Appl. Math. Mech. -Engl. Ed., 2008, 29(6):787–800 DOI 10.1007/s10483-008-0610-x c Editorial Committee of Appl. Math. Mech. and Springer-Verlag 2008

Applied Mathematics and Mechanics (English Edition)

Stabilization and control of subcritical semilinear wave equation in bounded domain with Cauchy-Ventcel boundary conditions ∗ A. Kanoune, N. Mehidi (Laboratory of Applied Mathematics, Department of Mathematics, University of Bejaia, 06000 Bejaia, Algeria) (Communicated by ZHOU Zhe-wei)

Abstract We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of RN with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy-Ventcel type. Under suitable and natural assumptions on the nonlinearity, we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p < 5. The results obtained in R3 and RN by B. Dehman, G. Lebeau and E. Zuazua on the inequalities of the classical energy (which estimate the total energy of solutions in terms of the energy localized in the exterior of a ball) and on Strichartz’s estimates, allow us to give an application to the stabilization controllability of the semilinear wave equation in a bounded domain of RN with a subcritical nonlinearity on the domain and its boundary, and conditions on the boundary of Cauchy-Ventcel type. Key words stabilization, exact controllability, limit problems, semilinear, subcritical, partial differential equations, Cauchy-Ventcel Chinese Library Classification O175.29, O175.4, O231.3 2000 Mathematics Subject Classification 35A27, 35B30, 35G30, 93B05, 93C20

Introduction We study the following damped semilinear wave equation on RN (N ≥ 1):  in [0, +∞] × RN , u + f (u) + a(x)∂t u = 0 ∂t u(0, x) = u1 (x) ∈ L2 (RN ), u(0, x) = u0 (x) ∈ H 1 (RN ),

(1)

where  = (∂t2 − Δx ). The nonlinearity f is a function from R to R, of class C 3 , satisfying the following subcritical conditions: f (0) = 0,    (j)  p−j , f (s) ≤ C (1 + |s|)

(2) for

j = 0, 1, 2, 3,

(3)

where C > 0, p is a real number such that 1 ≤ p ≤ 5, and sf (s) ≥ cs2

∀s ∈ R,

for

c > 0.

∗ Received Apr. 19, 2007 / Revised Apr. 30, 2008 Corresponding author Aomar Kanoune, E-mail: [email protected]

(4)

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The damping potential a = a(x) is assumed to be in L∞ (RN ), almost everywhere nonnegative, and such that it satisfies for some R > 0 and c0 > 0, a(x) ≥ c0 > 0

for

|x| ≥ R.

The energy of u at time t is defined by     1 2 2 |∂t u(t, x)| + |∇x u(t, x)| dx + Eu (t) = F (u(t, x))dx, 2 RN RN 

where

(5)

(6)

u

f (s)ds.

F (u) =

(7)

0

We obtain the following results[1] : Theorem 1 Under the assumptions above, for every E0 > 0, there exist C > 0 and γ > 0 such that the inequality Eu (t) ≤ C e−γt Eu (0), t > 0, (8)  0 1 holds for every solution u of system (1) with the initial data u , u satisfying     1 2    1 u (x) + ∇x u0 (x)2 dx + F (u0 (x))dx ≤ E0 . (9) Eu (0) = 2 RN RN Theorem 2

Assume that the conditions above are satisfied. Also assume that f (s) = cs + f1 (s)

with f1 such that there exists δ > 0 so that f1 (s)s ≥ (2 + δ)F1 (s),

∀s ∈ R,

s where F1 (s) = 0 f1 (z)dz. Then, there exist C > 0 and γ > 0 such that the inequality (8) holds for every solution u of (1). For the demonstration of the above theorems, see Refs. [1–4]. The result of the stabilization of Theorem 1 allows us to establish an exact controllability result for the semilinear subcritical wave equation on a bounded open domain of RN (N ≥ 1). More precisely, let Ω be a bounded smooth open domain of RN (N ≥ 1) and ω a neighborhood of its boundary ∂Ω = Γ in RN . Furthermore, let f : R −→ R be a function of class C 3 satisfying (2), (3) and sf (s) ≥ 0. (10) Let g : R −→ R be a function of class C 3 satisfying g(0) = 0,    (j)  g (s) ≤ C(1 + |s|)p−j ,

(11) for j = 0, 1, 2, 3,

(12)

and sg(s) ≥ 0. Finally, let



θ1 (x) be a non-negative function in C ∞ (Ω), θ2 (x) be a non-negative function in C ∞ (∂Ω).

(13)

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We consider the following Hilbert spaces : H = L2 (Ω) × L2 (Γ),

 V = v = (v1 , v2 ) ∈ H 1 (Ω) × H 1 (Γ) such that v1|Γ = v2 with the canonical norms

12 v H = |v1 |2L2 (Ω) + |v2 |2L2 (Γ)

and

12 v V = |v1 |2H 1 (Ω) + |v2 |2H 1 (Γ) .

We prove the following theorem: Theorem 3 Under the assumptions above, for every given E0 > 0, there exists a time T > 0 such that for every data (u0 , u1 ) and (y 0 , y 1 ) in H × V, satisfying  0 1    (u , u ) ≤ E0 and (y 0 , y 1 )H×V ≤ E0 , H×V there exists v = (v1 , v2 ) ∈ L1 ([0, T ] , V ) with support in [0, T ] × ω, and there exists a unique u = (u1 , u2 ) in C 0 ([0, +∞] , H) ∩ C 1 ([0, +∞] , V ), solution of the system: ⎧ in [0, +∞] × Ω, ⎪ ⎨u1 + θ1 (x)f (u1 ) = v1 (t, x) (14) ∂t2 u2 + ∂ν u1 − T u2 + θ2 (x)g(u2 ) = v2 (t, x) on [0, +∞] × Γ, ⎪ ⎩ 0 1 u(0) = u , ∂t u(0) = u in Ω satisfying

u(T, ·) = y 0

and

∂t u(T, ·) = y 1 .

As an immediate consequence, the following holds: Corollary 4 We consider the system (14) with θ1 ≡ 1 and θ2 ≡ 1, i.e., without cutting off the nonlinearity. Then, under the assumptions above, the same result as in Theorem 3 above holds with controls v = (v1 , v2 ) ∈ L1 (0, T ; Hloc )∩L∞ (0, T ; V 6/5 ), except for the uniqueness of the solution, where Hloc = L2loc (Ω) × L2loc (Γ) and V 6/5 = L6/5 (Ω) × L6/5 (Γ). This result improves that in Ref. [1] taking into account here other conditions on the boundary and other spaces; which brings us back to another consideration of energy. It also improves that in Ref. [5] which are valid under the more restrictive assumptions on the nonlinearity. We refer to Ref. [6] for a proof of the exact controllability in uniform time for the (1 − d) wave equation with a nonlinear term that grows at infinity in a slightly superlinear way. The proof of this exact controllability result, which is based on the stabilization result of Theorem 1, is roughly as follows. First, we show by means of a perturbation argument that, due to the exact controllability property of the linear wave equation in the geometric setting of Theorem 2, small data are controllable for the nonlinear equation too, i.e., given sufficiently small initial and final data the solution can be derived from the initial state to the final one. Then we adapt the proof of Theorem 1[1] to the case of the bounded open set. Then, for proof of the exact controllability, we use the fixed point theorem. The rest of this article is organized as follows. In next section, the subcritical wave equation in a bounded domain is given. Global existence and uniqueness are shown. The stabilization and exact controllability in a non-uniform time are given. In the sequel we assume that 4 ≤ p ≤ 5. The other cases 1 ≤ p < 4 can in fact be treated in a simpler way following the same arguments.

790

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Subcritical wave equation in a bounded domain

In this section, we consider the subcritical nonlinear wave equation in a bounded domain of RN (N ≥ 1) with a nonlinearity on the domain and its boundary. First, in Section 1.1, we prove the global existence and uniqueness result. In this respect, it is important to note that the mixed problem is in general well posed for most semilinearities[6] . However, we can deal with subcritical nonlinearities (p < 5) on the domain and on the boundary. In Section 1.2, we prove the stabilization result. In Section 1.3, we prove the controllability results of Theorem 3 and Corollary 4 in a non-uniform time guaranteeing that every initial state can be driven to any final state if time is large enough, depending on the size of the data to be controlled. 1.1 Global existence and uniqueness Let Ω be a smooth, open bounded set of RN (N ≥ 1). Consider a nonlinear function f : R −→ R verifying conditions (2), (3) and (10) and a nonlinear function g : R −→ R verifying (11), (12) and (13). Let θ1 (x) ∈ C0∞ (Ω) and θ2 (x) ∈ C0∞ (Γ) be two non-negative functions. We have the following result: Theorem 5 For every pair of functions (v1 , v2 ) ∈ L1 ([0, +∞] , L2 (Ω))×L1 ([0, +∞] , L2 (Γ)) and every pair of initial data (u0 , u1 ) ∈ H × V , the system ⎧ in [0, +∞] × Ω, ⎪ ⎨u1 + θ1 (x)f (u1 ) = v1 ∂t2 u2 + ∂ν u1 − T u2 + θ2 (x)g(u2 ) = v2 on [0, +∞] × Γ, (15) ⎪ ⎩ 0 1 ∂t u(0) = u in Ω u(0) = u , has a unique solution u = (u1 , u2 ) in the space C 0 ([0, +∞] , H) ∩ C 1 ([0, +∞] , V ). Moreover this solution satisfies the following Strichartz estimates. For every finite T > 0, r ≥ 2, q given by 1q = 12 − 1r and χ ∈ C0∞ (Ω), there exists a constant C > 0 such that   χ(x)u Lq ([0,T ];H) ≤ C (v1 , v2 ) L1 ([0,T ];H)×L1 ([0,T ];V ) , Eu (0) (16) for every v = (v1 , v2 ) and every initial data as before. Here and in the sequel, Eu stands for the energy of solution u of this system, i.e.,       1 1 2 2 2 2 |∇u| + |ut | dx + |∇T u| + |ut | dγ Eu (t) = 2 Ω 2 Γ   + θ1 (x)F (u)dx + θ2 (x)G(u)dγ, Ω



where

(17)

Γ



u

F (u) =

f (s)ds 0

and G(u) =

u

g(s)ds. 0

Proof We proceed in three steps. Step 1 Existence We decouple system (15), by cutting off the initial data (u0 , u1 ) and the right-hand side term v. Let ν1 be a neighborhood of the compact set supp (θ1 ) such that ν1 ⊂ Ω; and ν2 be a neighborhood of the compact supp (θ2 ) such that ν2 ⊂ Γ.

Stabilization and control for subcritical semilinear wave equation

791

Let ψ ∈ C0∞ (Ω) such that ψ = 1 on ν1 . Define ω1 = ψv1 and ω1 = (1 − ψ)v1 and ξ ∈ C0∞ (Γ) such that ξ = 1 on ν2 . Define ω2 = ξv2 and ω2 = (1 − ξ)v2 in such a manner that ω1 = 0 on supp (θ1 ) and ω2 = 0 on supp (θ2 ). Consider the two following systems: ⎧ in [0, +∞] × Ω, ⎨v + θ1 (x)f (v) = ω1 ∂t2 v + ∂ν v − T v + θ2 (x)g(v) = ω2 on [0, +∞] × Γ, (18) ⎩ (v(0), ∂t v(0)) = ψ(x)(u0 , u1 ) in Ω; ⎧ in [0, +∞] × Ω, ⎨w = ω1 ∂t2 w + ∂ν w − T w = ω2 on [0, +∞] × Γ, (19) ⎩ 0 1 in Ω. (w(0), ∂t w(0)) = (1 − ψ)(u , u ) Let T0 = min(d1 , d2 , d3 , d4 ), where d1 = distance(supp (ψ), Γ) and d2 = distance(supp (1 − ψ), supp (θ1 )) d3 = distance(supp (ξ), Γ) and d4 = distance(supp (1 − ψ), Γ), supp (θ2 )). Then we solve the two systems above (18) and (19) on the time interval [0, T0 ] . Because of the finite speed propagation of waves (=1 in the present model), it is clear that: (i) For this time interval, the solution of (18) coincides, in the support of ψ, with that of the Cauchy problem in the free space RN . Indeed, the solution of the latter vanishes on the boundary because of the fact that the initial data and the right hand side have been confined to Supp (ψ) and T0 ≤ d1 . (ii) For 0 ≤ t ≤ T0 , supp (w) ⊂ Ω  supp (θ1 ),

supp (w|Γ ) ⊂ Γ  supp (θ2 ),

i.e., w = 0 on supp (θ1 ) and w|Γ = 0 on supp (θ2 ). This clearly gives θ1 (x)f (v) = θ1 (x)f (v + w),

θ2 (x)g(v) = θ2 (x)g(v + w).

The function u = v + w, constructed as above, belongs to C 0 ([0, T ] ; H) ∩ C 1 ([0, T ] ; V ) and solves (15) for 0 ≤ t ≤ T0 . Indeed, the fact that it solves the equation above, is an obvious consequence of the previous discussion. The continuity in time of the solution is the consequence of the fact that both components v and w are indeed continuous. The continuity in time of v is a consequence of the fact that it coincides, on the time interval 0 ≤ t ≤ T0 , with the solution of the Cauchy problem in the whole space, which is known to be continuous in time with values in energy space. Step 2 Energy and Stricharts estimates Adding the classical energy estimate for each of the problems (18) and (19) above, one obtains that   Eu (t) ≤ C (v1 , v2 ) L1 (H) + Eu (0) , for 0 ≤ t ≤ T0 . Taking into account that the time T0 depends only on the geometry of the problem (i.e., ω and the supports of ψ, ξ, θ1 and θ2 ), it is clear that one may iterate this process to obtain a global solution with respect to time. On the other hand, let χ(x) be a cut-off function and assume (to simplify the notation) that χ = 1 in the support of θ1 . The function u  = χ(x)u solves the free system:   u + θ1 (x)f ( u) = χv1 + [, χ] u ∈ L1loc ([0, +∞] , L2 (RN )), (20) (0)) ∈ H 1 (RN ) × L2 (RN ), ( u(0), ∂t u

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which, combined with the energy estimate for the system[1]  u = F ∈ L1 ([0, +∞] , L2 (RN )), (u(0), ∂t u(0)) ∈ H 1 (RN ) × L2 (RN ), provides the Strichartz estimates (16). Step 3 Uniqueness We now prove the uniqueness of the solution. For that, we need the following lemma. Lemma 6 Let u and v be two solutions of (15). Then, for every T > 0 and 1 ≤ α ≤ there exist C1 , C2 > 0, satisfying θ1 (x) (f (u) − f (v)) Lα ([0,T ];L2 (Ω)) ≤ C1 u − v L∞ (0,T ;H 1 (Ω)) ,         θ2 (x) g u|Γ − g v|Γ  α ≤ C2 u|Γ − v|Γ L∞ (0,T ;H 1 (Γ)) . L ([0,T ];L2 (Γ))

2 p−3 ,

(21) (22)

Assuming for the moment that this lemma holds, we show the uniqueness of the solution. Let u and v be two solutions of (15). The function u − v solves the system: ⎧ in [0, +∞] × Ω, (u − v) + θ1 (x)(f (u) − f (v)) = 0 ⎪ ⎪ ⎨ ∂t2 (u − v) + ∂ν (u − v) − T (u − v) + θ2 (x)(g(u) − g(v)) = 0 on [0, +∞] × Γ, (23) ⎪ ⎪ ⎩ in Ω. (u − v)(0) = ∂t (u − v)(0) = 0 The energy inequality guarantees that u − v L∞ (0,T ;H 1 (Ω)) ≤ C1 θ1 (x) (f (u) − f (v)) L1 ([0,T ];L2 (Ω)) 1

≤ C1 T β θ1 (x) (f (u) − f (v)) Lα ([0,T ];L2 (Ω)) with

1 α

+

(24)

1 β

= 1, and         u|Γ − v|Γ  ∞ ≤ C2 θ2 (x) g u|Γ − g v|Γ L1 ([0,T ];L2 (Γ)) L (0,T ;H 1 (Γ))      1  ≤C2 T β θ2 (x) g u|Γ − g v|Γ Lα ([0,T ];L2 (Γ)) .

(25)

And using (21) and (22), we obtain 1

and

u − v L∞ (0,T ;H 1 (Ω)) ≤ C1 T β u − v L∞ (0,T ;H 1 (Ω))

(26)

   1  u|Γ − v|Γ  ∞  ∞ β u ≤ C T − v , 2 |Γ |Γ 1 L (0,T ;H (Γ)) L (0,T ;H 1 (Γ))

(27)

which yields to the result: u = v on Ω and u|Γ = v|Γ . 1 1 Then u = v by taking max(C1 T β , C2 T β ) < 1. Now we come back to the proof of Lemma 6: By hypotheses (3) and (12), we can write     f (u) − f (v) = (u − v)G1 (u, v) and g u|Γ − g v|Γ = (u|Γ − v|Γ )G2 (u|Γ , v|Γ ), where G1 and G2 verify

  p−1 p−1 , + |v| G1 (u, v) ≤ 1 + |u|   p−1  p−1  G2 (u|Γ , v|Γ ) ≤ 1 + u|Γ  . + v|Γ 

(28) (29)

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793

So, by H¨ older’s inequality it follows that  T α θ1 (x) (f (u) − f (v)) (t) L2 (Ω) dt 0

 ≤

T

0

α



(u − v)(t) L6 (Ω)

Ω

 α3 3 |θ1 G1 | dx dt

 α  ≤ (u − v)(t) L∞ (0,T ;H 1 (Ω))



T

Ω

0

 α3 |θ1 G1 | dx dt. 3

We also obtain   ≤

T

       θ2 (x) g u|Γ − g v|Γ (t)α 2 dt L (Γ)

T

  (u|Γ − v|Γ )(t)α 6 L (Γ)

0

0



Γ

3

|θ2 G2 | dγ

 α   ≤ (u|Γ − v|Γ )(t)L∞ (0,T ;H 1 (Γ))

T

0

 α3

 Γ

dt 3

|θ2 G2 | dγ

 α3 dt.

To complete the proof of the lemma, it is necessary to get a suitable upper bound last  for the  α 3 integrals. Obviously, the last integrals can be bounded above in terms of the L 0, T ; L (Ω)  norm and of the Lα 0, T ; L3(Γ) -norm respectively for θ1 G1 (u, v) and θ2 G2 (u|Γ , v|Γ ), which   may be estimated in terms of the Lα(p−1) 0, T ; L3(p−1) (supp(θ1 ) -norm and of the Lα(p−1)   0, T ; L3(p−1)(supp(θ2 ) -norm respectively for u, v and u|Γ , v|Γ . All these norms can be easily estimated in terms of the Strichartz-norms in (16). Indeed, 2r it is acceptable to set r = p − 1 with r ≥ 2. Then the exponent q = r−2 , corresponding to this choice of r in the Strichartz-norm (16), coincides with q = 2(p−1) p−3 which is greater than 2 α (p − 1) provided p−3 ≥ α. This is precisely the range of exponents in the statement of the lemma. 1.2 Stabilization The stabilization result in the case of bounded domains is as follows:

Proposition 7 Assume that the hypotheses of the previous theorem are satisfied. Let the set ω = {x ∈ Ω / a(x) ≥ c0 > 0} be a neighborhood of the boundary of Ω, i.e., the intersection with Ω of a neighborhood of Γ = ∂Ω in RN . Then the local stabilization property holds for the system: ⎧ in [0, +∞] × Ω, ⎪ ⎨u + a(x)∂t u + θ1 (x)f (u) = 0 ⎪ ⎩

∂t2 u + ∂ν u − T u + θ2 (x)g(u) = 0

on [0, +∞] × Γ,

(30)

(u(0), ∂t u(0)) ∈ V × H.

More precisely, for every E0 > 0, there exist C > 0 and γ > 0 such that inequality (8) holds for the energy Eu in (17) provided Eu (0) ≤ E0 . Remarks (a) Note that, in this proposition, the assumptions (4) and (13) on the nonlinearity may be respectively relaxed to sf (s) ≥ 0

and sg(s) ≥ 0.

(31,32)

(b) It would be interesting to investigate if a global stabilization result as that in Theorem 2 would be true in this case. For the proof of Proposition 7, we follow the same approach as that of Theorem 1[1] . We use contradiction reasoning.

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Let un be a sequence of solutions of ⎧ un + a(x)∂t un + θ1 (x)f (un ) = 0 in [0, +∞] × Ω, ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨ ∂t un + ∂ν un − T un + θ2 (x)g(un ) = 0 on [0, +∞] × Γ,  T Eun (0) 2 ⎪ ), a(x) |∂t un | dtdx −→ 0 (≤ ⎪ ⎪ n ⎪ Ω 0 ⎪ ⎪ ⎩ 1/2 αn = (Eun (0)) −→ α. Then, we obtain for vn =

(33)

un αn ,

⎧ 1 ⎪ vn + a(x)∂t vn + θ1 (x)f (αn vn ) = 0 in [0, +∞] × Ω, ⎪ ⎪ ⎪ α n ⎪ ⎪ ⎨ 1 θ2 (x)g(αn vn ) = 0 on [0, +∞] × Γ, ∂t2 vn + ∂ν vn − T vn + α ⎪ n ⎪ ⎪   T ⎪ ⎪ 1 2 ⎪ ⎩ a(x) |∂t vn | dtdx −→ 0 (≤ ). n Ω 0

(34)

We examine, again, separately the cases α > 0 and α = 0. We denote by v the weak limit of the sequence {vn } . First case α > 0 Letting n −→ ∞ in (33), We find that the limit u of the sequence un satisfies ⎧ ⎪ ⎪u + θ1 (x)f (u) = 0 in [0, +∞] × Ω, ⎪ ⎪ ⎨∂ν u − T u + θ2 (x)g(u) = 0 on [0, +∞] × Γ, ⎪ ∂t u = 0 on [0, T ] × ω, ⎪ ⎪ ⎪ ⎩ u ∈ L∞ ([0, T ] , V ) , ∂t u ∈ L∞ ([0, T ] , H) . Let χ(x) ∈ C0∞ (Ω) such that χ = 1 on supp(θ1 ) and supp (∇χ) ⊂ ω. The function u  = χu verifies    ⎧ u + θ1 (x)f ( u) = ∇χ · ∇u + (χ) · u ∈ L1 [0, T ] , L2 RN , ⎪ ⎪ ⎪ ⎪ ⎨∂ν u  − T u  + θ2 (x)g( u) = 0 on [0, +∞] × Γ,   ⎪ ∂t u =0 in [0, T ] × RN  Ω , ⎪ ⎪ ⎪       ⎩ u  ∈ L∞ [0, T ] , H 1 RN , ∂t u  ∈ L∞ [0, T ] , L2 RN . Then u  has bounded Strichartz norms. By the regularity theorem[1] , we find that u  as well as f  ( u) are bounded. Then w = ∂t u  satisfies  u)w = 0 in [0, T ] × RN , w + θ1 (x)f  ( w=0 in [0, T ] × (|x| > R) , where R is large enough. By unique continuation we deduce that w ≡ 0. Thus, u = u(x) ∈ V for t ∈ [0, T ] and it satisfies −u + θ1 f (u) = 0 in Ω. Multiplying this equation by u and integrating over Ω, we obtain       2 2 |∇u| + θ1 uf (u) dx + |∇T u| + θ2 ug(u) dγ = 0, Ω

Γ

which implies u ≡ 0, because of the good-sign assumptions (4) and (13).

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Consequently, un −→ 0 in H 1 ([0, T ] × Ω) . Here, we use again an argument based on microlocal defect measures. Let μ be an m.d.m associated to un in H 1 ([0, T ] × Ω) . It is easy to see from (34) that μ = 0 in [0, T ] × ω. To complete the argument, we use the propagation property of the m.d.m in Ω (away from the boundary). This gives μ = 0 everywhere; hence un −→ 0 in H 1 ([0, T ] × Ω) , which contradicts the fact that α > 0. Then this case is excluded. Second case α = 0 Letting n −→ ∞, we find that the limit v of the sequence vn satisfies ⎧ v + θ1 (x)f  (0)v = 0 in [0, T ] × Ω, ⎪ ⎪ ⎪ 2 ⎨ ∂t v + ∂ν v − T v + θ2 (x)g  (0)v = 0 on [0, T ] × Γ, ⎪ on [0, T ] × ω, ⎪∂t v = 0 ⎪ ⎩ ∞ ∂t v ∈ L∞ ([0, T ] , H) . v ∈ L ([0, T ] , V ) , The existing results the on unique continuation applied to this system after differentiation with respect to time allow us to show that v = 0. The rest of the proof is very close to the corresponding one of Theorem 1. Indeed, a suitable set of truncating functions replaces the problem under consideration with a global one in the whole space and the same arguments apply. The proof of the stabilization result on the domain Ω is now complete. 1.3 Exact controllability in non-uniform time In this section, we give the proofs of Theorem 3 and Corollary 4. We first prove Theorem 3 and then indicate how Corollary 4 may be obtained. Proof of Theorem 3 We can prove the exact controllability to zero for small data. For this, we will use a nonlinear variant of Lions’ Hilbert uniqueness method (H.U.M)[7] following closely the proof developed in Ref. [5]. We first consider the linearized system: ⎧  in [0, +∞] × Ω, ⎪ ⎨u + θ1 (x)f (0)u = v1 (t, x)1ω 2  (35) ∂t u + ∂ν u − T u + θ2 (x)g (0)u = v2 (t, x)1ω on [0, +∞] × Γ, ⎪ ⎩ 0 1 ∂t u(0) = u in Ω. u(0) = u , Here and in the sequel, ω denotes the neighborhood of the boundary where the control is supported and 1ω denotes its characteristic function. 0 1 This system is exactly  controllable  T >22R. Indeed, for any (u , u ) ∈ V × H, there  in time 2 2 2 exists v = (v1 , v2 ) in L 0, T ; L (ω) × L 0, T ; L (Γ) such that the solution of (35) satisfies u(T ) = ut (T ) = 0. Moreover, the control v = (v1 , v2 ) of minimal norm is unique and depends continuously on the initial data (u0 , u1 ) in the corresponding norms. More precisely, the control v = (v1 , v2 ) is given by v1 = 1ω Φ, v2 = Φ|Γ , where Φ ∈ C ([0, T ] ; H) ∩ C 1 ([0, T ] ; V  ) is the unique solution of ⎧  in [0, +∞] × Ω, ⎪ ⎨Φ + θ1 (x)f (0)Φ = 0 2 ∂t Φ + ∂ν Φ − T Φ + θ2 (x)g  (0)Φ = 0 on [0, +∞] × Γ, ⎪ ⎩ ∂t Φ(0) = Φ1 in Ω. Φ(0) = Φ0 ,

(36)

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  ∀ Φ0 , Φ 1 ∈ H × V  . We then solve ⎧  in [0, +∞] × Ω, ⎪ ⎨Ψ + θ1 (x)f (0)Ψ = v1 2 ∂t Ψ + ∂ν Ψ − T Ψ + θ2 (x)g  (0)Ψ = v2 on [0, +∞] × Γ, ⎪ ⎩ in Ω. Ψ(T ) = ∂t Ψ(T ) = 0

(37)

Clearly Ψ ∈ C 0 ([0, T ] ; V ) ∩ C 1 ([0, T ] ; H) .   The operator Λ : H ×V  −→ V ×H defined by Λ Φ0 , Φ1 = (−∂t Ψ(0), Ψ(0)) is an isomorphism. Indeed,       Λ Φ0 , Φ1 , Φ0 , Φ1  = (−∂t Ψ(0), Ψ(0)) , Φ0 , Φ1  = (−∂t Ψ(0), Ψ(0)) , (Φ(0), ∂t Φ(0)). To obtain this, it in necessary to multiply the equation Ψ + θ1 (x)f  (0)Ψ = 1ω Φ by Φ and integrate on [0, T ] × Ω to get the equality 

T

0

 Ω

 Ψ · Φdxdt +

T

0

 Ω



θ1 (x)f  (0) · Ψ · Φdxdt =



T

ω

0

|Φ|2 dxdt.

Then by integrations by parts and using the equations verified by Ψ and Φ, we obtain   (−∂t Ψ(0), Ψ(0)) , Φ0 , Φ1  =



T 0

 ω



2

|Φ| dxdt +

0

T

 Γ

2

|Φ| dγdt.

On the other hand, taking into account that T > 2R, we can prove the existence of a constant C > 0 such that  T    T  0 1  2 2  Φ ,Φ  ≤C |Φ| dxdt + |Φ| dγdt H×V  0

ω

0

Γ

for every solution Φ of (36). This can be done using methods[7] .   multiplier 0 1 0 1 Thus, given any (u , u ) ∈ V × H, there exists Φ , Φ in H × V  such that   Λ Φ0 , Φ1 = (−u1 , u0 ), which is precisely equivalent to saying that the solution u of (35) with control given by Φ, coincides with Ψ and therefore, in particular, satisfies the condition u(T ) = ut (T ) = 0. Now, after solving (36) for Φ, we solve ⎧ in [0, +∞] × Ω, ⎪ ⎨u + θ1 (x)f (u) = Φ1ω ∂t2 u + ∂ν u − T u + θ2 (x)g(u) = Φ|Γ on [0, +∞] × Γ, (38) ⎪ ⎩ in Ω. u(T ) = ∂t u(T ) = 0   The problem is then to show that the operator A defined by A Φ0 , Φ1 = (−∂t u(0), u(0)) on H × V  , with values in its dual V × H, is onto a small neighborhood of the origin. Note that the function v = u − Ψ, where Ψ is the solution of the corresponding linear problem (37), belongs to C 0 ([0, T ] ; V ) ∩ C 1 ([0, T ] ; H) (in fact, both u and Ψ do belong to this space).

Stabilization and control for subcritical semilinear wave equation

Moreover v satisfies ⎧  in [0, +∞] × Ω, ⎪ ⎨v + θ1 (x)f (0)v = −θ1 (x)R1 (u) 2  ∂t v + ∂ν v − T v + θ2 (x)g (0)v = −θ2 (x)R2 (u) on [0, +∞] × Γ, ⎪ ⎩ in Ω, v(T ) = ∂t v(T ) = 0 where

R1 (u) = f (u) − f  (0)u,

797

(39)

R2 (u) = g(u) − g  (0)u.

We have u = Ψ + v and therefore       A Φ 0 , Φ1 = Λ Φ0 , Φ1 + K Φ 0 , Φ1 ,   where K Φ0 , Φ1 = (−∂t v(0), v(0)) . Take into account that Λ : H × V  −→ V × H is an isomorphism, then solve the equation   A Φ0 , Φ1 = (−u1 , u0 ),

(40)

which is equivalent to finding the control for the data (u0 , u1 ). It is also equivalent to solving       (41) B Φ0 , Φ1 = −Λ−1 K Φ0 , Φ1 + Λ−1 (−u1 , u0 ) = Φ0 , Φ1 . Therefore the problem is to find a fixed point for the operator B, defined from H × V  into itself. For this, it is necessary to prove that the operator B is compact which in turn can be done by verifying that K, as an operator from H × V  into its dual, is compact. To show this fact, we observe that, 5, the Strichartz norms of u are bounded   by Theorem on the support of θ1 by the norm of Φ0 , Φ1 in H × V  . Applying the regularity theorem[1] , we obtain that θ1 (x)R(u) ∈ L1 ([0, T ] , Hε (Ω)) for some ε > 0 small enough, where Hε (Ω) = H ε (Ω) × H ε (Γ). This leads to   v ∈ C 0 [0, T ] ; V 1+ε (Ω) ∩ C 1 ([0, T ] ; Hε (Ω)) , 

ε+1 ε+1 with a bound on v in that space , v ) ∈ H (Ω) × H (Γ) /v = v where V 1+ε (Ω) 1 2 2 1|Γ  =0 (v  1   in terms of Φ , Φ H×V  . This completes the proof of the compactness property. Therefore, in order to obtain the fixed point, we may apply the Schauder fixed point theorem. To do this, it is necessary to find a constant ρ > 0, such that   0 1      B Φ , Φ  ≤ ρ, ∀ Φ0 , Φ1 ∈ H × V  :  Φ0 , Φ1 H×V  ≤ ρ. (42) H×V  We are going to show that this ρ > 0 exists provided (u0 , u1 ) is sufficiently small in V × H, which leads us to show that there exists ρ > 0 such that   0 1   0 1   0 1   K Φ , Φ  ∈ H × V ≤ ρ, ∀ Φ , Φ : Φ , Φ H×V  ≤ ρ. (43) V ×H For this, we write the energy inequality for system (39). Define     1 1 2 2 |∂t v| + |∇v| dx + θ1 (x)f  (0)v 2 dx Ev (t) = 2 Ω 2 Ω     1 1 2 2 |∂t v| + |∇T v| dγ + + θ2 (x)g  (0)v 2 dγ. 2 Γ 2 Γ

798

A. Kanoune and N. Mehidi

Differentiating Ev (t), and using integrations by parts together with the equations verified by v, we obtain   dEv (t) =− θ1 (x) · R1 (u)(t) · ∂t v(t)dx − θ2 (x) · R2 (u)(t) · ∂t v(t)dγ. (44) dt Ω Γ But θ1 (x) · R1 (u)(t) · ∂t v(t) L1 (Ω) ≤ ∂t v(t) L2 (Ω) · θ1 (x) · R1 (u)(t) L2 (Ω) , θ2 (x) · R2 (u)(t) · ∂t v(t) L1 (Γ) ≤ ∂t v(t) L2 (Γ) · θ2 (x) · R2 (u)(t) L2 (Γ) . Hence 1/2

· θ1 (x) · R1 (u)(t) L2 (Ω) ,

1/2

· θ2 (x) · R2 (u)(t) L2 (Γ) .

θ1 (x) · R1 (u)(t) · ∂t v(t) L1 (Ω) ≤ C1 · (Ev (t)) θ2 (x) · R2 (u)(t) · ∂t v(t) L1 (Γ) ≤ C2 · (Ev (t)) So

θ1 (x) · R1 (u)(t) · ∂t v(t) L1 (Ω) + θ2 (x) · R2 (u)(t) · ∂t v(t) L1 (Γ)   1/2 ≤ C · (Ev (t)) · θ1 (x) · R1 (u)(t) L2 (Ω) + θ2 (x) · R2 (u)(t) L2 (Γ) . Then   dEv (t) 1/2 ≥ −C · (Ev (t)) · θ1 (x) · R1 (u)(t) L2 (Ω) + θ2 (x) · R2 (u)(t) L2 (Γ) . dt Integrating on the time interval [t, T ] and taking into account v(T ) = ∂t v(T ) = 0, we obtain  T dEv (s) ds ≥ −C · AT (t), ds t where

 AT (t) =

T

t

   (Ev (s))1/2 · θ1 (x) · R1 (u)(s) L2 (Ω) + θ2 (x) · R2 (u)(s) L2 (Γ) ds,

which is equivalent to Ev (T ) − Ev (t) ≥ −C · AT (t). But Ev (T ) = 0, because v(T ) = ∂t v(T ) = 0. Hence Ev (t) ≤ C · AT (t). Moreover



AT (t) ≤ C · [ θ1 (x) · R1 (u) L1 (0,T ;L2 (Ω)) + θ2 (x) · R2 (u) L1 (0,T ;L2 (Γ)) ] ·

T

t

1/2

(Ev (s))

ds.

Then Ev (t) ≤ C · (T − t) [Ev (t)]

1/2

[ θ1 (x) · R1 (u) L1 (0,T ;L2 (Ω)) + θ2 (x) · R2 (u) L1 (0,T ;L2 (Γ)) ].

But (v(t), ∂t v(t)) V ×H 1/2       2 2 2 2 2 2 = |v(t)| + |v(t)| + |∇v(t)| + |∇T v(t)| + |∂t v(t)| + |∂t v(t)| Ω

≤ C · [Ev (t)]

Γ

1/2

.

Ω

Γ

Ω

Γ

Stabilization and control for subcritical semilinear wave equation

799

  (v(t), ∂t v(t)) V ×H ≤ C · θ1 (x) · R1 (u) L1 (0,T ;L2 (Ω)) + θ2 (x) · R2 (u) L1 (0,T ;L2 (Γ)) .

(45)

Hence

Since we have the injection H 1 ⊂−→ L6 and that Ev (t) is decreasing, then     2 p , θ1 (x) · R1 (u) L2 (Ω) ≤ C θ1 |u| · θ1 |u|  2 L (Ω)

and

   2 p θ2 (x) · R2 (u) L2 (Γ) ≤ C θ2 |u| · θ2 |u| 



L2 (Γ)

=⇒ θ1 (x) · R1 (u) L2 (Ω) + θ2 (x) · R2 (u) L2 (Γ)     14  14  3/2 3/2 2 2(2p−3) 2 2(2p−3) 2 2 . ≤ C u L6 (Ω) + u L6 (Ω) θ1 |u| dx + u L6 (Γ) + u L6 (Γ) θ2 |u| dγ Ω

Γ

Hence (v(t), ∂t v(t)) V ×H      3/2 3/2 1/2 1/2 ≤ C · sup u(t) L6 (Ω) + u(t) L6 (Γ) · sup u(t) L6 (Ω) + u(t) L6 (Γ) 0≤t≤T



T



+ Ω

0

 14  θ12 |u|2(2p−3) dx +

T



0≤t≤T

Γ

0

θ22 |u|2(2p−3) dγ

 14  .

(46)

Furthermore, by comparison with the Strichartz norms of u, we have  0

T

 Ω

θ12

2(2p−3)

|u|

 14  dx + 0

T

 Γ

θ22

2(2p−3)

|u|

 14 dγ

  ≤ C T, Φ L1 0,T ;H .

On the other hand, applying the same energy inequality to u, we have   1/2 ≤ C · Φ L1 (0,T ;H) ≤ C ·  Φ0 , Φ1 H×V  , ∀t ∈ [0, T ] . (Eu (t)) Using the embedding from V into L6 (Ω) × L6 (Γ), we obtain, by combining (46) and (47),  3/2 sup (v(t), ∂t v(t)) V ×H ≤ C ·  Φ0 , Φ1 H×V  .

0≤t≤T

This nonlinear estimate immediately yields (43). The proof of Theorem 3 is now complete. Proof of Corollary 4 We consider arbitrary initial and final data: (u0 , u1 ), (y 0 , y 1 ) ∈ V × H. According to Theorem 3, there exist T > 0 and a control v = (v1 , v2 ) ∈ L2 ([0, T ] × Ω) × L2 ([0, T ] × Γ) with support in ([0, T ] × ω) × ([0, T ] × Γ) , such that the unique solution of (14) satisfies u(T ) = y 0 ,

ut (T ) = y 1 .

(47)

800

A. Kanoune and N. Mehidi

We now introduce the control v = (v1 , v2 ), where v1 = v1 + (1 − θ1 )f (u),

v2 = v2 + (1 − θ2 )g(u).

This control has also its support in ([0, T ] × ω) × ([0, T ] × Γ) . This is guaranteed by taking θ1 ≡ 1 in Ω  ω and θ2 ≡ 1 in Γ  (Γ ∩ ω) . Furthermore, the solution u of (14) satisfies ⎧ in [0, +∞] × Ω, ⎪ ⎨u + f (u) = v1 2 ∂t u + ∂ν u − T u + g(u) = v2 on [0, +∞] × Γ, (48) ⎪ ⎩ 0 1 in Ω. u(0) = u , ∂t u(0) = u Note that we are not in the exact conditions to guarantee that the finite energy solution u is unique since we do not know whether Strichartz inequalities hold in the domain Ω up to the boundary. But the existence is guaranteed. In fact, u solution of (14) solves (48) too. In order to conclude the proof of Corollary 4, it is fitting to analyze the regularity of v. We know that v = (v1 , v2 ) ∈ L2 ([0, T ] × Ω) × L2 ([0, T ] × Γ) . Thus, it is sufficient to analyze the regularity of (1 − θ1 )f (u) and (1 − θ2 )g(u).   10 The function u has finite Strichartz norms in the interior of Ω. In particular, u ∈ L5 0, T ; Hloc 10 10 10 = L10 where Hloc loc (Ω) × Lloc (Γ) (take q = 5 and r = 3 in the Strichartz norms). Consequently,   2 ((1 − θ1 )f (u), (1 − θ2 )g(u)) ∈ L1 0, T ; Hloc , 2 = L2loc (Ω) × L2loc (Γ). where Hloc Furthermore, taking into account the fact that u has finite energy, it is easy to see that   ((1 − θ1 )f (u), (1 − θ2 )g(u)) ∈ L∞ 0, T ; H 6/5 ,

where H 6/5 = L6/5 (Ω) × L6/5 (Γ). The proof of Corollary 4 is now complete.

References [1] Dehman B, Lebeau G, Zuazua E. Stabilization and control for the subcritical semilinear wave equation[J]. Annales Scientifiques de l’Ecole Normale Sup´erieure, S´erie 4, 2003, 36(4):525–551. [2] Bardos C, Lebeau G, Rauch J. Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[J]. SIAM Journal on Control and Optimization, 1992, 30(5):1024–1065. [3] Gerard P. Oscillation and concentration effects in semilinear dispersive wave equation[J]. J Funct Anal, 1996, 41(1):60–98. [4] Rauch J, Taylor M. Exponential decay of solutions to hyperbolic equations in bounded domains[J]. Indiana University Mathematical Journal, 1974, 24(1):79–86. [5] Zuazua E. Exact controllability for the semilinear wave equation[J]. J Math Pures Appl, 1990, 69(1):33–55. [6] Lions J L. Quelques m´ethodes de r´esolution des probl`emes aux limites non-lin´eaires[M]. Paris: Dunod, 1969 (in French). [7] Lions J L. Contrˆ olabilit´e exacte, stabilisation et perturbations de syst`emes distribu´es[M]. Tome 1, in: RMA, Vol 8, Paris: Masson, 1988 (in French).