Exact controllability of the superlinear heat equation

Transcription

1 Exact controllability of the superlinear heat equation Youjun Xu 1,2, Zhenhai Liu 1 1 School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan , P R China 2 School of Mathematics and Physics, University of South China, Hengyang, Hunan , P R China Abstract. In this paper, we consider the controllability of a semilinear heat equation with a nonlinearity that has a superlinear growth at infinity with Dirichlet boundary conditions in a bounded domain of R N. The proof of the main result in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments. Keywords. Controllability; Parabolic equation; Superlinearity. AMS (MOS) subject classification: 35K55; 35K05; 93B05. 1 Statement of the problem and main results Let Ω R N, N 1, be an open and bounded set with boundary Ω C 2. Let ω be a open and non-empty subsets of Ω. For T > 0, we denote = Ω (0, T ) and Σ = Ω (0, T ). We consider the following quasi-linear parabolic system: y t y + f(x, y) = ξ + v1 ω in, y = 0 on Σ, (1.1) y(x, 0) = y 0 (x) in Ω, where y t = y t, ξ Lr (), y 0 L 2 (Ω) are given and v is a control function to be determined in L r (), 1 ω denotes the characteristic function of the set w. r > 1 + N, if N 2; r = 2 if N = 1. (1.2) 2 Let us consider an ideal trajectory y, solution of the problem without control yt y + f(x, y ) = ξ in, y = 0 on Σ, y (x, 0) = y0(x) in Ω, (1.3) 0 Corresponding author: Youjun Xu, addresses: xuyoujun7618@yahoo.com.cn, zhhliu@mail.csu. edu.cn(z Liu). 0 The authors were supported financially by the National Natural Science Foundation of China ( ), The author(y. Xu) was supported financially by the Scientific Research Fund of Hunan Provincial Educational Department (06B080).

2 2 Y Xu, Z. Liu where y 0 L 2 (Ω) and ξ L r () with r as in (1.4). Then we know that under condition (1.2), (1.3) has one local solution in time T (see [15] and [16]). Moreover, there exists T > 0 such that for T < T, the solution y of (1.5) satisfies y C([0, T ]; L 2 (Ω)) L 2 (0, T ; H 1 0 (Ω)). The controllability of linear and semilinear parabolic systems has been extensively considered (see[1, 2, 3, 4, 5, 9, 10, 13, 14, 17]). [9] [10] [5] [4] concerning null controllability, [3] [17][10][4] for approximate controllability, [10] [13] and [14] for exact controllability to the trajectories. In the study of the null controllability of semilinear parabolic systems with superlinear nonlinearities, additional technical difficulties arise. Let us recall what happens in the simpler case of one semilinear heat equation. In [1], Barbu obtained null L r(n) -controls v such that v L r(n) () C y 0 L 2 However, this technique can only be applied to the null controllability of the superlinear heat equation when N < 6. The main goal of this paper is to generalize some previous results under the Dirichlet boundary conditions, in particular, those in [10] and [1]. We overcome such difficulty and obtain the controllability of system in the Dirichlet boundary conditions. The main result in this paper is the following one: Theorem 1.1. Assume that f : Ω R is locally Lipschitz-continuous in y, measurable in x. g(x, s) lim = 0, (1.4) s log 3/2 (1+ s ) where g(x, s) = 1 d 0 ds f(x, σs)dσ and ξ Lr (), and r satisfies (1.2). Then (1.1) is exactly controllable to the trajectory at time T. The proof of Theorem 1.1 is based on the null controllability of a linear problem (see Theorem 3.1) which is obtained from observability inequalities (see Theorem 2.1). The idea of combining the controllability of a linearized system and a fixed point argument in the proof will be applied in this paper. It is introduced in [14] (also see [1, 4, 7, 8]) in the context of the boundary controllability of the semilinear wave equation. In the following, we state some technical results which will be used later. They are known results on the local regularity for the solutions to the linear heat equation. Nevertheless, they will be essential in our analysis. First, let us present the following notation, which is used all along this paper. For r [1, ) and a given Banach space X, Lr (X) will denote the norm in L r (0, T ; X). For simplicity, the norm in L r () will be represented by L r, for r [1, ), and will denote the norm in L (). For r [1, ), and any open set V R N, we will consider the Banach space, and its norm, defined by X r (0, T ; V ) = {u L r (0, T ; W 2,r (V )) : u t L r (0, T ; L r (V ))} u X r (0,T ;V )= u t L r (W 2,r (V )) + u L r (L r (V )) In particular, we will consider the space X r (0, T ; Ω) and its norm, denoted by X r. The norm in the space L 2 (0, T ; H 2 (Ω)) C([0, T ]; H 1 (Ω)) will be denoted by L 2 (H 2 C(H 1 )). Lemma 1.2 [8]. Let a L () and F L 2 () be given. Let us consider a solution

3 Y Xu, Z. Liu 3 y L 2 (0, T ; H 2 (Ω)) C([0, T ]; H 1 (Ω)) to y t y + ay = F in, y = 0 on Σ, y(x, 0) = 0 in Ω, (1.5) (a) Let V Ω (resp. B Ω) be an open set. Let us suppose that F L r (0, T ; L r (V)) (resp. F L r (0, T ; L r (Ω\ B))), with r (2, ). Then, for any open set V V(resp. B B Ω) one has y X r (0, T ; V ) (resp.y X r (0, T ; Ω\ B ). Moreover, there exist positive constant C = C(Ω, T, N, r, V, V )(resp. C = C(Ω, T, N, r, B, B )) and K = K (N) such that y Xr (0,T ;V ) C(1+ a ) K [ F Lr (L r (V) + y L2 (H 2 ) C(H 1 )]. (resp. y Xr (0,T ;Ω\ B ) C(1+ a ) K [ F Lr (L r (Ω\ B ) + y L2 (H 2 ) C(H 1 )]. (b) Assume, in addition, that F L r (0, T ; W 1,r (V )), with r as above, and a L γ () N, with max(r, N γ = 2 + 1), if r N 2 + 1, N ɛ, if r = N 2 + 1, and ɛ being an arbitrarily small positive number. Then, for any open set V V, one has y L r (0, T ; W 3,r (V )), y t L r (0, T ; W 1,r (V )) and, for a new positive constant C = C(Ω, T, N, r, V, V ), the following estimate holds y Lr W 3,r (V ) + y t Lr W 3,r (V ) CH [ F Lr (W 1,r (V)) + y L2 (H 2 ) C(H 1 )] where H = H (N, a, a L γ ) = (1+ a ) K +1 (1+ a L γ ). Lemma 1.3 [8]. Let Ω R N, N 1, be an open set with Ω C 2. The following continuous embeddings hold: (i) If r < N 2 + 1, then Xr L p (), where 1 p = 1 r 2 N + 2. (ii) If r = N 2 + 1, then Xr L q (), q <. (iii) If N < r < N + 2, then Xr C β, β N ( ), with β = 2. r (iv) If r = N + 2, then X r C l, l 2 ( ), l (0, 1). (v) If r > N + 2, then X r 1+α 1+α, N + 2 C 2 ( ), where α = 1. r The rest of this paper is organized as follows: in Section 2, we present an observability inequality; in Section 3, we prove Theorem 1.1.

4 4 Y Xu, Z. Liu 2 The observability inequality In this section we present an observability inequality that is a generalization of the one given in [10] to the case of linear systems with Dirichlet boundary conditions, which will be essential to prove Theorem 1.1. Before giving the proof of Theorem 1.1, we have to present some technical results. Let us consider following problem p t + p = f in, p = 0 on Σ, (2.1) p(x, 0) = p 0 (x) in Ω. Lemma 2.1 [11]. Let ω be a nonempty open subset of Ω and ω 0 be an open subset of ω such that ω 0 ω. Then there exists a function Φ(x) C 2 (Ω) such that Φ(x) > 0 in Ω, Φ = 0 on Ω and Φ > 0 in Ω\ω 0. For λ > 0, for any (x, t), we set α(x, t) = exp(λφ(x)) exp(2λ Φ C(Ω)), ϕ(x, t) = exp(λφ(x)) t(t t) t(t t) Lemma 2.2 [11]. Assume p be a solution of (2.1) associated to p 0 L 2 (Ω) and f L 2 (). Let ω be nonempty open subset of Ω. Then there exist positive constant λ 0 and a function s 0 (λ) R +, such that s 1 exp(2sα)t(t t)( p t 2 + p 2 ) + s exp(2sα)t 1 (T t) 1 p 2 + s 3 exp(2sα)t 3 (T t) 3 p 2 dxdt C λ (s 3 exp(2sα)t 3 (T t) 3 p 2 dxdt + exp(2sα) f 2 dxdt) ω (0,T ) (2.2) for s s 0, with α as in Lemma 2.1. In the sequel, unless otherwise specified, C will stand for a generic positive constant only depending on Ω and ω, whose value can change from line to line. Let us introduce the following (adjoint)system q t q + aq = 0 in, q = 0 on Σ, (2.3) q(x, T ) = q T in Ω, Theorem 2.1. For any a L () and q T L 2 (Ω), one has q(x, 0) 2 L 2 (Ω) exp(ck(t, a )( q r dxdt) 2/r (2.4) ω (0,T ) where K(T, a ) = T + (T 1/2 + T ) a + a 2/3, q is the solution of system (2.3), 1 r + 1 r = 1.

5 Y Xu, Z. Liu 5 Proof of the Theorem 2.1. Let ω ω, applying (2.2), then we get s 3 exp(2sα)t 3 (T t) 3 q 2 dxdt C λ (s 3 exp(2sα)t 3 (T t) 3 q 2 dxdt ω (0,T ) + exp(2sα) aq 2 dxdt) (2.5) for all s s 0. We can estimate the second term in the right as follows: exp(2sα) aq 2 dxdt 2 6 T 6 a 2 exp(2sα)t 3 (T t) 3 q 2 dxdt (2.6) together with (2.5), we have exp(2sα)t 3 (T t) 3 q 2 dxdt C for all s s 1 = max{s 0, CT 2 a 2/3 } On the other hand, we easily know that ω (0,T ) exp(2sα)t 3 (T t) 3 q 2 dxdt (2.7) exp(2sα)t 3 (T t) T 6 exp( CsT 2 ) (x, t) (2.8) exp(2sα)t 3 (T t) 3 ( 16 3 )3 T 6 exp( CsT 2 ) (x, t) Ω ( T 4, 3T 4 ) (2.9) together with (2.7), set M s = s max{exp(2λ Φ C(Ω) ) exp(λφ(x))}, then exp( M s t(t t) ) q 2 dxdt T 6 exp(2sα)t 3 (T t) 3 q 2 dxdt (2.10) ω Let a function ξ(x) D(ω) such that ξ = 1 in ω. We set p = ξv(t)q where q is the solution of (2.3) and v(t) = exp(sα)t 3/2 (T t) 3/2. Note that p(t ) = p(0) = 0. Then, we have p t p = aξvq ξv t q 2v ξ q ξqv in, p = 0 on Σ, (2.11) p(x, T ) = 0 in Ω, For simplicity of the computations, we set p(x, t) = p(x, T t) for any (x, t). In a similar way, we introduce the functions ã, ṽ, q. Thus, we have p t p = ãξṽ q ξṽ t q 2ṽ ξ q ξ qṽ in, p = 0 on Σ, (2.12) p(x, 0) = 0 in Ω,

6 6 Y Xu, Z. Liu Thanks to the regularizing effect of the heat equation, we have for any t > 0, 1 r 1, r 2 S(t)u L r 1 (ω) Ct N 2 ( 1 r 2 1 r 1 ) 1 2 u L r 2 u L r2, (2.13) S(t)u W 1,r 1 (ω) Ct N 2 ( 1 r 2 1 r 1 ) u L r 2 u L r2, (2.14) where {S(t)} t 0 be the semigroup generated by the heat equation with Dirichlet boundary conditions. Applying the L 2 L r regularing effect of the heat equation, we obtain Note that t p(, t) L2 (ω ) C(1+ a ) t + C + C 0 t 0 0 v (t τ) N 2 ( r ) q(, τ) L r (ω) dτ v (t τ) N 2 ( r ) 1 2 q(, τ) L r (ω) dτ v t (t τ) N 2 ( r ) q(, τ) L r (ω) dτ for s σ(ω, ω)t 2. Together with (2.15), we have (2.15) ṽ CT 3 exp( sct 2 ), (x, t) (2.16) ṽ t CT 6 (s + T 2 )exp( sct 2 ) (2.17) p(, t) L2 (ω ) CT 3 (1 + T 1/2 + T 1/2 a )exp( sct 2 ) t 0 t 0 (t τ) N 2 ( r ) 1 2 q(, τ) L r (ω) dτ + CT 1/2 T 6 (s + T 2 )exp( sct 2 ) (t τ) N 2 ( r ) 1 2 q(, τ) L r (ω) dτ (2.18) If r satisfies N 2 (1 2 1 r ) + 1 r < 1 that is to say r 2(N + 2) > (2.19) N + 4 Thus, by Young s inequality and estimate the L 2 (0, T ; L 2 (ω ))-norm of p, we obtain p(, t) L 2 (0,T ;L 2 (ω )) CT α (1 + s + T 2 + T 1/2 a ) exp( sct 2 (2.20) )( q(, τ) r 2/r dτ) ω (0,T ) where C depends on Ω, ω, ω, r, N; α depends on r, N; s σ(ω, ω)t 2. Note that N < 4 and r as in (1.2) and (2.19) is satisfied. For N 4, we continuous this process and obtain (2.20) for r sufficiently small. Combined with (2.10), we get exp( M s t(t t) ) q 2 dxdt T 6 exp(2sα)t 3 (T t) 3 q 2 dxdt CT α (1 + s + T 2 (2.21) ω (0,T ) + T 1/2 a )exp( sct 2 )( q(, τ) r dτ) 2/r ω (0,T )

7 Y Xu, Z. Liu 7 Let θ 0 C 1 [0, 1] be a function such that, θ 0 [0, 1], θ 0 = 1 in [0, 1 4 ], θ 0 = 0 in [ 3 4, 1]. Now, we define a function θ(t) = θ 0 ( t T ) and rewrite (2.3) for θ(t)q, then (θq) t (θq) + aθq = qθ t in Ω (0, 3T 4 ), θq = 0 on Ω (0, 3T 4 ), (2.22) θq(x, 3T 4 ) = 0 in Ω, Multiplying (2.11) by θq and integrating in Ω, we have d 2dt θq (θq) a(θq) 2 dx = Ω Ω qθ t θqdx Thus d dt θq (θq) a θq 2 dx + 2 qθ t θqdx Ω Ω By using Hölder and Young inequalities, then we have d dt (exp(2 a t) θq 2 2) exp(2 a t) θ t θq 2 dx (2.23) For any t 0, integrating this inequality with respect to time in [0, t] with t [ 3T 4, T ], note that θ 1, θ t C/T, we obtain q(x, 0) 2 L 2 (Ω) 1 T exp(3t 2 a ) q 2 dxdt (2.24) Ω ( T 4, 3T 4 ) Together with (2.9) and (2.7), we obtain q(x, 0) 2 L 2 (Ω) 1 T exp(3t 2 a ) ω (0,T ) By (2.20), we have q(x, 0) 2 L 2 (Ω) exp(ck(t, a ))( Ω exp(2sα)t 3 (T t) 3 q 2 dxdt (2.25) ω (0,T ) q(, τ) r dτ) 2/r (2.26) where K(T, a ) = T + (T + T 1/2 ) a + a 2/3. This completes the proof. 3 Proof of Theorem 1.1 In this section, we will prove Theorem 1.1. The structure of the proof is similar to that of Theorem 1.1 in [10]. Firstly, we consider the controllability of a linear heat equation under Dirichlet boundary conditions. For given a L () and ξ L r (), we analyze the linear system y t y + ay = ξ + v1 ω in, y = 0 on Σ, (3.1) y(x, 0) = y 0 in Ω,

8 8 Y Xu, Z. Liu Theorem 3.1. Assume that T > 0, a L (). Then, there exists a positive constant M (depending on Ω, ω, T ) such that, for any ξ L r () verifying M exp( t(t t) ) ξ 2 dxdt < (3.2) one can find a control function ˆv L r (ω (0, T )) such that the corresponding solution ŷ of (3.1) satisfying ŷ(x, T ) = 0 in Ω (3.3) Furthermore, ˆv can be chosen in such a way that ˆv Lr (ω (0,T )) exp(ck(t, a ))( y 0 L2 (Ω) + ξ Lr ()) (3.4) where K(T, a ) is given by Theorem 2.1. Proof of Theorem 3.1. For every ɛ > 0, q T L 2 (Ω), we consider the functional J ɛ with J ɛ (q T ) = 1 2 ( q r dxdt) 2/r + ɛ q T 2 L 2 (Ω) + q(x, 0)y 0 (x)dx + ξqdxdt, ω (0,T ) Ω (3.5) where q is the solution of (2.3) associated to q T L 2 (Ω). It is easy to see that J ɛ is a continuous and strictly convex functional. Moreover, By (2.7), it verifies the unique continuation property, i.e. Arguing as in [3], J ɛ is coercive. In fact, we have q = 0, in ω (0, T ), then q 0. (3.6) lim inf q T L 2 J ɛ (q T ) q T L 2 ɛ (3.7) therefore, J ɛ achieves its minimum at a unique point ˆq T,ɛ L 2 (Ω). let ˆq ɛ be the solution of (2.3) associated to ˆq T,ɛ. Arguing as in [3], we take in (3.1) v = v ɛ where ˆv ɛ = sgn(ˆq ɛ ) ˆq ɛ r 1 ω then, we find a solution ŷ ɛ satisfies ŷ ɛ (, T ) L 2 ɛ (3.8) On the other hand, at the minimum ˆq T,ɛ, we have J ɛ (ˆq T,ɛ ) J ɛ (0) = 0. By (3.5), we obtain 1 2 ( q r dxdt) 2/r ˆq ɛ (x, 0)y 0 (x)dx ξˆq ɛ dxdt ω (0,T ) Ω ˆq ɛ (x, 0) L2 (Ω) y 0 (x) L 2 (Ω) M + ( exp( t(t t) ) ξ 2 dxdt) 1/2 ( By (2.10), (2.21) and (2.26), we obtain exp( M t(t t) ) q 2 dxdt) 1/2 (3.9) ˆv ɛ L r (ω (0,T )) exp(ck(t, a ))( y 0 L 2 (Ω) + ξ L 2 ()) (3.10)

9 Y Xu, Z. Liu 9 Since ˆv ɛ is uniformly bounded in L r (ω (0, T )), we choose an subsequence still denoted by itself, we deduce that ˆv ɛ ˆv weakly in L r (ω (0, T )) (3.11) where ˆv L r (ω (0, T )) and satisfies (3.4). Accordingly, ŷ ɛ (, T ) ŷ(, T ) where ŷ is the solution of (3.1) and associated to ˆv. Let ɛ 0 in (3.8), then we obtain ŷ(, T ) = 0 in Ω. This ends the proof. Remark 3.1. Applying same argument of [10], it is also possible to obtain the control in L (ω (0, T )). Proof of Theorem 1.1. Let us consider a trajectory y, solution of (1.5). We let u = y y, where y is the solution of (1.1). Then, we have that u t u + F (x, t; u) = v1 ω in, u = 0 on Σ, (3.12) u(x, 0) = u 0 (x) in Ω, where u 0 = y 0 y 0, and F (x, t; s) = f(x, y (x, t)+s) f(x, y (x, t)) for all (x, t), s R. Note that F (x, t; s) = g(x, t; s)s, where By (1.4), we easily know that lim s g(x, t; s) = 1 0 f s (x, y (x, t) + λs)dλ 1 1 log 3/2 (x, 1+ s ) f 0 s (x, s 0 + λs)dλ = 0, (3.13) uniformly in s 0 K, for every compact set K R. Theorem 1.1 will be proved if we show that, if for each u 0 L 2 (Ω), there exists v L r (ω (0, T )) such that u(x, T ) = 0 (3.14) We first consider the case in which u 0 L (Ω) and g C 0 (R). Then we have for each ɛ > 0, there exists C ɛ > 0 such that g(x, t; s) 2/3 C ɛ + ɛlog(1+ s ) (x, t), s R (3.15) Let us set Z = C 0 ( ) L r (0, T ; W 1,r (Ω)) and let R > 0 be a constant whose value will be determined below. We consider the truncation function T R : R R which is given as follows: T R (s) = { s if s R, Rsgn(s) otherwise, (3.16)

10 10 Y Xu, Z. Liu for each z Z, we consider the linear systems u t u + g(x, t; T R (z))u = v1 ω in, u = 0 on Σ, u(x, 0) = u 0 (x) in Ω, (3.17) We have that (3.17) is of the form (3.1), with { a = az = g(x, t; T R (z)) L () (3.18) Consequently, we can apply Theorem 3.1 to (3.17). In fact, we will apply this result in a time interval [0, T z ]. Where T z = min{t, g(x, t; T R (z)) 2/3, g(x, t; T R (z)) 1/3 } (3.19) here the subindex z denotes that it depends on z. This is a key point in this proof that will derive to approximate estimates (the idea is taken from [4]). By Theorem 3.1, we obtain the existence of the control ˆv z L r (ω (0, T z )), such that the solution û z of (3.17) in Ω (0, T z ) with v = ˆv z satisfies û z (x, T z ) = 0 in Ω. (3.20) Moreover, ˆv z Lr (ω (0,T z)) exp[ck(t, a ))] u 0 L2 (Ω) (3.21) Let us extend by zero û z and ˆv z to the whole cylinder = Ω (0, T ), which we call still û z and ˆv z. It is clear that û z is the corresponding solution of (3.17) associated to ˆv z and û z (x, T ) = 0 in Ω. (3.22) Note that the null controllability problem of (3.17) is equivalent to p t p + g(x, t; T R (z))p = η (t)u + v1 ω in, p = 0 on Σ, p(x, 0) = 0 in Ω, (3.23) verifies where p = u η(t)u, η(t) C 0 ([0, T ]) satisfies p(x, T ) = 0 in Ω. (3.24) η 1 in [0, T 3 ], η 0 in [2T 3, T ] and U solves (3.17) with v = 0. Suppose that there exists a control ṽ L 2 (ω (0, T )) solving (3.23) with supp ṽ B 0 [0, T ], B 0 ω, is nonempty open set, and let p is the corresponding state, then p = (1 θ(x)) p together v = θ(x)η (t)u + 2 θ p + θ p (3.25) solves the null controllability problem (3.23) where θ D(ω) verifies θ 1 in B 0. By classical energy estimates, we have p Y = L 2 (0, T ; H 2 (Ω) H 1 0 (Ω)) C([0, T ]; H 1 0 (Ω)), p Y exp[c(1+ g(x, t; T R (z)) 2/3 )]( u ṽ 2 )

11 Y Xu, Z. Liu 11 Applying Lemma 1.2 and let B 0 B 1 ω, then we have p X r (0, T ; Ω\ B 1 ), p X r (0,T ;Ω\ B 1) exp[c(1+ g(x, t; T R (z)) 2/3 ]( u ṽ 2 ) On the other hand, arguing as in [8], we obtain that ṽ 2 exp[c(1+ g(x, t; T R (z)) ] u 0 2 Then, since supp (1 θ(x)) Ω\ B 1, we have p X r and p X r exp[c(1+ g(x, t; T R (z)) 2/3 )] u 0 2 Applying Lemma 1.3, the space X r, r > N compactly embedded into C0 (). Then we have p Z and p Z exp[c(1+ g(x, t; T R (z)) 2/3 ] u 0 2 (3.26) By (3.25), we obtain with v L r and v L r exp[c(1+ g(x, t; T R (z))) ] u 0 2 (3.27) For any given v L r (ω (0, T z ), p v Z be the solution of (3.23) in. Let us defined A : Z A (z) L r () A (z) = {v L r (ω (0, T z )) : p satisfies (3.23), p(, T ) = 0, v verifies (3.27)} and let Λ be the set-valued mapping defined on Z as follows: Λ(z) ={p v : p v satisfies (3.23), v A (z), p v verifies (3.26)} (3.28) Let us prove that Λ fulfill the assumptions of Kakutani s Fixed Point Theorem. Firstly, we can check that Λ(z) is a nonempty set, moreover, z Z, then Λ(z) is a uniformly bounded closed convex subset of X r. By Lemma 1.3, X r compactly embedded into C β, β 2 (), β = 2 N+2 r. Then there exists compact set K Z such that Λ(z) K, z B(0, R). (3.29) Let us now prove that Λ is an upper hemicontinuous multivalued mapping, that is to say, for any bounded linear form µ Z, the real-valued function z B(0, R) sup µ, p p Λ(z) is upper semicontinuous. Correspondingly, let us know that B λ,µ = {z B(0, R) : sup µ, p λ} p Λ(z) is a closed subset of Z, for any λ R, µ Z. To this end, we consider a sequence {z n } B λ,µ such that z n z in Z. Our aim is to prove that z B λ,µ. Since all sets Λ(z n ) are compact sets, then for any n 1, there exists p n Λ(z n ), such that µ, p n = sup µ, p λ (3.30) p Λ(z)

12 12 Y Xu, Z. Liu From the definition of Λ(z n ) and A (z), let v n A (z), p n Λ(z n ) solves (3.23) with control v n, such that p n p strongly in Z, v n v weakly in L r () Since g(x, t; T R (z)) are continuous functions, we have g(x, t; T R (z n )) g(x, t; T R (z)) in C 0 (), Let n, then we obtain p solves (3.23) with control function v. Moreover, p and v satisfies (3.26)(3.21), that is, v A (z), p Λ(z). Then, let n, we have, sup µ, p µ, p λ (3.31) p Λ(z) So z B λ,µ and hence, Λ(z) is upper hemicontinuous. Finally, we have that there exists R > 0 such that Λ( B(0, R)) B(0, R) (3.32) For any z B(0, R) Z, from (3.26) and (3.15), we obtain p Z exp[c(1 + C ɛ + 2ɛlog(1 + 2R))] u 0 2 exp[c(1 + C ɛ )](1 + R) Cɛ u 0 (3.33) Choosing ɛ = 1 2C, we get p Z C(1 + R) 1/2 u 0 Thus, we get (3.32) if R is large enough. Applying the Kakutani Fixed-point Theorem, we obtain that there exists p Z such that p Λ(z). Note that the definition of p and (3.18), we conclude that there exists u Z such that u {u v : v Lr (ω (0,T )) exp[c(1+ g(x, t; T R (z)) ] u 0 2, u v Z exp[c(1+ g(x, t; T R (z)) 2/3 ] u 0 2 } That is to say, there exists a u Z such that u solves (3.12) and (3.14) holds. In the following, if u 0 L 2 (Ω), for δ > 0 small enough. Set v 0 for t (0, δ). Applying the regularizing effect of the heat equation, we conclude that the corresponding solution u of (3.17) satisfies u(, δ) L (Ω). Then, we argue as above for p(, δ) in [δ, T ] and we get control v L r (0, T ; L r (ω)) such that (3.22) holds (see [15],[16]). If g(x, t; T R (z)) are not continuous functions, we introduce a function ρ R such that ρ 0 in R, supp ρ B(0, 1) and ρ(s)ds = 1. We consider the functions ρ n, g n (n 1), with R ρ n (s) = nρ(ns) s R, g n = ρ n g

13 Y Xu, Z. Liu 13 Then it is not difficult to know that the following properties of g n hold: 1. g n C 0 (R) n Set f n (x, t; s) = g n (x, t; s)s for all (x, t), s R, then f n f uniformly in the compact sets of R. 3. F or any given M > 0, C(M) > 0 such that sup g n (x, t; s) C(M) n 1. s M 4. T he functions g n verifying (1.4) unformly in n, that is to say, ɛ > 0, there exists M(ɛ) > 0 such that g n (x, t; s) ɛlog 3/2 (1+ s ), whenever s M(ɛ) n 1. For every n, we can argue as above and find a control v n L r (ω (0, T )) such that the system (u n ) t u n + F (x, t; u n ) = v n 1 ω in, u n = 0 on Σ, (3.34) u n (x, 0) = u 0 (x) in Ω, possesses at least one solution u n Z satisfying u n (x, T ) = 0 in Ω. Thanks to the estimates obtained as above and the properties of g n, we obtain v n Lr (ω (0,T )) exp[c(1+ g(x, t; T R (z)) ] u 0 2, u n Z exp[c(1+ g(x, t; T R (z)) 2/3 ] u 0 2 for all n 1. Arguing as above, we let n in (3.34), then can find a control v L r (ω (0, T )) such that (3.12) possesses a solution u satisfying (3.14). This ends the proof of Theorem References [1] V. Barbu, Exact controllability of the superlinear heat equation. Appl. Math. Opt. 42 (2000) [2] Fernández-Cara and E. Zuazua, Controllability for blowing up semilinear parabolic equations. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000) [3] C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) [4] A. Doubova, E. Fernández-Cara, M. Gonzaáez-Burgos and E. Zuazua, On the controllability of parabolic system with a nonlinear term involving the state and the gradient. SIAM J. Control Optim, 41 (3) (2003) [5] S. Anita and V. Barbu, Null controllability of nonlinear convective heat equation. ESAIM: COCV 5 (2000) [6] Bodart, O., Gonzaáez-Burgos, M., Peéez-Garc a, R. Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient. Nonlinear Anal. 57 (5-6) (2004)

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