a bounded domain with boundary of class C2, let T > 0 be given and assume f: IR 7→ IR is locally Lipschitz-continuous. We will consider semilinear parabolic systems of the form      yt−∆y+f(y) = v1ωin Ω ×(0, T) y= 0 on ∂Ω×(0, T) y(x, 0) = y0(x) in Ω. (1.1) In (1.1), y=y(x, t) is the state and v=v(x, t) is a control that acts on the system through the non-empty open set ω⊂Ω. 1ωdenotes the characteristic function of the set ω. We shall denote by Qthe cylinder Ω ×(0, T) and by Σ its lateral boundary ∂Ω×(0, T). We will assume that y0∈L2(Ω) and v∈L∞(ω×(0, T)). In most part of this paper, we will also assume that, for some y0and v, system (1.1) admits a solution globally defined in the time interval [0, T]. Of course, this holds immediately when f(0) = 0,(1.2) in which case y≡0 solves (1.1), with y0= 0 and v≡0. In the sequel, Cdenotes a generic positive constant. For instance, the equality C=C(Ω, ω) means that Conly depends of Ω and ω. For simplicity, we will require fto satisfy |f0(s)| ≤ C(1 + |s|p) a.e., with p≤1+4/d. (1.3) Under this condition, system (1.1) possesses exactly one local (in time) solution. In accordance with the results in [CH1], under the growth condition |f(s)| ≤ C(1 + |s|log(1 + |s|)) ∀s∈IR,(1.4) the solutions of (1.1) are globally defined in [0, T]. More precisely, one has y∈C([0, T]; L2(Ω)) ∩L2(0, T;H1 0(Ω)). This is also true if, instead of (1.4), we assume a “good-sign” condition, like the following: f(s)s≥ −C(1 + |s|2)∀s∈IR.(1.5) The main goal of this paper is to analyze the controllability of (1.1) when blow-up occurs, i.e. without imposing any of these conditions (1.4) or (1.5). It will be assumed that ω6= Ω. Otherwise, when ω= Ω, the problem can be reduced to the controllability of the (linear) heat equation since the nonlinear term is absorbed by the control in a trivial way. 2 We will first analyze the so called null controllability property. System (1.1) is said to be null-controllable at time Tif, for any y0∈L2(Ω) and any globally defined bounded trajectory y∗(corresponding to the data y∗ 0∈L2(Ω) and v∗∈L∞(ω×(0, T))), there exists a control v∈L∞(ω×(0, T)) such that the corresponding solution of (1.1) is also globally defined in [0, T] and satisfies y(x, T) = y∗(x, T) in Ω.(1.6) Obviously, when (1.1) is linear, this is equivalent to say that, for each y0∈L2(Ω), there exists v∈L∞(ω×(0, T)) such that the corresponding solution ysatisfies y(x, T) = 0 in Ω.(1.7) This justifies the definition we have introduced of null-controllable systems. Notice that, if (1.6) holds, extending vfor t≥Tas the control v∗associated to y∗, we obtain a solution ythat coincides with y∗as long as y∗exists. In particular, if y∗is a stationary solution of (1.1) or, more generally, y∗is defined for all t≥0, then yis also globally defined and coincides with y∗for all t≥T. Hence, from the viewpoint of applications, it is very important to know whether or not (1.1) is null-controllable. Our first main result is of negative nature. Theorem 1.1 There exist locally Lipschitz-continuous functions fsuch that f(0) = 0, satisfying |f(s)| ∼ |s|logp(1 + |s|)as |s| → ∞ (1.8) with p > 2, for which system (1.1) fails to be null-controllable for all T > 0. Remark 1.1 For the proof of Theorem 1.1, we choose p > 2 and f(s) = Z|s| 0 logp(1 + |σ|)dσ ∀s∈IR (1.9) and we prove a localized estimate (in Ω \¯ω) that shows that the control cannot compensate the blow-up phenomena occuring in Ω \¯ω. Arguments of this kind are well known. For instance, see J. Henry [H] for the proof of the lack of approximate controllability of the heat equation with nonlinear absorption terms; see also O. Yu. Imanuvilov [I] and A. Fursikov and O. Yu. Imanuvilov [FI] for examples of systems that fail to be null-controllable with power-like nonlinearities, i.e. in the more restrictive class of nonlinear terms growing at infinity like |s|p with p > 1. The function fin (1.9) is such that f(0) = 0. Therefore, y∗≡0 is a stationary solution of (1.1) corresponding to the control v∗≡0 and, as mentioned above, the null controllability problem makes sense. 3 It is important to observe that the solutions to (1.1) can blow up in the absence of control as soon as |f(s)|grows at infinity as in (1.8) with p > 1. Therefore, Theorem 1.1 does not exclude the null controllability of a whole range of nonlinear systems for which blow-up occurs. Namely, when (1.8) is satisfied with 1 < p ≤2. In our second main result, we establish conditions under which (1.1) is null-controllable: Theorem 1.2 Let T > 0. Assume that (1.1) admits at least one globally defined and bounded solution y∗, corresponding to the data y∗ 0∈L2(Ω) and v∗∈L∞(ω×(0, T)). Assume that f: IR 7→ IR is locally Lipschitz-continuous and satisfies (1.3) and f(s) |s|log3/2(1 + |s|)→0as |s| → ∞.(1.10) Then (1.1) is null-controllable at time T. Remark 1.2 There is an extensive literature on semilinear parabolic and elliptic problems analyzing the existence of global and/or stationary solutions. At this respect the following result by H. Brezis et al. [BCMR] is worth mentioning: Assume that, for some σ≥0, one has f(σ)<0and Z∞ σ 1 f(s)ds > −∞.(1.11) Also, assume there exists y0∈L∞(Ω), with y0≥0, such that the system (1.1) corresponding to this initial y0and v≡0possesses one global classical solution (defined for all t≥0). Then there exists at least one weak stationary solution of (1.1) corresponding to v≡0. Recall that (1.11) guarantees the existence of solutions of (1.1) that blow up in finite time. Roughly speaking, this result shows that, in the class of nonlinearities in which blow-up arises, for v≡0, the existence of positive global solutions implies that weak stationary solutions exist. It is a routine argument to show that, under the growth conditions we are imposing on f, these weak solutions are bounded. Remark 1.3 The proof of Theorem 1.2 provides estimates on the size of the control needed to achieve null controllability. Furthermore, by inspection of the proof, one sees that null controllability still holds under slightly more general conditions. More precisely, for each globally defined and bounded solution y∗, there exists `(y∗)>0 such that, if lim sup |s|→∞ |f(s)| |s|log3/2(1 + |s|)≤`(y∗)<+∞ and y0is given, we can find v∗∈L∞(ω×(0, T)) such that (1.6) is satisfied. 4 Remark 1.4 In particular, Theorem 1.2 says that, for each y0∈L2(Ω), there exists a control vsuch that the corresponding solution of (1.1) is globally defined in [0, T]. In other words, if a globally defined trajectory exists, then for each y0a control vcan be found such that the corresponding solution is globally defined as well. Of course, this cannot be guaranteed to hold for any right hand side and any initial datum, since we are in the range in which blow-up may occur. Remark 1.5 Without the assumption (1.3) on f, the uniqueness of a solution to (1.1) for a given vis not guaranteed. In these conditions, arguing as in the proof of Theorem 1.2, we can deduce the existence of a control vsuch that system (1.1) admits at least one solution that is globally defined in [0, T] and satisfies (1.6). We refer to [K] for a similar discussion. Theorem 1.2 asserts that system (1.1) is null-controllable under the growth condition (1.8) provided p < 3/2. Obviously, there is a gap between Theorems 1.1 and 1.2: We do not know whether or not system (1.1) is null-controllable when fsatisfies (1.8) with 3/2≤p≤2. As we shall see, the proof we give of Theorem 1.1 does not hold for p≤2. The same can be said about the proof of Theorem 1.2 when p≥3/2. Thus, the case 3/2≤p≤2 is an interesting open problem. We shall return to it in Section 6. In [F], it was shown that zero controllability holds under the more restrictive condition f(s) |s|log(1 + |s|)→0 as |s| → ∞. (1.12) Note that, under condition (1.12), the solutions to (1.1) cannot blow up. Recently, V. Barbu [B] has proved the zero controllability of (1.1) under the growth condition (1.10), but imposing additional “good-sign” conditions similar to (1.5) (see also [AB]). Therefore, to our knowledge, Theorem 1.2 is the first result in the literature on the null controllability of blowing-up semilinear heat equations. Recall that, in the context of the semilinear wave equation, due to the finite speed propagation property, if blow-up occurs, exact controllability cannot hold (see [Z3]). Thus, Theorem 1.2 holds due to the parabolic nature of the equation under consideration. For the proof of Theorem 1.2, we will use the explicit estimates on the cost of controllability obtained in [FZ] and the fixed point method introduced in the context of the semilinear wave equation in [Z1] and later applied to semilinear heat equations in [FPZ], [F] and [Z2] (see 5 also [FI]). Recall that the estimates in [FZ] were obtained by adapting the global Carleman inequalities in [FI]. However, when applying the fixed point argument, we introduce a new ingredient to avoid blow-up to occur. Indeed, as usual, we first linearize the system and show its controllability analyzing how the control depends of the size of the potential of the linearized equation. Usually, one takes Tas the control time for all the linearized equations (see e.g. [Z1]). However, in the proof of Theorem 1.2, the control time is chosen depending on the size of the potential so that, roughly speaking, it decreases as this size increases and tends to zero as the size tends to infinity. This is made in order to avoid blow-up phenomena to occur. Note that this strategy is in agreement with common sense: In the presence of blow-up phenomena, one has to act on the system very fast, before blow-up occurs. This idea of taking short control times has been used in [K] for the one-dimensional heat equation with nonlinearities that behave sublinearly at infinity. Let us now analyze the approximate controllability property. System (1.1) is said to be approximately controllable at time Tif, for any y0∈L2(Ω), y1∈L2(Ω) and ε > 0, there exists a control v∈L∞(ω×(0, T)) such that the solution of (1.1) is globally defined in [0, T] and satisfies ky(·, T)−y1kL2(Ω) ≤ε. (1.13) In other words, system (1.1) is approximately controllable if the set of reachable states is dense in L2(Ω). In the context of linear heat equations, approximate controllability is a consequence of the null controllability property (see for instance [FZ]). But this is not necessarily true for semilinear equations. Our third main result is the following: Theorem 1.3 There exist locally Lipschitz-continuous functions fsatisfying (1.8) with p > 2, such that, whatever T > 0is, system (1.1) is not approximately controllable at time T. Remark 1.6 The proof of Theorem 1.3 is similar to the proof of Theorem 1.1. In this case, we introduce the following function f: f(s) = Zs 0 logp(1 + |σ|)dσ ∀s∈IR. The absorption effect of the nonlinearity allows to prove that, for any y0∈L2(Ω), the set of reachable states are uniformly bounded away from the control subdomain ω. Recall that, when f(s) = |s|r−1sfor some r > 1, this was already observed by A. Bamberger (see for instance [H]). 6 Remark 1.7 It would also be interesting to know whether blow-up phenomena can be an obstruction to approximate controllability even when the initial data are small. In particular, let us assume that y0= 0 and let fbe as in (1.9) with p > 2. We do not know if the following is true: For any y1∈L2(Ω) and any ε > 0, there exists a control v∈L∞(ω×(0, T)) such that the corresponding solution of (1.1) is globally defined in [0, T]and satisfies (1.13). We also have the following result: Theorem 1.4 Let T > 0and let the assumptions of Theorem 1.2 be satisfied. Then (1.1) is approximately controllable at time T. Let us briefly describe the strategy of proof of Theorem 1.4. We will argue as follows. Let y0,y1and ε > 0 be given and assume that y1is regular enough. We divide the time interval [0, T] in two parts. In the first (large) subinterval [0, T −δ], we choose v1such that ysatisfies (1.6), with y∗being the globally defined solution whose existence is assumed. In the second one, [T−δ, T ], we apply a control v2that drives the solution from y∗(·, T −δ) to w(·, T) exactly. Here, wis the solution of the auxiliary problem      wt−∆w+f(w) = 0 in Ω ×(T−δ, T) w= 0 on ∂Ω×(T−δ, T ) w(x, T −δ) = y1(x) in Ω. The existence of v1and v2is implied by Theorem 1.2. Furthermore, if δis sufficiently small, we have kw(·, T)−y1kL2(Ω) ≤ε. (1.14) Consequently, by setting v=v1for t∈(0, T −δ) and v=v2for t∈(T−δ, T) we will have found a control vsuch that the corresponding solution to (1.1) satisfies (1.13). Analyzing this proof, we see that the main ingredients needed for the proof to work are (a) there exists a trajectory starting from y0defined in the whole time interval [0, T −δ] and (b) the system is null-controllable in any arbitrarily short time interval. Since (1.4) avoids the presence of any blow-up phenomena, we find that an immediate consequence of Theorem 1.4 is the following: Corollary 1.1 Let T > 0. Assume that fis locally Lipschitz-continuous and satisfies (1.3) and (1.4). Then (1.1) is approximately controllable at time T. The rest of this paper is organized as follows. Section 2 is devoted to prove Theorem 1.1. In Section 3, we prove Theorem 1.2. In Section 4, we prove Theorem 1.3. Section 5 is concerned with the proof of Theorem 1.4. Finally, in Section 6, we discuss in detail the open problem mentioned above (fsatisfies (1.8) with 3/2≤p≤2) and other related issues. 7 2 Proof of the lack of null controllability This section is devoted to prove Theorem 1.1. Let us introduce the following function f: f(s) = Z|s| 0 logp(1 + σ)dσ ∀s∈IR,(2.1) with p > 2. Obviously, fis convex and f(s)s < 0 for s < 0. On the other hand, f(s)∼ |s|logp(1 + |s|) as |s| → ∞. Therefore, fis in the range of nonlinearities in which blow-up occurs in the absence of control, i.e. with v≡0. The proof of Theorem 1.1 is based on the fact that there are initial data which lead to blow-up before time T, whatever the control vis. We proceed as in [FI], Section I.5 (see also [H] for a similar argument in the context of approximate controllability). Thus, we introduce a nonnegative function ρ∈ D(Ω) such that ρ= 0 in ω, ZΩ ρ dx = 1.(2.2) Let v∈L∞(ω×(0, T)) be given and let ybe the solution to (1.1). Let us multiply by ρthe equation satisfied by yand let us integrate over Ω. Taking into account that the control is supported by ω×(0, T), we obtain: d dt ZΩ ρy dx =ZΩ ρ(∆y)dx −ZΩ ρf(y)dx. (2.3) Moreover, ZΩ ρ(∆y)dx =ZΩ (∆ρ)y dx. (2.4) From (2.3), (2.4) and the definition of f, we have d dt −ZΩ ρy dx=−ZΩ (∆ρ)y dx +ZΩ ρf(|y|)dx. (2.5) Taking into account that fis convex, we can introduce its convex conjugate f∗. For the moment, let us assume that ρf∗(2∆ρ/ρ)∈L1(Ω) (2.6) 8 (we shall return to (2.6) later on). Then, from Young’s inequality, we have ZΩ (∆ρ)y dx≤ZΩ ρ ∆ρ ρ|y|dx ≤1 2ZΩ ρf∗(2|∆ρ|/ρ)dx +1 2ZΩ ρf(|y|)dx. (2.7) Let us set k:= 1 2ZΩ ρf∗(2|∆ρ|/ρ)dx, (2.8) which is finite according to (2.6). From (2.5), (2.7) and (2.8), the following is deduced: d dt −ZΩ ρy dx≥ − ZΩ (∆ρ)y dx+ZΩ ρf(|y|)dx ≥ −k+1 2ZΩ ρf(|y|)dx. From Jensen’s inequality, we also have ZΩ ρf(|y|)dx ≥fZΩ ρ|y|dx. On the other hand, taking into account that fis increasing on [0,∞), we know that fZΩ ρ|y|dx≥fZΩ ρy dx=f−ZΩ ρy dx. Thus, if we set z(t) = −ZΩ ρ(x)y(x, t)dx for all tand z0=−ZΩ ρ(x)y0(x)dx, we find that    z0(t)≥ −k+1 2f(z(t)), z(0) = z0. (2.9) We are now going to use (2.9) to prove that, for appropriate initial data, zblows up at a finite time. More precisely, let y0∈L2(Ω) be such that z0=−ZΩ ρ(x)y0(x)dx > 0, f(z0)>2k and assume that z: [0, T∗)7→ IR is a C1function satisfying (2.9). Let us see that T∗<+∞. The function zis nondecreasing. Furthermore, if we set G(z0;s):=Zs z0 2 f(σ)−2kdσ ∀s≥z0, 9 3.3 Null controllability of the linear heat equation with a potential For the proof of Theorem 1.2, we will use a fixed point argument below. One of the main ingredients is to obtain explicit estimates of the norms of the controls needed to achieve the null controllability of the linear heat equation with a potential. Let us consider the system:      yt−∆y+ay =v1ωin Q y= 0 on Σ y(x, 0) = y0(x) in Ω, (3.13) where a∈L∞(Q). The following holds: Theorem 3.1 For any T > 0, any a∈L∞(Q)and any y0∈L2(Ω), there exist controls v∈L∞(ω×(0, T)) such that the corresponding solution of (3.13) satisfies y(x, T) = 0 in Ω.(3.14) Furthermore, vcan be chosen such that the following estimate holds: kvkL∞(ω×(0,T)) ≤exp C1 + 1 T+T+ (T1/2+T)kak∞+kak2/3 ∞ky0kL2(Ω) .(3.15) Proof. Let us fix T > 0, a∈L∞(Q) and y0∈L2(Ω). For any ε > 0, we consider the following functional: Jε(ϕ0) = 1 2"ZZω×(0,T) |ϕ|dx dt#2 +εkϕ0kL2(Ω) +ZΩ ϕ(x, 0)y0(x)dx ∀ϕ0∈L2(Ω).(3.16) Here, for each ϕ0∈L2(Ω), ϕis the corresponding solution of (3.1). It is not difficult to see that ϕ07→ Jε(ϕ0) is a continuous and strictly convex function on L2(Ω). Moreover, Jεis coercive. In fact, proceeding as in [FPZ], it can be checked that lim inf kϕ0kL2(Ω)→∞ Jε(ϕ0) kϕ0kL2(Ω) ≥ε. Therefore, Jεachieves its minimum at a unique b ϕ0 ε∈L2(Ω). Let b ϕεbe the associate solution of (3.1). Again arguing as in [FPZ], it is easy to see that, for some vε∈ ZZω×(0,T) |b ϕε|dx dt!sgn ( b ϕε), 16 the solution yεof (3.13) satisfies kyε(·, T)kL2(Ω) ≤ε. (3.17) We claim that, for a suitable C=C(Ω, ω)>0, the following holds: kvεkL∞(ω×(0,T)) ≤exp C1 + 1 T+T+ (T1/2+T)kak∞+kak2/3 ∞ky0kL2(Ω) .(3.18) Indeed, at the minimum b ϕ0 ε, we have Jεb ϕ0 ε≤Jε(0) = 0. In accordance with (3.16), we find that 1 2"ZZω×(0,T) |b ϕε|dx dt#2 ≤ZΩb ϕε(x, 0)y0(x)dx ≤ k b ϕε(·,0)kL2(Ω)ky0kL2(Ω) and, therefore, ZZω×(0,T) |b ϕε|dx dt ≤exp C1 + 1 T+T+ (T1/2+T)kak∞+kak2/3 ∞ky0kL2(Ω) (3.19) (we have used here (3.3)). From (3.19), we obtain (3.18) taking into account that kvεkL∞(ω×(0,T)) =ZZω×(0,T) |b ϕε|dx dt. In view of the uniform bound (3.18), by extracting an appropriate subsequence, we deduce that vε→vweakly-∗in L∞(ω×(0, T)), where vsatisfies (3.15). Since we have (3.17) for all ε > 0, we deduce that vis such that the solution of (3.13) satisfies (3.14). This ends the proof. Remark 3.1 An immediate consequence of Theorem 3.1 is that we can also estimate the minimal norm of a control vneeded to drive an initial datum y0to a final state z(·, T), with z being the solution of (3.13) corresponding to an initial datum z0∈L2(Ω) and a right hand side w∈L∞(ω×(0, T)). Indeed, by means of the change of variable p=y−z, we see that vdrives y from y0to z(·, T) if and only if the control ν=v−wdrives pfrom y0−z0to zero. In accordance with Theorem 3.1, we can find such a νsatisfying kνkL∞(ω×(0,T)) ≤exp C1 + 1 T+T+ (T1/2+T)kak∞+kak2/3 ∞ky0−z0kL2(Ω) . 17 Hence, we have      kvkL∞(ω×(0,T)) ≤exp C1 + 1 T+T+ (T1/2+T)kak∞+kak2/3 ∞ky0−z0kL2(Ω) +kwkL∞(ω×(0,T)) . 3.4 The fixed point method: Conclusion Let us complete the proof of Theorem 1.2. Let y∗be a bounded and globally defined solution of (1.1), associated to y∗ 0∈L2(Ω) and v∗∈L∞(ω×(0, T)). Let us perform the change of variable p=y−y∗. Then, ysolves (1.1) if and only if psatisfies      pt−∆p+f(y∗+p)−f(y∗) = ν1ωin Q p= 0 on Σ p(x, 0) = p0(x) in Ω, (3.20) where ν=v−v∗and p0=y0(x)−y∗ 0. Our task is then to show that, for each p0∈L2(Ω), there exists ν∈L∞(ω×(0, T)) such that p(x, T) = 0 in Ω.(3.21) We will argue as follows. Let us assume that |y∗(x, t)| ≤ k∗in Ω ×(0, T) and let us set M(k∗) = max |a|≤k∗|f(a)|.(3.22) We will first consider the case in which p0∈C0,β(Ω) for some β∈(0,1) and fis C1in [−k∗, k∗]. Let us set e g(a, s) =    f(a+s)−f(a) sfor |a| ≤ k∗,s6= 0 f0(a) for |a| ≤ k∗,s= 0. (3.23) Then e gis continuous. We claim that, for each η > 0, there exists Cη>0 (which only depends of η,k∗and the function f) such that (|e g(y∗(x, t), s)|2/3≤Cη+ηlog(1 + |s|) ∀s∈IR,∀(x, t)∈Q. (3.24) 18 Indeed, it will be sufficient to prove that, for each η > 0, one has |e g(y∗(x, t), s)| ≤ Cη+ηlog3/2(1 + k∗+|s|) (3.25) for all s∈IR and (x, t)∈Q. Let ηbe given and let s(η) be such that s(η)≥k∗+ 1 and  f(s) s≤2η 3log3/2(1 + |s|) for |s| ≥ s(η). (3.26) For |s| ≤ 2s(η), we have |e g(y∗(x, t), s)| ≤ L(k∗+ 2s(η)) ∀(x, t)∈Q, (3.27) where L(k∗+2s(η)) is a Lipschitz constant for fin [−k∗−2s(η), k∗+2s(η)]. On the other hand, for |s|>2s(η), we see that |e g(y∗(x, t), s)| ≤  f(y∗(x, t) + s) s+1 2s(η)|f(y∗(x, t))| ≤max |s| 2≤|σ|≤|s|+k∗ f(σ) s+1 2s(η)max |a|≤k∗|f(a)| ≤3 2max |s| 2≤|σ|≤|s|+k∗ f(σ) σ+1 2k∗M(k∗). Here, we have used that s(η)> k∗and, consequently, |σ|≤|s|+k∗and |s|>2s(η) imply |σ|<3|s| 2. We deduce that, for |s|>2s(η), |e g(y∗(x, t), s)|ηlog3/2(1 + k∗+|s|) + 1 2k∗M(k∗)∀(x, t)∈Q. (3.28) Combining (3.27) and (3.28), it is not difficult to deduce (3.24), with Cηonly depending of η, k∗and f. Notice that two different functions ffor which the restrictions to IR \(−k∗−1, k∗+ 1) and the quantities M(k∗) and L(k∗+ 1) (a Lipschitz constant in [−k∗−1, k∗+ 1]) coincide lead to the same constants Cηin (3.25) and (3.24). This will be used below. Let us set Z=L∞(Q). Let R > 0 be a constant whose value will be determined below. We will use the truncation function TR, which is given as follows: TR(s) = (sif |s| ≤ R Rsgn(s) otherwise. 19 For each z∈Z, we will consider the linear system      pt−∆p+e g(y∗, TR(z))p=ν1ωin Q p= 0 on Σ p(x, 0) = p0(x) in Ω. (3.29) Obviously, (3.29) is of the form (3.13), with a=e g(y∗, TR(z)) ∈L∞(Q). Let us set T∗ z= min hT, ke g(y∗, TR(z))k−2/3 ∞,ke g(y∗, TR(z))k−1/3 ∞i.(3.30) According to Theorem 3.1, there exist controls νz∈L∞(ω×(0, T ∗ z)) such that the solution of (3.29) in Ω ×(0, T∗ z) with ν=νzsatisfies p(x, T∗ z) = 0 in Ω and kνzkL∞(ω×(0,T∗ z)) ≤C0(Ω, ω, T ∗ z,ke g(y∗, TR(z))k∞)kp0kL2(Ω) ,(3.31) where C0(Ω, ω, T ∗ z,ke g(y∗, TR(z))k∞) = eC1+ 1 T∗ z+T∗ z+((T∗ z)1/2+T∗ z)keg(y∗,TR(z))k∞+keg(y∗,TR(z))k2/3 ∞.(3.32) Let A(z)⊂L∞(ω×(0, T)) be the family formed by the extensions by zero to the whole interval [0, T] of all these controls. Assume that νz∈A(z). From the definition (3.30) of T∗ zand (3.31)–(3.32), it is clear that kνzkL∞(ω×(0,T)) ≤exp hC1 + ke g(y∗, TR(z))k2/3 ∞ikp0kL2(Ω) ,(3.33) where C=C(Ω, ω, T ). Now, let Λ(z)⊂L∞(Q) be the family of the solutions of (3.29) corresponding to the controls νz∈A(z). Obviously, if pz∈Λ(z), one has pz≡0 in Ω ×(T∗ z, T). In particular, pz(x, T) = 0 in Ω.(3.34) In this way, we have been able to introduce a set-valued mapping z7→ Λ(z). We will now check that, for some R, this mapping possesses at least one fixed point psuch that kpk∞≤R. (3.35) 20 Of course, this will imply the existence of a control ν∈L∞(ω×(0, T)) such that the solution of (3.20) satisfies (3.21). To this end, we will first see that Kakutani’s Fixed Point Theorem can be applied to Λ. Indeed, it is not difficult to see that Λ(z) is, for each z∈L∞(Q), a nonempty closed convex set. From parabolic regularity, we also see that there exists a fixed compact subset K⊂L∞(Q) such that Λ(z)⊂K∀z∈L∞(Q) (we are using here the hypothesis p0∈C0,β(Ω)). Furthermore, z7→ Λ(z) is upper hemicontinuous, i.e. the real-valued function z7→ sup p∈Λ(z) hµ, pi is upper semicontinuous for each bounded linear form µon L∞(Q). Consequently, Kakutani’s Theorem will imply the existence of a fixed point of Λ if we prove that, whenever Ris sufficiently large, any fixed point of Λ must satisfy (3.35). We will use (1.10) (and (3.24); notice that, up to now, this has been ignored). Thus, let pbe a fixed point, associated to the control νp∈A(p). From classical L∞estimates on the solutions of (3.29), we have kpk∞≤eT∗ pkeg(y∗,TR(p))k∞kp0kL∞(Ω) +T∗ peT∗ pkeg(y∗,TR(p))k∞kvpkL∞(ω×(0,T)) . Consequently, taking again into account the definition of T∗ pand using (3.24) and (3.33), we deduce that kpk∞≤eC(1+keg(y∗,TR(p))k2/3 ∞)kp0k∞+kvpkL∞(ω×(0,T)) ≤eC(1+Cη+ηlog(1+kTR(p)k∞))kp0k∞ =eC(1+Cη)kp0k∞(1 + kTR(p)k∞)ηC ≤eC(1+Cη)kp0k∞(1 + R)ηC , where C=C(Ω, ω, T ). Taking η > 0 small enough to have ηC < 1, we deduce the existence of R > 0 such that kpk∞≤R. As mentioned above, this proves that, for some ν∈L∞(ω×(0, T)), the corresponding solution to (3.20) satisfies (3.21). Hence, our assertion is proved when y0∈ C0,β(Ω) and fis C1in [−k∗, k∗]. Notice that, in the previous argument, Rcan be chosen depending only of Ω, ω,T, the restriction of fto IR \(−k∗, k∗), M(k∗), L(k∗+ 1) and kp0kL∞(Ω) . Thus, we have solved the controllability problem (3.20)–(3.21) with pand νrespectively bounded in L∞(Q) and L∞(ω×(0, T)) by constants which only depend of these data. Now, let us assume that fis only locally Lipschitz-continuous and p0∈C0(Ω). We can put f= lim n→∞ fnuniformly in IR, 21 for some locally Lipschitz-continuous functions fnwhich are C1in [−k∗, k∗], coincide with f outside (−k∗−1, k∗+ 1) and have the same Lipschitz constants in [−k∗−1, k∗+ 1]. We can also put p0= lim n→∞ pn 0uniformly in Ω, for some functions pn 0∈C0,β(Ω). For each n≥1, we can argue as before and find a control νn∈L∞(ω×(0, T)) such that the solution pnto      pn t−∆pn+fn(y∗+pn)−fn(y∗) = νn1ωin Q pn= 0 on Σ pn(x, 0) = pn 0(x) in Ω (3.36) satisfies pn(x, T) = 0 in Ω. From the estimates we have found, we deduce that it can be assumed that kνnkL∞(ω×(0,T)) and kpnk∞ are uniformly bounded. Accordingly, we can take limits in (3.36) (at least on a subsequence). In other words, it can be assumed that pnconverges (at least) strongly in L2(Q) and a.e. and νn converges weakly-∗in L∞(ω×(0, T)). This provides a control ν∈L∞(ω×(0, T)) such that the corresponding solution to (3.20) satisfies (3.21). This proves our assertion when p0∈L∞(Ω). Finally, let us consider the general case, where p0∈L2(Ω) and fis only locally Lipschitzcontinuous. Let δ > 0 be sufficiently small and let us set ν≡0 for t∈(0, δ). Then, by parabolic regularity, the corresponding (local) solution to (3.20) satisfies p(·, δ)∈C0(Ω). Now, we can apply the arguments above to p(·, δ) in the interval [δ, T]. Of course, this provides a control ν∈L∞(ω×(δ, T )) such that (3.21) holds. This completes the proof of Theorem 1.2. Remark 3.2 The arguments used for the proof of Theorem 1.2 also lead to new proofs of some known facts on the local and large time null controllability of (1.1). For instance, it can be shown that, if fsatisfies (1.2) and (1.3), for each T > 0 we can find ρ(T)>0 such that, whenever ky0kL2(Ω) ≤ρ(T), we have (1.7) for some v. On the other hand, assuming that fsatisfies a good-sign condition and f(0) = 0, we can prove zero controllability for large T. Indeed, due to the nonlinear absorption effect, solutions without control enter a small ball of L2(Ω) at a uniform time. Then, making use of local null controllability, solutions are driven to zero (see [F], [FI] and [I] for more details concerning these and other related questions). 22 4 Proof of the lack of approximate controllability This section is devoted to prove Theorem 1.3. The proof is similar to the one of Theorem 1.1. This time, it will be seen that, for an appropriate function satisfying (1.8) with p > 2, once the initial datum y0and a compact set F⊂Ω\ωare fixed, the restrictions to Fof the reachable states are uniformly bounded in L1(F). We proceed as in [H]. Let us introduce the function f, with f(s) = Zs 0 logp(1 + σ)dσ ∀s∈IR and p > 2. Let us also introduce a nonnegative function ρ∈ D(Ω) satisfying (2.2). Let y0∈L2(Ω) and v∈L∞(ω×(0, T)) be given and let ybe the solution to (1.1). Let us multiply by ρsgn(y) the equation satisfied by yand let us integrate over Ω. Then d dt ZΩ ρ|y|dx =ZΩ ρsgn(y)(∆y)dx −ZΩ ρ|f(y)|dx. From Kato’s inequality, we know that ZΩ ρsgn(y)(∆y)dx ≤ZΩ (∆ρ)|y|dx. We also have ZΩ (∆ρ)|y|dx≤1 2ZΩ ρf(|y|)dx +1 2ZΩ ρe f(2|∆ρ|/ρ)dx, (4.1) where e fis the conjugate of the convex function s7→ f(|s|). Arguing as in the proof of Theorem 1.1, we see that, by choosing ρappropriately (depending of Ω and ω), the last integral in the right hand side of (4.1) is finite (because p > 2). Consequently, d dt ZΩ ρ|y|dx +1 2ZΩ ρf(|y|)dx ≤C, where C=C(Ω, ω, p). In view of the convexity of the function s7→ f(|s|), this gives d dt ZΩ ρ|y|dx +1 2fZΩ ρ|y|dx≤C. In other words, setting z(t) = ZΩ ρ(x)|y(x, t)|dx for all t, we see that z0(t) + 1 2f(z(t)) ≤C. 23 This implies that z(T) = ZΩ ρ(x)|y(x, T)|dx is bounded independently of the control v. Therefore, once y0∈L2(Ω) and an arbitrary compact set F⊂Ω\ωare fixed, the restrictions to Fof the reachable states are indeed uniformly bounded in L1(F). This is obviously in contradiction with the density of the set of reachable states in L2(Ω). 5 Proof of the approximate controllability result The goal of this section is to prove Theorem 1.4. We will use Theorem 1.2. More precisely, we will deduce that (1.1) is approximately controllable by driving the solution exactly to the final point of an uncontrolled trajectory. In order to compensate the fact that we are in the range of nonlinearities for which blow-up may occur, we have to assume again that a globally defined solution y∗exists. This hypothesis, in combination with the null controllability property, allows to keep the trajectory well defined during a large part of the time interval [0, T]. First of all, notice that |f(s)| ≤ C(1 + |s|log3/2(1 + |s|)) ∀s∈IR,(5.1) as a consequence of (1.10). Let us fix T > 0, y0∈L2(Ω), y1∈C1(Ω) (for instance) and ε > 0 and let us try to find v∈L∞(ω×(0, T)) such that the solution of (1.1) satisfies (1.13). Obviously, it will be sufficient to consider final data y1∈C1(Ω), since this space is dense in L2(Ω). For any small δ > 0, we will consider the following auxiliary system:      wt−∆w+f(w) = 0 in Ω ×(T−δ, T) w= 0 on ∂Ω×(T−δ, T ) w(x, T −δ) = y1(x) in Ω. (5.2) There exists exactly one solution wto (5.2) which is defined in the whole interval [T−δ, T ]. Furthermore, wis bounded and there exists δ > 0 (small enough) such that kw(·, T)−y1kL2(Ω) ≤ε. (5.3) δdepends of Ω, ky1kL∞(Ω) ,εand the constant Cin (5.1). We fix δand wfrom now on such that (5.3) is satisfied. Our strategy is as follows: •For t∈(0, T −δ), we set v=v1, where v1∈L∞(ω×(0, T −δ)) is such that (1.1) is driven to y∗(·, T −δ) at time t=T−δ. In view of Theorem 1.2, such a control v1exists. This defines yin Ω ×(0, T −δ), with y(x, T −δ) = y∗(x, T −δ) in Ω. (5.4) 24 •We see from (5.3) that, in [T−δ, T], the problem is reduced to drive our system from y∗(T−δ) to the state w(·, T) exactly. Again in view of Theorem 1.2, there exists v2∈ L∞(ω×(T−δ, T )) such that the solution to      yt−∆y+f(y) = v21ωin Ω ×(T−δ, T) y= 0 on ∂Ω×(T−δ, T ) y(x, T −δ) = y∗(x, T −δ) in Ω (5.5) satisfies y(x, T) = w(x, T) in Ω. Now, we set v=v2for t∈(T−δ, T ). Obviously, vhas the desired property. This completes the proof of Theorem 1.4. 6 Further comments, results and open problems 6.1 On the assumption of existence of a globally defined trajectory The positive results of this paper, both in what concerns null and approximate controllability, require the existence of a globally defined and bounded solution for suitable y0and v. Obviously, this assumption is necessary since, otherwise, no global solution exists and the controllability problems do not make sense since they require the trace of the solution at time t=Tto be well defined. The existence of global and/or stationary solutions for semilinear parabolic problems like (1.1) with v≡0 has been the objective of intensive research. See for instance [BCMR], [CH1], [CH2], [G], [GV] and the references therein. However, the wider issue of whether global solutions exist for suitable right hand sides vlocalized in a given subdomain ωseems to be much more open. Of course, it would be interesting to have at our disposal sharp sufficient conditions guaranteeing the existence of globally defined solutions of (1.1) with possibly v6= 0. 6.2 On the optimality of the null controllability results Let us first discuss the optimality of the negative result in Theorem 1.2. Its proof is based on a localization in space of classical estimates for semilinear heat equations. Therefore, the critical growth conditions are determined by the interaction between the elliptic operator −∆ and the nonlinearity f. For second order O.D.E.’s, the critical growth condition is f(s)∼ |s|log2|s|. 25