Electronic Journal of Qualitative Theory of Differential Equations 2003, No. 16, 1-32; http://www.math.u-szeged.hu/ejqtde/

October 22, 2003

REMARKS ON NULL CONTROLLABILITY FOR SEMILINEAR HEAT EQUATION IN MOVING DOMAINS

by

Silvano B. de Menezes – Juan Limaco – L.A. Medeiros

Abstract

We investigate in this article the null controllability for the semilinear heat operator u′ − Δu + f(u) in a domain which boundary is moving with the time t.

Key Words: heat operator, semilinear, moving boundary, null controllability.

Mathematical Subject Classification: 35B99, 35K05, 93B05.

1. Introduction and Main Result

In this article we consider semilinear parabolic problems in domains which are moving with the time t. Given a time T > 0, the state equation is posed in an open set Q̂ of ℝ^{n} × (0, T) contained in ℝ^{n+1} = ℝ_{x}^{n} × ℝ_{t}. The open set Q̂ is the union of open sets Ω_{t} of ℝ^{n}, for 0 < t < T, which are images of a reference domain Ω_{0} by a diffeomorphism τ_{t}: Ω_{0} → Ω_{t}.

We identify Ω_{0} to a bounded open set Ω of ℝ^{n} and its points are represented by y = (y_{1}, y_{2}, …, y_{n}) and those of Ω_{t} by x = (x_{1}, x_{2}, …, x_{n}) are such that x = τ_{t}(y). We also employ the notation τ(y, t) instead of τ_{t}(y).

Thus, the noncylindrical domain Q̂ of ℝ^{n+1} is defined by

Q̂ = ⋃_{0<t<T}{Ω_{t} × {t}}.

The boundary of Ω_{t} is represented by Γ_{t} and the lateral boundary of Q̂, denoted by Σ̂, is given by

Σ̂ = ⋃_{0<t<T}{Γ_{t} × {t}}.

Let Q be the cylinder

Q = Ω × (0, T),

Ω the reference domain. We have the natural diffeomorphism between Q and Q̂ given by

(y, t) ∈ Q → (x, t) ∈ Q̂, (x, t) = (τ_{t}(y), t) = (τ(y, t), t).

We will develope the article under the following assumptions.

(A1) For all 0 ≤ t ≤ T, τ_{t} is a C^{2}-diffeomorphism from Ω to Ω_{t}. (A2) τ(y, t) ∈ C^{0}([0, T]; C^{2}(\overline{Ω}));

We assume Ω ⊂ ℝ^{n} bounded and of class C^{2}. It could be Lipschitz continuous and unbounded.

In this article we will work with the following state equation

(1.1)  | \begin{matrix}u′ − Δu + f(u) = h(x, t)χq̂ in Q̂ \\ y = 0 on Σ̂ \\ u(x, 0) = u_{0}(x) in Ω. \end{matrix}

In (1.1) we have u = u(x, t), u′ = \frac{∂u}{∂t} ; Δ is the Laplace’s operator in ℝ^{n}; q̂ is an open, non-empty, subset of Q̂. We also denote by w_{t} the cross section of q̂ at any 0 < t < T; χq̂ the characteristic function of q̂. The function h(x, t) is the control that acts on the state u(x, t) localized in q̂. The nonlinear function f is real and globally Lipschitz such that f(0) = 0. This means that there exists a constant K_{0} , called Lipschitz constant, such that

(1.2)  |f(ξ) − f(η)| ≤ K_{0}|ξ − η|

for all ξ, η ∈ ℝ.

As we will see later, if u_{0} ∈ L^{2}(Ω), h ∈ L^{2}(Q̂) the system (1.1) has a unique solution

u ∈ C^{0}([0, T]; L^{2}(Ω_{t})) ∩ L^{2}(0, T; H_{0}^{1}(Ω_{t})).

The null controllability problem for (1.1) can be formulated as follows: Give T > 0 and u^{0} ∈ L^{2}(Ω), to find a controll h ∈ L^{2}(Q̂) such that the solution u = u(x, t) of (1.1) satisfies the conditions:

• u(x, T) = 0 for all x ∈ Ω_{T} , (1.3) (1.4) • |h|_{L^{2}(Q̂)} ≤ c|u^{0}|_{L^{2}(Ω)} , for all u^{0} ∈ L^{2}(Ω).

There is a large literature on the null controllability for heat equations in cylindrical domains. See for instance, and the bibliography therein, Lions [20,21,22], Fabre-Puel-Zuazua [12], Fernandez-Cara and Zuazua [14], Cabanillas-Menezes-Zuazua [4], Zuazua [38]. In the context of noncylindrical domain, Limaco-Medeiros-Zuazua [17], proved null controllability for linear heat equation.

The main result of the present paper is the following:

Theorem 1.1. Assume f is C^{1} and satisfies (1.2) with f(0) = 0. Then, for all T > 0 and for every u^{0} ∈ L^{2}(Ω), there exists h ∈ L^{2}(Q̂) such that the solution u = u(x, t) of (1.1) satisfies (1.3). Moreover, (1.4) holds for a suitable constant C > 0 independent of u^{0}. In other words, system (1.1) is null controllable for T > 0.

The methodology of the proof of the Theorem 1.1 is based in the fixed point method, see Zuazua [39,40]. There is however a new difficulty related to the fact that Q̂ is noncylindrical. To set up this point we employ the idea contained in [27]. The first step on the fixed point method is to study the null controll for the linearized system. This problem is reduced, by duality, to obtain a observability inequality for the adjoint system. This is get as an application of Carlemann inequalities as in Imanuvilov-Yamamoto [17].

This work is organized as follows: Section 2 is devoted to prove the null controllability for the linearized system. In Section 3 we prove Theorem 1.1 by a fixed point method.

To close this section we mention some basic references on the analysis of Partial Differential Equations in noncylindrical domains. Among many references we mention the following: Lions [19]; Cooper and Bardos [9]; Medeiros [6]; Inoue [18]; Rabello [33]; Nakao and Narazaki [34], for wave equations. Bernardi, Bonfanti and Lutteroti [2], Miranda, Medeiros [9] for Schrödinger equations; Cheng-He and Ling-Hsiano [7] for Euler equation; Miranda and Limaco [33] for Navier-Stokes equations; Chen and Frid [6] for hyperbolic systems of conservation law. Note that in [29] and [30] they considered τ_{t}(y) = K(t)y. In [2] the authors considered τ_{t}(y) = K(t)y + h(t), h(t) a vector of ℝ^{n}.

2. Analysis of the Linear Problem

The main result of this article will be proved in Section 3 by means of a fixed point argument. As an step preliminary we need to analyse the null controllability of the following linearized system:

(2.1)  | \begin{matrix}u′ − Δu + a(x, t)u = h(x, t) in Q̂ \\ u = 0 on Σ̂ \\ u(x, 0) = u^{0} in Ω, \end{matrix}

where the potential is assumed to be in L^{∞}(Q̂).

First of all we study the existence and uniqueness of solution of the system (2.1).

2.1 Strong and Weak Solutions

We distinguish three classes of solutions for the system (2.1), as follows: strong, weak and ultra weak solutions defined by transposition.

Definition 2.1. a) A real function u = u(x, t) defined in Q̂ is said to be a strong solution for the boundary value problem (2.1) if

(2.2) u ∈ C^{0}([0, T]; H_{0}^{1}(Ω_{t})) ∩ L^{2}(0, T; H^{2}(Ω_{t}) ∩ H_{0}^{1}(Ω_{t})) ∩ H^{1}(0, T; L^{2}(Ω_{t}))

and the three conditions in (2.1) are satisfied almost everywhere in their corresponding domains. b) We say that the function u is a weak solution of (2.1) if

(2.3)  u ∈ C^{0}([0, T]; L^{2}(Ω_{t})) ∩ L^{2}(0, T; H_{0}^{1}(Ω_{t}))

and

(2.4)  \begin{matrix}∫_{0}^{T} ∫_{Ω_{t}} uφ′ dxdt + ∫_{0}^{T} ∫_{Ω} u^{0}(x)φ(x, 0) dxds + \\ + ∫_{0}^{T} ∫_{Ω_{t}} ∇_{x}u ∇_{x}φ dxdt = ∫_{0}^{T} ∫_{Ω_{t}} hφ dxdt \end{matrix}

for all φ ∈ L^{2}(0, T; H_{0}^{1}(Ω_{t})) ∩ C^{1}([0, T]; L^{2}(Ω_{t})) such that φ(T) = 0.

Theorem 2.1. Assume that the noncylindrical domain Q̂ satisfies the conditions of the Section 1. Then, if u^{0} ∈ H_{0}^{1}(Ω), a(x, t) ∈ L^{∞}(Q̂) and h ∈ L^{1}(0, T; L^{2}(Ω_{t})), the problem (2.1) has a unique strong solution.

Moreover, there exists a positive constant C (depending on Q̂ but independent of u^{0} and h) such that

(2.5) ||u||_{L^{∞}(0,T;H_{0}^{1}(Ω_{t}))} + |u′|_{L^{2}(Q̂)} + |u|_{L^{2}(0,T;L^{2}(Ω_{t}))} ≤ C (||u^{0}||_{H_{0}^{1}(Ω)} + |h|_{L^{2}(Q̂)})

and for f ∈ L^{2}(0, T; H_{0}^{1}(Ω_{t})),

(2.6)  ||u||_{L^{∞}(0,T;H_{0}^{1}(Ω_{t}))} + |u|_{L^{2}(0,T;H^{2}(Ω_{t})))} ≤ C (||u^{0}||_{H_{0}^{1}(Ω)} + |h|_{L^{2}(0,T;H_{0}^{1}(Ω_{t}))})

Proof: As in [27] we employ the argument consisting in transforming the heat equation in the noncylindrical domain Q̂, into a variable coefficients parabolic equation in the reference cylinder Q by means of the diffeomorphism (x, t) = (τ_{t}(y), t) = (τ(y, t), t) for x ∈ Ω_{t}, y ∈ Ω and 0 ≤ t ≤ T, i.e., for (x, t) ∈ Q̂ and (y, t) ∈ Q.

In fact we set

(2.7)  v(y, t) = u(τ_{t}(y), t) = u(τ(y, t), t), for y ∈ Ω, 0 ≤ t ≤ T,

or equivalently

(2.8)  u(x, t) = v(τ_{t}^{−1}(x), t) = v(ρ(x, t), t), x ∈ Ω_{t} , 0 ≤ t ≤ T.

Here and in the following τ_{t}^{−1} denotes the inverse of τ_{t}, which, according to assumption (A1) is a C^{2} map from Ω_{t} to Ω, for all 0 ≤ t ≤ T. This map will be denoted by ρ_{t}. We shall also employ the notation ρ(x, t) = ρ_{t}(x), y_{j} = ρ_{j}(x, t), 1 ≤ j ≤ n.

We obtain,

\frac{∂}{∂t} u(x, t) = u′(x, t) = \frac{∂v}{∂t} (ρ(x, t), t) + ∇_{y} v(ρ(x, t), t) · \frac{∂}{∂t} ρ(x, t),

where · is the scalar product in ℝ^{n}. In other words,

\frac{∂}{∂t} u(x, t) = u′(x, t) = \frac{∂v}{∂t} (y, t) + ∇_{y} v(y, t) · b̃(y, t)

where b̃(y, t) denotes a vector field

(2.9)  b̃(y, t) = \frac{∂}{∂t} ρ(x, t).

Note that according to the assumption (A2),

(2.10)  b̃ ∈ C^{1}(\overline{Ω}).

On the other hand,

\frac{∂}{∂x_{i}} u(x, t) = \frac{∂}{∂y_{j}} v(y, t) \frac{∂y_{j}}{∂x_{i}} = ∇_{y} v(y, t) · \frac{∂}{∂x_{i}} ρ(x, t),

and

\frac{∂^{2}}{∂x_{i}^{2}} u(x, t) = \frac{∂^{2}}{∂y_{j}∂y_{k}} v(y, t) \frac{∂y_{j}}{∂x_{i}} \frac{∂y_{k}}{∂x_{i}} + \frac{∂}{∂y_{j}} v(y, t) \frac{∂^{2}y_{j}}{∂x_{i}^{2}}

or

Δu(x, t) = ∑_{i=1}^{n} \frac{∂^{2}v}{∂y_{j}∂y_{k}} \frac{∂ρ_{j}}{∂x_{i}} \frac{∂ρ_{k}}{∂x_{i}} + ∑_{i=1}^{n} \frac{∂v}{∂y_{j}} \frac{∂^{2}ρ_{j}}{∂x_{i}^{2}} ·

Thus by the mapping x = τ^{−1}(y) that takes Q̂ into the cylinder Q we transform (2.1) in an equivalent problem (2.11) given by

(2.11)  | \begin{matrix}\frac{∂v}{∂t} − ∑_{i=1}^{n} \frac{∂^{2}v}{∂y_{j}∂y_{k}} \frac{∂ρ_{j}}{∂x_{i}} \frac{∂ρ_{k}}{∂x_{i}} − ∑_{i=1}^{n} \frac{∂v}{∂y_{j}} Δρ_{j} + \\ + b̃ · ∇_{y}v + a(y, t)v = h(y, t) in Q \\ v = 0 on Σ, lateral boundary of Q \\ v(y, 0) = u^{0}(y) in Ω \end{matrix}

Analysis of the operator A(t)v = − ∑_{i=1}^{n} \frac{∂^{2}v}{∂y_{j}∂y_{k}} \frac{∂ρ_{j}}{∂x_{i}} \frac{∂ρ_{k}}{∂x_{i}}

For v, φ ∈ L^{2}(0, T; H_{0}^{1}(Ω)) and Gauss’ Lemma we obtain the bilinear form a(t, v, φ) defined by

a(t, v, φ) = (A(t)v, φ) = ∑_{i=1}^{n} ∫_{Ω} \frac{∂v}{∂y_{j}} \frac{∂φ}{∂y_{k}} \frac{∂ρ_{j}}{∂x_{i}} \frac{∂ρ_{k}}{∂x_{i}} dx.

This bilinear form is bounded because ρ ∈ C^{2}(−\overline{Ω}) by assumption (A2). Let us prove that it is H_{0}^{1}(Ω)-coercive. In fact, set φ = v ∈ H_{0}^{1}(Ω). We have

a(t, v, v) = ∑_{i=1}^{n} ∫_{Ω} (\frac{∂v}{∂y_{k}} \frac{∂ρ_{k}}{∂x_{i}} ) (\frac{∂v}{∂y_{j}} \frac{∂ρ_{j}}{∂x_{i}} ) dy =

= ∑_{i=1}^{n} ∫_{Ω} ∑_{k=1}^{n} (\frac{∂v}{∂y_{k}} \frac{∂ρ_{k}}{∂x_{i}} )^{2} dy.

Note that (\frac{∂v}{∂y_{k}} \frac{∂ρ_{k}}{∂x_{i}} )_{1≤k≤n} is a vector of ℝ^{n}. Let us consider the n × n matrix M given by

M = (\frac{∂ρ_{k}}{∂x_{i}} )_{1≤i,k≤n}

and the vector ξ of ℝ^{n} defined by

ξ = (\frac{∂v}{∂y_{k}} )_{1≤k≤n}

We observe that

∑_{i=1}^{n} ∑_{k=1}^{n} (\frac{∂v}{∂y_{k}} \frac{∂ρ_{k}}{∂x_{i}} )^{2} = ||Mξ||_{ℝ^{n}}^{2} .

But, by assumption, M is bounded and invertible what comes from assumptions on x = τ_{t}(y). Then,

||M^{−1}ξ||_{ℝ^{n}}^{2} ≤ C_{0}||ξ||_{ℝ^{n}}^{2} , C_{0} > 0

or

||M ξ||_{ℝ^{n}}^{2} ≥ \frac{1}{C^{0}} ||ξ||_{ℝ^{n}} .

Thus, returning to the quadratic form, we obtain

a(t, v, v) = ∫_{Ω} ||M ξ||_{ℝ^{n}}^{2} dy ≥ \frac{1}{C_{0}} ∫_{Ω} |ξ|_{ℝ^{n}}^{2} dy

or

a(t, v, v) ≥ \frac{1}{C_{0}} ||v||_{H_{0}^{1}(Ω)}^{2} .

In (2.11) set

b(y, t) = b̃(y, t) − Δρ(y, t), b ∈ [L^{∞}(Q)]^{n}.

Thus, from (2.11) we obtain for (2.1) in Q the following system

(2.12)  | \begin{matrix}v′ + A(t)v + b · ∇_{y}v + a(y, t)v = h(y, t) in Q \\ v = 0 on Σ \\ v(y, 0) = u^{0}(y) in Ω \end{matrix}

Note that (2.12) is a linear parabolic system with variable coefficients in a cylinder Q = Ω × (0, T), Ω a regular bounded open set of ℝ^{n}. Since A(t) is coercive the boundary value problem (2.12) is a classical problem studies in Lions-Magenes [24]. If we take u^{0} ∈ H_{0}^{1}(Ω) and h ∈ L^{2}(0, T; L^{2}(Ω)) then (2.12) has strong solution v ∈ C^{0}([0, T]; H_{0}^{1}(Ω)) ∩ L^{2}(0, T; H^{2}(Ω)) ∩ H^{1}(0, T; L^{2}(Ω)). Otherwise, if u^{0} ∈ L^{2}(Ω) and h ∈ L^{2}(0, T; L^{2}(Ω)), then (2.14) has a weak solution u ∈ C^{0}([0, T]; L^{2}(Ω) ∩ L^{2}(0, T; H_{0}^{1}(Ω)). In both cases we have uniqueness.

From the assumption (A1) and (A2) the transformation y → x from Q in Q̂ maps the space C^{0}([0, T]; H_{0}^{1}(Ω)) ∩ L^{2}(0, T; H^{2}(Ω)) ∩ H^{1}(0, T; L^{2}(Ω)) into the space C^{0}([0, T]; H_{0}^{1}(Ω_{t})) ∩ L^{2}(0, T; L^{2}(Ω_{t})) ∩ H^{1}(0, T; L^{2}(Ω_{t})).

To prove the estimate (2.5) we first establish the classical energy estimate. In fact, multiplying (2.1) by u integrating for x ∈ Ω_{t} and 0 < t < T, we get

(2.13)  \begin{matrix}∫_{0}^{t} ∫_{Ω_{t}} u′u dxds + ∫_{0}^{t} |∇u(s)|_{L^{2}(Ω_{s})}^{2} ds = \\ = − ∫_{0}^{t} ∫_{Ω_{t}} au^{2} dxds + ∫_{0}^{t} ∫_{Ω_{t}} hu dxds ≤ \\ ≤ ||a||_{L^{∞}(Q̂)} ∫_{0}^{t} ∫_{Ω} u^{2} dxds + C(ε)|h|_{L^{2}(0,T;H^{−1}(Ω_{t}))}^{2} + \\ + ε|u|_{L^{2}(0,T;H_{0}^{1}(Ω_{t}))}^{2} \end{matrix}

Note that since u vanishes on the lateral boundary Σ̂ of Q̂ we have (cf. Duvaut [11], p.26):

(2.14)  \begin{matrix}∫_{0}^{t} ∫_{Ω_{t}} uu′ dxdt = \frac{1}{2} ∫_{0}^{t} ∫_{Ω_{t}} \frac{∂}{∂t} |u(x, t)|^{2} dxdt = \\ = \frac{1}{2} [∫_{Ω_{t}} u^{2}(x, t) dx − ∫_{Ω}(u^{0}(x))^{2} dx] \end{matrix}

As a consequence of the assumptions (A1) and (A2) it follows that the Poincaré inequality is satisfied, uniformly, in the domain Ω_{t} for all 0 ≤ t ≤ T. Thus, in view of (2.13) and (2.14) we have

(2.15)  \begin{matrix}\frac{1}{2} ∫_{0}^{t} ∫_{Ω_{t}} \frac{∂}{∂t} |u(x, t)|^{2} dxdt + \frac{1}{2} ∫_{0}^{t} |∇u(s)|_{L^{2}(Ω_{s})}^{2} ds ≤ \\ ≤ c|h|_{L^{2}(0,T;H^{−1}(Ω_{t}))}^{2} + ||a||_{L^{∞}(Q̂)} ∫_{0}^{t} ∫_{Ω_{t}} u^{2} dxdt \end{matrix}

Then,

(2.16)  ∫_{Ω_{t}} u^{2} dx ≤ C (|u^{0}|_{L^{2}(Ω)}^{2} + |h|_{L^{2}(0,T;H^{−1}(Ω_{t}))}^{2})

By (2.14), (2.15) and (2.16) we have

(2.17) |u(t)|_{L^{2}(Ω_{t})}^{2} + \frac{1}{2} ∫_{0}^{t} |∇u(s)|_{L^{2}(Ω_{s})}^{2} ds ≤ C [|u^{0}|_{L^{2}(Ω)}^{2} + |h|_{L^{2}(0,T;H^{−1}(Ω_{t}))}^{2}] ,

for a constant C > 0.

In particular, strong solutions satisfies the energy estimate

(2.18)  |u|_{L^{2}(0,T;L^{2}(Ω_{t}))}^{2} + |u|_{L^{2}(0,T;H_{0}^{1}(Ω_{t}))}^{2} ≤ C [|u^{0}|_{L^{2}(Ω)}^{2} + |h|_{L^{2}(0,T;H^{−1}(Ω_{t}))}^{2}] ,

with C > 0 constant independent of the solution.

Now we multiply (2.1) by −Δu and integrate. We have

(2.19)  − ∫_{Ω_{t}} u′ Δu dx + ∫_{Ω_{t}} |Δu|^{2} dx − ∫_{Ω_{t}} auΔu dx = − ∫_{Ω_{t}} hΔu dx.

Moreover

(2.20) − ∫_{Ω_{t}} u′Δu dx = ∫_{Ω_{t}} ∇u · ∇u′ dx = \frac{d}{dt} ∫_{Ω_{t}} |∇u|^{2} dx − ∫_{Γ_{t}} |∇u|^{2} w · n_{t} dσ,

where n_{t} denote the unit outward normal vector to Ω_{t} and w is the velocity field w = [\frac{∂τ}{∂t} ] ρ(x, t) [cf. Duvaut [11] p.26]. Note that according to the assumption (A1) and (A2), by uniform (with respect to t) elliptic regularity, classical trace results and interpolation, [cf. Lions-Magenes [24] p.49], we obtain

(2.21)  \begin{matrix}|∫_{Γ_{t}} |∇u|^{2} w · n_{t} dσ| ≤ c ∫_{Γ_{t}} |∇u|^{2} dσ ≤ \\ ≤ c_{α} [∫_{Ω_{t}} |Δu|dx]^{α} [∫_{Ω_{t}} |∇u|^{2} dx]^{1−α} \end{matrix}

for all α ≥ \frac{1}{2} ·

EJQTDE, 2003 No. 16, p. 11

Combining (2.19)-(2.21) and by Cauchy-Schwarz’s inequality we deduce that

(2.22)

\frac{d}{dt} ∫_{Ω_{t}} |∇u|^{2} dx ≤ −\frac{1}{2} ∫_{Ω_{t}} |Δu|^{2} dx + |∇h|_{L^{2}(Ω_{t})} · |∇u|_{L^{2}(Ω_{t})} +

+ c|∇u|_{L^{2}(Ω_{t})}^{2} + |a|_{L^{∞}(Q̂)} |∇u|_{L^{2}(Ω_{t})}^{2} =

= −\frac{1}{2} ∫_{Ω_{t}} |Δu|^{2} dx + |∇h|_{L^{2}(Ω_{t})} · |∇u|_{L^{2}(Ω_{t})} + C|∇u|_{L^{2}(Ω_{t})}^{2} .

Solving this differential inequality we deduce the existence of a constant C such that

(2.23)  |u|_{L^{2}(0,T;H_{0}^{1}(Ω_{t}))}^{2} + |u|_{L^{2}(0,T;H^{2}(Ω_{t}))}^{2} ≤ C [|u^{0}|_{H_{0}^{1}(Ω)}^{2} + ||h||_{L^{1}(0,1;L^{2}(Ω_{t}))}] .

A variation of this argument alows also to get

(2.24)  ||u||_{L^{∞}(0,T;H_{0}^{1}(Ω_{t}))}^{2} + |u|_{L^{2}(0,T;H^{2}(Ω_{t}))} ≤ C [|u^{0}|_{H_{0}^{1}(Ω_{t})}^{2} + |h|_{L^{2}(Q̂)}^{2}]

In fact, to obtain (2.24) instead of (2.23) it is sufficient to estimate the term ∫_{Ω_{t}} hΔu dx as follows

∫_{Ω_{t}} hΔu dx ≤ \frac{1}{2} ∫_{Ω_{t}}(|h|^{2} + |Δu|^{2})dx.

This complete the proof of the Theorem 2.1.

Remark 2.1 Note that we could also have obtained the above estimates using existence results for the variable coefficients parabolic equation satisfied by v and then doing the change of variables x → y of Q̂ into Q.

Theorem 2.2 Given u^{0} ∈ L^{2}(Ω) and f ∈ L^{2}(0, T; H^{−1}(Ω_{t})), there exists a unique weak solution of (2.1). Moreover, there exists a constant C > 0 (depending on Q̂ but independent of u^{0} and h) such that

(2.25)  ||u||_{L^{∞}(0,T;L^{2}(Ω_{t}))} + |∇u|_{L^{2}(Q̂)} ≤ C [|u^{0}|_{L^{2}(Ω)} + |h|_{L^{2}(0,T;H^{−1}(Ω_{t}))}]

A similar argument allows to replace h ∈ L^{2}(0, T; H^{−1}(Ω_{t})) by the assumption h ∈ L^{1}(0, T L^{2}(Ω_{t})) and to obtain the estimate

(2.26)  ||u||_{L^{∞}(0,T;L^{2}(Ω_{t}))} + |∇u|_{L^{2}(Q̂)} ≤ C [|u^{0}|_{L^{2}(Ω)} + ||h||_{L^{1}(0,T;L^{2}(Ω_{t}))}]

Proof: We follow the argument of reference [27]. We proceed by steps.

Step 1 (Existence). Let u_{m}^{0} ∈ H_{0}^{1}(Ω) and h_{m} ∈ L^{2}(Q̂) be a sequence of regularized initial data and right hand side terms, respectively, such that u_{m}^{0} → u^{0} strongly in H_{0}^{1}(Ω), h_{m} → h strongly in L^{2}(0, T; H^{−1}(Ω_{t})). Then, for each m ∈ ℕ, let us consider the unique strong solution u_{m} of (2.1) with initial data u_{m}^{0} and right side h_{m}. Thus, for any n, k ∈ ℕ we have

(2.27)  | \begin{matrix}(u_{n} − u_{k})′ − Δ(u_{n} − u_{k}) + a(x, t)(u_{n} − u_{k}) = h_{n} − h_{k} a.e. in Q̂, \\ (u_{n} − u_{k}) = 0 on Σ̂ \\ (u_{n} − u_{k})(0) = u_{n}^{0} − u_{k}^{0} in Ω \end{matrix}

By the energy estimate (2.18) we obtain that (u_{m}) is a Cauchy sequence in the space

C^{0}([0, T]; L^{2}(Ω_{t})) ∩ L^{2}(0, T; H_{0}^{1}(Ω_{t})).

Thus it converges, as m → ∞, to a limit u ∈ C([0, T]; L^{2}(Ω_{t})) ∩ L^{2}(0, T; H_{0}^{1}(Ω_{t})). The limit u is a weak solution of (2.1) satisfying (2.4) and the estimate (2.25). In fact, u_{m} is strong solution for every m ∈ ℕ. Then, multiplying the equation with u_{m} by a test function φ and integrating by parts we deduce that u_{m} satisfies the weak formulation (2.4).

The convergence of u_{m} to u in the space

C^{0}([0, T]; L^{2}(Ω_{t})) ∩ L^{2}(0, T; H_{0}^{1}(Ω_{t}))

allows to pass to the limit in the weak formulation to conclude that u satisfy (2.4).

Step 2 (Uniqueness). Assume that the system (2.2) admits two weak solution u and û satisfying (2.4). Introduce w = u−û. Then, w belongs to C^{0}([0, T]; L^{2}(Ω_{t}))∩ L^{2}(0, T; H_{0}^{1}(Ω_{t})) and satisfies

∫_{0}^{T} ∫_{Ω_{t}} wφ′ dxdt + ∫_{0}^{T} ∫_{Ω_{t}} ∇_{x}w · ∇_{x}φ dxdt + ∫_{0}^{T} ∫_{Ω_{t}} a(x, t)uφ dxdt = 0

for all test function φ. In order to conclude that w = 0, it is sufficient to consider w = φ as a test function. Of course we cannot do it directly since w is not a test function. It is justified by regularization and cut-off argument.

In this way we complete the proof obtaining the energy estimate for w what guarantees that

∫_{0}^{t} ∫_{Ω_{t}} \frac{∂}{∂t} |w|^{2} dxdt + ∫_{0}^{t} |∇w(s)|^{2} ds ≤ ||a||_{L^{∞}(Q̂)} ∫_{0}^{t} ∫_{Ω_{t}} |w|^{2} dxdt.

Then w = 0 because w(0) = 0.

Step 3. To prove the estimate (2.26), it suffices to employ in (2.18) the estimate

|∫_{0}^{T} ∫_{Ω_{t}} ρh dxdt| ≤ |h|_{L^{1}(0,T;L^{2}(Ω))} · ||u||_{L^{∞}(0,T;L^{2}(Ω_{t}))} .

2.2 Ultra Weak Solutions by Transposition Method

In this section we address the question of finding solutions u of the boundary value problem

(2.28)  | \begin{matrix}u′ − Δu + a(x, t)u = 0 in Ω_{t} , for 0 < t < T \\ u = 0 on Γ_{t} for 0 < t < T \\ u(0) = u^{0} in Ω \end{matrix}

when u^{0} ∈ H^{−1}(Ω) and a(x, t) ∈ L^{∞}(Q̂).

We employ the transposition method as in Lions-Magenes [23]. First of all we define what we understand by ultra weak solution by this method.

A function u = u(x, t) is said to be ultra solution of (2.28) or solution by transposition if

(2.29)  u ∈ C^{0}([0, T]; H^{−1}(Ω_{t})) ∩ L^{2}(0, T; L^{2}(Ω_{t}))

and

(2.30)  ∫_{0}^{T} ∫_{Ω_{t}} u(x, t)h(x, t) dxdt = ⟨u^{0}, φ(0)⟩ for all h ∈ L^{2}(Q̂)

where φ is the unique strong solution of the adjoint system

(2.31)  | \begin{matrix}− φ′ − Δφ + aφ = h in Ω_{t} for 0 < t < T \\ φ = 0 on Γ_{t} for 0 < t < T \\ φ(x, T) = 0 in Ω \end{matrix}

Here, ⟨ , ⟩ denotes the duality passing between H^{−1}(Ω) and H_{0}^{1}(Ω).

According to Theorem 2.1, the system (2.31) admits a unique strong solution φ. Thus the definition of ultra weak solution makes sense.

Note that the strong solution φ satisfies the following estimates:

(2.32)  ||φ||_{L^{∞}(0,T;H_{0}^{1}(Ω_{t}))} ≤ c|h|_{L^{2}(Q̂)}

and

(2.33)  ||φ||_{L^{∞}(0,T;H_{0}^{1}(Ω_{t}))} ≤ c||h||_{L^{1}(0,T;H_{0}^{1}(Ω_{t}))}

These estimates were proved in Theorem 2.1. Indeed, it is sufficient to make the change of variables t → T − t to reduce the system (2.31) to (2.1).

By Riesz-Fréchet theorem we deduce that there exists a unique ultra weak solution in the class (2.29). More precisely, in view of (2.32) we deduce the existence of unique solution u ∈ L^{2}(Q̂) and the second estimate (2.33) provides the aditional regularity u ∈ L^{∞}(0, T; H^{−1}(Ω_{t})). Moreover, one deduces the existence of a constant, independent of u^{0}, such that

(2.34)  ||u||_{L^{∞}(0,T;H^{−1}(Ω_{t}))} + |u|_{L^{2}(Q̂)} ≤ c|u^{0}|_{H^{−1}(Ω)} .

In order to show that u ∈ C^{0}([0, T]; H^{−1}(Ω_{t})) we use a classical density argument. When u^{0} is smooth enough, u is a weak or strong solution, therefore u is continuous with respect to time with values in H^{−1}(Ω_{t}). According to (2.34), by density, we deduce that u ∈ C^{0}([0, T]; H^{−1}(Ω_{t})) whenever u^{0} ∈ H^{−1}(Ω).

To complete this section, we observe that when u^{0} is smooth so that exist weak or strong solution then they are also ultra weak solutions. It is sufficient to integrate by parts in the strong formulation of (2.1) or consider the weak formulation.

2.3 Observability of the Linearized Adjoint System

As we said before we employ a fixed point argument in order to prove our results in the semilinear case. However, first we analyse the null controllability for the following linearized system:

(2.35)  | \begin{matrix}u′ − Δu + a(x, t)u = h(x, t)χq̂ in Q̂ \\ u = 0 on Σ̂ \\ u(x, 0) = u_{0}(x) in Ω, \end{matrix}

where the potential a = a(x, t) is assumed to be in L^{∞}(Q̂). Remember we denote by û_{t} the cross section of q̂ at any 0 < t < T.

As we know, the null controllability of (2.35) is equivalent to a suitable observability property for the adjoint system of (2.35).

EJQTDE, 2003 No. 16, p. 16

Thus, let us consider the adjoint system

(2.36)  | \begin{matrix}− φ′ − Δφ + aφ = 0 in Q̂ \\ φ = 0 on Σ̂ \\ φ(T) = φ^{0} in Ω_{T} \end{matrix}

for which we have the following observability property.

Proposition 2.1. For all T > 0 and R > 0 there exists a positive constant C > 0 such that

(2.37)  |φ(0)|_{L^{2}(Ω)}^{2} ≤ C ∫q̂ |φ(x, t)|^{2} dxdt,

for every solution of (2.36) and for any a ∈ L^{∞}(Q̂) such that ||a||_{L^{∞}(Q̂)} ≤ R.

Remark 2.2. The constant C in (2.37) will be referred to as the observability constant. It depends on Q̂, q̂ the time T and the size R of the potential but does not depend of the solution φ of (2.36).

Proof of the Proposition 2.1: The inequality (2.37) is a consequence of the results in [17]. In fact, by the change of variables x → y, from Q̂ into Q, the adjoint system (2.36) is transformed into a variable coefficient parabolic equation of the form

(2.38)  | \begin{matrix}− ψ′ + A(t)ψ + b · ∇ψ + aψ = 0 in Q \\ ψ = 0 in Σ \\ ψ(T) = ψ^{0} ìn Ω_{T} \end{matrix}

as in (2.12) with h̃ = 0. Thus the coefficient of the principal part A(t), according to the assumption (A1) and (A2) are of class C^{1} and a and b are bounded. Then, the observability inequalities in [17] guarantee that for every T > 0 and every open subset q of Q, there exists a constant C > 0 such that

(2.39)  |ψ(0)|_{L^{2}(Ω)} ≤ C ∫_{q} |ψ(y, t)|^{2} dydt.

In particular it is true for q ⊂ Q image of q̂ by x → y. Thus estimate (2.37) for φ is obtained from (2.39) for ψ by the change of variables y → x.

2.4 Approximate Controllability for the Linearized System

From the observability inequality (2.37) the null controllability result for the linearized system can be proved as the limit of an approximate controllability property.

In fact, given u^{0} ∈ L^{2}(Ω) and δ > 0 we introduce the quadratic functional

(3.40)  J_{δ}(φ^{0}) = \frac{1}{2} ∫q̂ φ^{2} dxdt + δ|φ^{0}|_{L^{2}(Ω_{T})} + ∫_{Ω} u^{0}φ(0) dx,

where φ is the solution of (2.36) with initial data φ^{0}. The functional J_{δ} is continuous and strictly convex in L^{2}(Ω_{t}). Moreover, J_{δ} is coercive. More precisely, in view of (2.37) we have

(2.41)  lim inf_{|φ^{0}|_{L^{2}(Ω_{T})}→∞} \frac{J_{δ}(φ^{0})}{|φ^{0}|_{L^{2}(Ω_{T})}} ≥ δ.

To prove (2.41) we follows the argument used in [27] which we will not repeat here.

Thus J_{δ} has a unique minimizer in L^{2}(Ω_{T}). Let us denote it by φ̂^{0,δ}. It is not difficult to prove that the control h_{δ} = φ̂^{δ}, where φ̂^{δ} is the solution of (2.36) associated to the minimizer φ̂^{0,δ} is such that the solution u_{δ} of (2.1) satisfies

(2.42)  |u_{δ}(T)|_{L^{2}(Ω_{T})} ≤ δ.

We refer to [12] for the details of the proof.

2.5 Null Controllability of the Linearized System The null controllability property may be obtained as the limit when δ tends to

EJQTDE, 2003 No. 16, p. 18 zero of the approximate controllability property above obtained. However, to pass to the limit we need a uniform bound of the control. To obtain this bound we observe that, by (2.37),

(2.43)  J_{δ}(φ^{0}) ≥ \frac{1}{2} ∫q̂ φ^{2} dxdt − C [∫q̂ φ^{2} dxdt]^{1/2} |u^{0}|_{L^{2}(Ω)} ,

when C > 0 is independent of δ. On the order hand,

(2.44)  J_{δ}(φ̂^{0,δ}) ≤ J_{δ}(0) = 0.

Writing (2.43) for φ̂^{0,δ} instead of φ, with φ^{0,δ} the minimizer of J_{δ} in L^{2}(Ω_{T}) and combining it with (2.44), we deduce that

(2.45)  |h_{δ}|_{L^{2}(0,T;L^{2}(Ω_{t}))} ≤ 2C|u^{0}|_{L^{2}(Ω)} ,

for all δ > 0.

In other words, h_{δ} remains bounded in L^{2}(0, T; L^{2}(Ω_{t})) as δ → 0. Note also that the constant C in (2.45) is independent of ||a||_{L^{∞}(Q̂)} ≤ R.

Extracting a sub net h_{δ} deduce that

(2.46)  h_{δ} ⇀ h, as δ → 0, weakly in L^{2}(Q̂),

for some h ∈ L^{2}(Q̂).

We can prove that the limit h is such that the solution u of (2.1) satisfies (1.3). Moreover, by the lower semicontinuity of the norm with respect to the weak topology and by (2.46) we deduce that

(2.47)  |h|_{L^{2}(Q̂)} ≤ lim inf_{δ→0} |h_{δ}|_{L^{2}(Q̂)} ≤ 2C|u^{0}|_{L^{2}(Ω)} .

By the process we complete the proof of the following result.

Theorem 2.3. Assume that the noncylindrical domain Q̂ satisfies the conditions fixed in Section 1 and that a(x, t) ∈ L^{∞}(Q̂). Then, for every T > 0 and u^{0} ∈ L^{2}(Ω).

EJQTDE, 2003 No. 16, p. 19 there exists h ∈ L^{2}(Q̂) such that the solution of (2.1) satisfies (1.3). Moreover, there exists a constant C > 0, depending on R > 0, but independent of u^{0}, such that (1.4) holds for every potential a = a(x, t) in L^{∞}(Q̂) such that ||a||_{L^{2}(Q̂)} ≤ R.

3. Proof of the Main Result

This section is devoted to prove Theorem 1.1. As we said, in the Introduction, it will be a consequence of Theorem 2.3 above and a fixed point argument.

In order to be self contained we will prove existence result.

Theorem 3.1. Assume that f: ℝ → ℝ is C^{1} and globally Lipschitz function, such that f(0) = 0. Let u^{0} ∈ L^{2}(Ω) and h ∈ L^{2}(Q̂). Then, there exists a unique solution u ∈ C^{0}([0, T]; L^{2}(Ω_{t})) ∩ L^{2}(0, T; H_{0}^{1}(Ω_{t})) of the problem (1.1).

Proof: As in the proof of Theorem 2.1 we transform the problem in a noncylindrical domain Q̂ into a parabolic problem with variables coefficients in the cylinder Q. In fact, the change of variables (2.7), equivalently (2.8), transforms (1.1) in

(3.1)  | \begin{matrix}v′ + A(t)v + b · ∇_{y}v + f(v) = h(y, t) for (y, t) ∈ Q \\ v = 0 for (y, t) ∈ Σ \\ v(y, 0) = v^{0}(y) for y ∈ Ω. \end{matrix}

Then we know that (3.1) admits a unique solution

v ∈ C^{0}([0, T]; L^{2}(Ω)) ∩ L^{2}(0, T; H_{0}^{1}(Ω)).

By the change of variable y → x we deduce the existence of a unique solution u of (1.1) in the class

u ∈ C^{0}([0, T]; L^{2}(Ω_{t})) ∩ L^{2}(0, T; H_{0}^{1}(Ω_{t})).

As in [12] we introduce the nonlinearity

(3.2)  g(s) = | \begin{matrix}\frac{f(s)}{s} if s ≠ 0 \\ f′(0) if s = 0. \end{matrix}

Note that g is uniformly bounded with ||g||_{∞} ≤ ||f′||_{∞}. Given any z ∈ L^{2}(Q̂) we consider the linearized system

(3.3)  | \begin{matrix}u′ − Δu + g(z)u = h χq̂ in Q̂ \\ u = 0 on Σ̂ \\ u(x, 0) = u^{0}(x) in Ω \end{matrix}

Observe that (3.3) is a linear systsem in the state u = u(x, t) with potential a = g(z) ∈ L^{∞}(Q̂), satisfying the condition

(3.4)  ||a||_{L^{∞}(Q̂)} ≤ ||f′||_{L^{∞}(ℝ)} .

With this notation, the system (3.3) can be written as

(3.5)  | \begin{matrix}u′ − Δu + au = h χq̂ in Q̂ \\ u = 0 on Σ̂ \\ u(x, 0) = u^{0}(x) in Ω. \end{matrix}

By the subsection 2.4, if δ > 0 is fixed, for each z ∈ L^{2}(Q̂) we can define a control h_{δ} = h_{δ}(x, t) ∈ L^{2}(Q̂) such that the solution u_{δ} of (3.5) satisfies

(3.6)  |u_{δ}(T)|_{L^{2}(Ω_{T})} ≤ δ,

see (2.42).

Moreover, for every R > 0 and all potential a = a(x, t) ∈ L^{∞}(Q̂) such that ||a||_{L^{∞}(Q̂)} ≤ R, we have

(3.7)  |h_{δ}|_{L^{2}(Q̂)} ≤ |u^{0}|_{L^{2}(Ω)} ,

EJQTDE, 2003 No. 16, p. 21 for all δ > 0. Therefore, the controls h_{δ} are uniformly bounded (with respect to z and δ) in L^{2}(Q̂).

This result allows to define a nonlinear mapping

(3.8)  N_{δ}(z) = u, from L^{2}(Q̂) into L^{2}(Q̂)

where u satisfies (3.5) and (3.6).

In this way, the approximate controllability problem for (1.1) is reduced to find a fixed point for the map N_{δ}. Indeed, if z ∈ L^{2}(Q̂) is such that N_{δ}(z) = u = z, u solution of (3.3) is solution of (1.1) Then the control h_{δ} = h_{δ}(z) is the one we were looking for, since, by construction, u_{δ} = u_{δ}(z) satisfies (3.6).

As we shall see, the nonlinear map N_{δ} satisfies the following properties:

(3.9)  N_{δ} is continuous and compact,

(3.10) The range of N_{δ} is bounded, i.e., exists M > 0 such that |N_{δ}(z)|_{L^{2}(Q̂)} ≤ M

for all z ∈ L^{2}(Q̂). Therefore, by (3.9), (3.10) and Schauder fixed point theorem, it follows that N_{δ}

is a fixed point.

By the moment, assume that (3.9) and (3.10) are true which proof comes after. Then if (3.9) and (3.10) are true it follows, by Schauder’s fixed point theorem, that we have a control h_{δ} in L^{2}(Q̂) such that the solution u_{δ} of

(3.11)  | \begin{matrix}u′_{δ} − Δu_{δ} + f(u_{δ}) = h_{δ}(x, t)χq̂ in Q̂ \\ u_{δ} = 0 on Σ̂ \\ u_{δ}(x, 0) = u^{0}(x) in Ω. \end{matrix}

satisfies

(3.12)  |u_{δ}(T)|_{L^{2}(Ω_{T})} ≤ δ,

EJQTDE, 2003 No. 16, p. 22 with an estimate of the form

(3.13)  |h_{δ}|_{L^{2}(Q̂)} ≤ C|u^{0}|_{L^{2}(Ω)}, ,

with C independent of δ.

Passing to the limit as δ → 0, as in Section 2, we deduce the existence of a limit control h ∈ L^{2}(Q̂) such that the solution u of (1.1) satisfies (1.3) and (1.4).

To complete the argument we need to prove (3.9) and (3.10).

Continuity of N_{δ} N_{δ} N_{δ}. Assume that z_{j} → z in L^{2}(Q̂). Then the potential a_{j} = g(z_{j}) is such that

(3.14)  a_{j} = g(z_{j}) → a = g(z) in L^{2}(Q̂).

In fact, we have

(3.15)  |g(z_{j})| = \frac{|f(z_{j})|}{|z_{j}|} ≤ K_{0} ,

by hypothesis, K_{0} the Lipschitz constant of f, then |g(z_{j})|^{p} ≤ K_{0}^{p}, 1 ≤ p < ∞.

We also have z_{j} → z in L^{2}(Q̂) and consequently a subsequence z_{j} → z a.e. in Q̂. Then

|g(z_{j})|^{p} = \frac{|f(z_{j})|^{p}}{|z_{j}|^{P}} → \frac{|f(z)|^{p}}{|z|^{p}} a.e. in Q̂.

It follows by Lebesgue’s bounded convergence theorem that

g(z_{j}) → g(z) in L^{p}(Q̂)

for all 1 ≤ p < ∞, that is a_{j} → a in L^{p}(Q̂). According to Theorem 2.3 the corresponding control are uniformly bounded, i.e.,

(3.16)  |h|_{L^{2}(Q̂)} ≤ C, for all y ≥ 1,

and, more precisely,

(3.17)  h_{j} = φ̂_{j} in q̂,

where φ̂ solves

(3.18)  | \begin{matrix}− φ′ − Δφ + g(z_{j})φ = 0 in Q̂ \\ φ = 0 on Σ̂ \\ φ(x, T) = φ̂_{j}^{0} in Ω_{T} , \end{matrix}

with the initial data φ̂_{j}^{0} that minimizes the correspondent functional J_{δ} in L^{2}(Q̂). We also have

(3.19)  |φ̂_{j}^{0}|_{L^{2}(Ω_{T})} ≤ C.

By extracting a subsequence (φ̂_{j}^{0}) we have

(3.20)  φ̂_{j}^{0} ⇀ φ̂^{0} weakly in L^{2}(Q_{T}).

From (3.18), (3.14) and (3.15) we have a subsequence still represented by (φ̂_{j}) such that

φ̂_{j} ⇀ χ weakly in L^{2}(0, T; H_{0}^{1}(Ω_{t})).

We will prove that χ = φ̂ and φ̂ solves

||||||||||

− φ′ − Δφ + g(z)φ = 0 in Q̂
φ = 0 on Σ̂
φ(x, T) = φ̂^{0} in Ω_{T} .

It is sufficient to prove that

g(z_{j})φ̂_{j} ⇀ g(z)φ̂ weakly in L^{2}(Ω × (0, T)).

In fact, with the change of variables y → x from Q̂ into Q the system (3.18) is transformed in one system in ψ_{j}(x, t) = φ_{j}(y, t) with y = τ_{t}(x), as follows:

||||||||||

− ψ′_{j} + A(t)ψ_{j} + a_{j}ψ_{j} + b · ∇ψ_{j} = 0 in Q
ψ_{j} = 0 on Σ
ψ_{j}(T) = ψ̂_{j}^{0} in Ω.

For the parabolic problem for ψ_{j} we obtain estimates in the cylinder which permits to employ compacteness argument of the type Lions-Aubin for ψ_{j}. When we change the variables y → x we obtain subsequence (φ̂_{j}) in L^{2}(Q̂) such that

φ̂_{j} → φ̂ strong L^{2}(Q̂).

This implies that

g(z_{j})φ̂_{j} → g(z)φ̂ weakly in L^{2}(Q̂).

Therefore,

h_{j} → h in L^{2}(Q̂)

where

h = φ̂ in q̂.

Note that u_{j} and u solve (3.11), what implies, by the estimates, that

u_{j} → u in L^{2}(Q̂),

where u solves

(3.21)  | \begin{matrix}u′ − Δu + g(z)u = h χq̂ in Q̂ \\ u = 0 on Σ̂ \\ u(x, 0) = u_{0}(x) in Ω \end{matrix}

and

(3.22)  |u(T)|_{L^{2}(Ω)} ≤ δ.

To complete the proof of the continuity of N_{δ} it is sufficient to check that the limit φ̂^{0} obtained in (3.20) is the minimizer of the functional J_{δ} associated to the limit control problem (3.21) and (3.22).

To do this, given ψ^{0} ∈ L^{2}(Ω_{T}) we have to show that

(3.23)  J_{δ}(φ̂^{0}) ≤ J_{δ}(ψ^{0}).

In fact, by weak lower continuity, we have

(3.24)  J_{δ}(φ̂^{0}) ≤ lim inf_{j→∞} J_{δ,j}(φ̂_{j}^{0}).

We also have

J_{δ}(ψ^{0}) = lim inf_{j→∞} J_{δ,j}(ψ^{0}), for all ψ^{0} ∈ L^{2}(Ω_{T}).

But,

(3.25)  J_{δ,j}(φ̂_{j}^{0}) ≤ J_{δ,j}(ψ^{0}) for all ψ^{0} ∈ L^{2}(Ω_{T}),

because φ̂_{j}^{0} is the minimizer of J_{δ,j}. Thus, by (3.25) and (3.24) we obtain (3.23).

Compactness of N_{δ} N_{δ} N_{δ}. The above argument says that when z varies in a bounded set B of L^{2}(Q̂) implies that u = N_{δ}(z) lies in a bounded set of L^{2}(Q̂) where u solves

(3.34)  | \begin{matrix}u′ − Δu = h χq̂ − g(z)u in Q̂ \\ u = 0 on Σ̂ \\ u(x, 0) = u^{0}(x) in Ω. \end{matrix}

Set β = h χq̂ − g(z)u and (3.34) can be write for u = w + v, with w fixe solution

of ||||||||||

w′ − Δw = 0 in Q̂
w = 0 on Σ̂
w(x, 0) = u^{0}(x) in Ω

and v solution of

||||||||||
v′ − Δv = β in Q̂
v = 0 on Σ̂
v(x, 0) = 0 in Ω

It follows that w is fixe and belongs to

L^{∞}(0, T; L^{2}(Ω_{t})) ∩ L^{2}(0, T; H_{0}^{1}(Ω_{t})).

As β is uniformly bounded in L^{2}(Q̂) we have v varies in a bounded set of L^{2}(0, T; H_{0}^{1}(Ω_{t})∩H^{2}(Ω_{t})) and v′ varies in a bounded set of L^{2}(0, T; L^{2}(Ω_{t})). Thus by Aubin-Lions compactness result, v varies in a relatively compact set of L^{2}(0, T; L^{2}(Ω_{t})). It then follows that u = w+v, with w in L^{2}(0, T; L^{2}(Ω_{t})) varies in a relatively compact set of L^{2}(0, T; L^{2}(Ω_{t})).

Acknowledgements

We take this oportunity to express our appreciation to Enrique Zuazua who calls our attention to this type of problem and for his permanent conversations about the development of our work.

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S.B. de Menezes Departamento de Matemática - UFPa 66075-110 Belém - PA, Brasil silvano@ufpa.br

J. Limaco Instituto de Matemática - UFF Rua São Pedro s/n 24210-110 Niterói, RJ, Brasil

L.A. Medeiros Instituto de Matemática - UFRJ Caixa Postal 68530 21945-970 Rio de Janeiro, RJ, Brasil lmedeiros@abc.org.br

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