Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 1, 1-9; http://www.math.u-szeged.hu/ejqtde/

Existence of global solution for a nonlocal parabolic problem

Abderrahmane El Hachimi & Moulay Rchid Sidi Ammi

Abstract

In this paper, we study a non-local initial boundary-value problem arising in Ohmic heating. By using a dynamical systems approach, some existence and uniqueness results are proved and the existence of a compact attractor is shown.

Mathematics Subject Classifications: 35K20, 35K35, 35K45, 35K60.

Key words: thermistor, a nonlocal, existence, attractor global.

1 Introduction

In this paper, we shall deal with the following nonlocal parabolic problem

\frac{∂u}{∂t} − △u = λ\frac{f(u)}{( ∫_{Ω} f(u) dx)^{2}} , in Ω×]0; T [,  (1.1)

u = 0 on ∂Ω×]0; T [,  (1.2)
u/_{t=0} = u_{0} in Ω,  (1.3)

where T > 0, Ω is a regular open bounded subset of ℝ^{N}, N ≥ 1, λ is a positive parameter and f a function from ℝ to ℝ satisfying the hypotheses (H_{1})−(H_{2}) below. Problem (1.1) − (1.3) represents, for example, static materiel such as thermistors [3, 6, 14, 15] and arises by reducing the system of two equations

u_{t} = ∇.(k(u)∇u) + σ(u)|∇φ|^{2},  (1.4)
∇(σ(u)∇φ) = 0,  (1.5)

to a simple but realistic equation (see [8]). More precisely, u represents the temperature produced by an electric current flowing through a conductor, φ the electric potential, σ(u) is the electrical conductivity and k(u) is the thermal conductivity. Taking the latter to be constant, problem (1.4) − (1.5) can then be reduced to the single nonlocal equation(1.1), where f(u) = σ(u) and λ = \frac{I^{2}}{|Ω|^{2}} ≥ 0, I is the electric current which is supposed to be constant and |Ω| is the measure of Ω.

Our goal here concerns the existence and uniqueness of weak solutions to (1.1)−(1.3). We shall also show existence of global attractor. Let us first recall that problem (1.1) − (1.3) has been the subject of variety of investigation in the past decade. Particularly, some results have been obtained by many authors in the case where N = 1 and f taking particular forms: Montesinos and Gallego [11] proved the existence of weak solution under

0 < σ_{1} ≤ σ(s) ≤ σ_{2}, ∀s ∈ ℝ.  (1.6)

Antontsev and Chipot [1] obtained also an existence and uniqueness results for (1.4)− (1.5) supposing that σ ∈ C^{0}(Ω) and (1.6); furthermore, a study of smoothness of solutions was treated in that paper under some assumptions on the conductivity and initial data. In [8, 9, 13], major emphasis is placed on cases where the spatial dimension N is 1 or 2 and f is of the form f(u) = exp(u)or exp(−u). In these works, additional regularity assumptions are made on u_{0} and a combination of usual Lyapounov functional and a comparison method is the main ingredient. Our purpose is to extend some of the results therein to problem (1.1) − (1.3), where here, the condition (1.6) is weakened to (H_{2}) below. Following the frame work of Fioas and Temam [12], we shall also deal with the asymptotic behaviour of the solutions of problem (1.1) − (1.3) via a dynamical systems approach. We start by proving the existence of absorbing sets in L^{∞}(Ω) and in H_{0}^{1}(Ω), which in turn paves the way for the existence of the global attractor. Cimatti [4] obtained similar results for particular cases, when N = 1, by constructing a Lyapounov functional. As a concluding result, we show that the attractor is bounded subset of H^{2}(Ω) under restrictive assumptions on data.

2 Existence and regularity of global attractor.

a) Existence and uniqueness. We assume the following

(H1) f : ℝ → ℝ is a locally Lipschitzian function .

(H2) There exist positive constants c_{1}, c_{2} and α such that, for all ξ ∈ ℝ

σ ≤ f(ξ) ≤ c_{1}|ξ|^{α+1} + c_{2}.

Let us denote by ∥.∥_{k} the norm in L^{k}(Ω) . We adopt the following weak formulation for (1.1) − (1.3): u is a solution of (1.1) − (1.3) if and only if

u ∈ L^{∞}(τ, +∞, H_{0}^{1}(Ω) ∩ L^{∞}(Ω)) with\frac{∂u}{∂t} ∈ L^{2}(τ, +∞, L^{2}(Ω))

for any τ > 0, and satisfying

∫_{0}^{T} ∫_{Ω} u\frac{∂}{∂t}ϕ − ∇u∇ϕ dxdt = ∫_{0}^{T} (\frac{λ}{( ∫_{Ω} f(u) dx)^{2}} ∫_{Ω} f(u)ϕdx)dt,

for any ϕ ∈ C^{∞}((0, ∞), Ω).

Now, we state our main result.

Theorem 2.1. Let hypotheses (H_{1}) − (H_{2}) be satisfied. Assume that u_{0} ∈ L^{k_{0}+2}(Ω) with k_{0} such that

k_{0} ≥ max (0, \frac{αN}{2} − 2).  (2.1)

Then, there exists d_{0} > 0 such that if ∥u_{0}∥_{k_{0}+2} < d_{0}, the problem (1.1) admits a solution u verifying for all τ > 0

u ∈ L^{∞}(τ, +∞, L^{k_{0}+2}(Ω)), |u|^{γ}u ∈ L^{∞}(τ, +∞, H_{0}^{1}(Ω)), with γ = \frac{k_{0}}{2} .

Moreover, if u_{0} ∈ L^{∞}(Ω), then u ∈ L^{∞}(τ, +∞, L^{∞}(Ω)) and is unique.

Remark. The value of d_{0} will be given in the course of the proof.

Proof. We use a Faedo-Galerkin method see [10]. Let u_{m} ⊆ D(Ω) be such that u_{0m} → u_{0} in H_{0}^{1}(Ω) and let (w_{j})_{j} ⊆ H_{0}^{1}(Ω) a special basis. We seek u to be the limit of a sequence (u_{m})_{m} such that

u_{m}(t) = Σ_{j=1}^{m}g_{jm}(t)w_{j},

where g_{jm} is the solution of the following ordinary differential system

\begin{cases}⟨u′_{m}, w_{j}⟩ + (u_{m}, w_{j}) = _{\frac{λ}{( ∫_{Ω} f(u_{m}) dx)^{2}}} ⟨f(u_{m}), w_{j}⟩, 1 ≤ j ≤ m, \\ u_{m}(0) = u_{om}. \end{cases}  (2.2)

It is easy to see that (2.2) has a unique solution u_{m} according to hypotheses (H_{1}) − (H_{2}) and Cartan’s existence theorem concerning ordinary differential equations (see [5]). This solution is shown to exist on a maximal interval [0; t_{m}[. The following estimates enable us to assert that it can be continued on the whole interval [0; T]. We shall denote by C_{i} different positive constants, depending on data, but not on m.

Lemma 2.2. For any τ > 0, there exists a constant c_{3}(τ), c_{4}(τ) such that

∥u_{m}(t)∥_{k_{0}+2} ≤ c_{3}(τ), ∀t ≥ τ,  (2.3)
∥u_{m}(t)∥_{∞} ≤ c_{4}(τ), ∀t ≥ τ.  (2.4)

Proof. (i) Multiplying the first equation of (2.2) by |u_{m}|^{k}g_{jm}, integrating on Ω, adding from j = 1 to m and using (H_{1}) − (H_{2}), yields

\frac{1}{k + 2} \frac{d}{dt}∥u_{m}∥_{k+2}^{k+2} + \frac{4}{(k + 2)^{2}} ∥∇|u_{m}|^{\frac{k}{2}} u_{m}∥_{2}^{2} ≤ c_{5}∥u_{m}∥_{k+α+2}^{k+α+2} + c_{6}.  (2.5)

By using well-known Sobolev and Gagliardo-Nirenberg’s inequalities, we have

∥u_{m}∥_{k_{0}+α+2}^{k_{0}+α+2} ≤ c_{7}∥u_{m}∥_{k_{0}+2}^{α}∥∇|u_{m}|^{γ}u_{m}∥_{2}^{2},  (2.6)

Thus, from (2.5) and (2.6), we obtain

\frac{1}{k_{0} + 2} \frac{d}{dt}∥u_{m}∥_{k_{0}+2}^{k_{0}+2} ≤ (c_{8}∥u_{m}∥_{k_{0}+2}^{α} − \frac{4}{(k_{0} + 2)^{2}} )∥∇|u_{m}|^{γ}u_{m}∥_{2}^{2} + c_{6}.  (2.7)

We shall make the following compatibility condition on u_{0}

∥u_{0}∥_{k_{0}+2} < (\frac{4}{c_{8}(k_{0} + 2)^{2}} )^{\frac{1}{α}} = d_{0}.  (2.8)

Then, there exists a small τ > 0 such that

∥u_{m}(t)∥_{k_{0}+2} < d_{0} for t ∈]0, τ[.  (2.9)

Hence

\frac{1}{k_{0} + 2} \frac{d}{dt}∥u_{m}∥_{k_{0}+2}^{k_{0}+2} + c_{9}∥∇|u_{m}|^{γ}u_{m}∥_{2}^{2} ≤ c_{6} ∀ 0 < t < τ.  (2.10)

By Poincaré’s inequality and after integrating, it follows that

∥u_{m}(t)∥_{k_{0}+2} ≤ c_{10}, ∀ 0 < t < τ,

Therefore, relation (2.3) is achieved by iterating successively the same process on intervals of periode τ such as [0, τ], [τ, t + τ], ..... (ii) By using Hôlder’s inequality, we get

∥u_{m}∥_{k+α+2}^{k+α+2} ≤ c_{11}∥u_{m}∥_{k+2}^{θ_{1}}∥u_{m}∥_{k_{0}+2}^{θ_{2}}∥u_{m}∥_{q}^{θ_{3}},  (2.11)

with θ_{1}, θ_{2} and θ_{3} satisfying

\frac{θ_{1}}{k + 2} + \frac{θ_{2}}{k_{0} + 2} + \frac{θ_{3}}{q} = 1 and θ_{1} + θ_{2} + θ_{3} = k + α + 2.

We require moreover

\frac{θ_{1}}{k + 2} + \frac{θ_{3}}{2(γ + 1)} = 1.

Using the boundedness of ∥u_{m}∥_{k_{0}+2}, the choice of q, Sobolev and Young’s inequalities and relation (2.11), we derive that

c_{5}∥u_{m}∥_{k+α+2}^{k+α+2} ≤ c_{12}∥u_{m}∥_{k+2}^{θ_{1}}∥∇|u_{m}|^{γ}u_{m}∥2^{\frac{θ_{3}}{γ+1}}

≤ c_{13}(k + 2)^{θ_{4}}∥u_{m}∥_{k+2}^{k+2} + \frac{2}{(k + 2)^{2}} ∥∇|u_{m}|^{γ}u_{m}∥_{2}^{2},

where θ_{4} is some positive constant. Hence (2.5) becomes

\frac{1}{k + 2} \frac{d}{dt}∥u_{m}∥_{k+2}^{k+2} + \frac{c_{14}}{(k + 2)^{2}} ∥∇|u_{m}|^{γ}u_{m}∥_{2}^{2} ≤ c_{15}(k + 2)^{θ_{4}}∥u_{m}∥_{k+2}^{k+2} + c_{5}.

Therefore, by applying lemma 4 ([7]) we conclude to (2.4).

Passage to the limit in (2.2) as m → ∞. Multiplying the jth equation of system (2.2) by g_{jm}(t), adding these equations for j = 1, ..., m and integrating with respect to the time variable, we deduce the existence of a subsequence of u_{m} such that

u_{m} → u weak star in L^{∞}(0, T; L^{2}(Ω)),
u_{m} → u weak in L^{2}(0, T; H_{0}^{1}(Ω)),
u_{mt} → u_{t} weak in L^{2}(0, T; H^{−1}(Ω)),
u_{m} → u strongly in L^{2}(0, T; L^{2}(Ω)) and a.e in Q_{T} .

Straightforward standard compactness arguments allow us to assert that u is a solution of problem (1.1) Uniqueness. Consider u_{1} and u_{2} two weak solutions of the problem (1.1) and define w = u_{1} − u_{2}. Substracting the equations verified by u_{1} and u_{2}, we obtain

\frac{dw}{dt} − △w = \frac{λ}{( ∫_{Ω} f(u_{1}) dx)^{2}} (f(u_{1}) − f(u_{2}))

+ λ\frac{( ∫_{Ω} f(u_{2}) − f(u_{1}) dx)( ∫_{Ω} f(u_{2}) + f(u_{1}) dx)}{( ∫_{Ω} f(u_{1}) dx)^{2}( ∫_{Ω} f(u_{2}) dx)^{2}}f(u_{2}). (2.12)

Taking the inner product of (2.12) by w and using (H_{1}) and (2.4), we get

\frac{1}{2} \frac{d}{dt}∥w(t)∥_{2}^{2} ≤ c_{16}∥w(t)∥_{2}^{2},

which implies that w = 0. Hence the solution is unique. □

b) We denote by {T (t), t ≥ 0} the continuous semi-group generated by (1.1) and defined by

\begin{aligned}T (t) : L^{∞}(Ω) & → & L^{∞}(Ω) \\ u_{0} & → & T (t)u_{0} = u(t, .). \end{aligned}

In this part, we refer to [12] for used concepts.

Theorem 2.3. Assume that (H_{1})−(H_{2}) are satisfied, Then T (t) possesses a maximal attractor which is bounded in H_{0}^{1}(Ω) ⋂ L^{∞}(Ω), compact and connected in L^{∞}(Ω).

Proof.

(i) (2.4) imply that there exists an absorbing set in L^{k}(Ω), 1 ≤ k ≤ ∞.

(ii) To obtain existence of absorbing sets in H_{0}^{1}(Ω) and the uniform compactness of T (t), multiply (2.2) by g′_{jm}(t), add from j = 1 to m and integrate on Ω by using Young’s inequality, it follows therefore that, for any t ≥ τ > 0

∫_{Ω}(\frac{∂u_{m}}{dt})^{2}dx + \frac{d}{dt}∥∇u_{m}∥_{2}^{2} ≤ c_{17}(τ),  (2.13)

which gives

\frac{d}{dt}∥∇u_{m}∥_{2}^{2} ≤ c_{17}(τ), ∀t ≥ τ > 0.  (2.14)

On the other hand, multiplying (2.2) by g_{jm}, adding and integrating on Ω × [t, t + τ] we get

∫_{t}^{t+τ} ∥∇u_{m}(s)∥_{2}^{2}ds ≤ c_{18}(τ), ∀t ≥ τ > 0.  (2.15)

Then, by the uniform Gronwall’s lemma (see [12], p.89) and the lower semicontinuity of the norm, we have

∥∇u(t)∥_{2}^{2} ≤ c_{19}(τ), ∀t ≥ τ.  (2.16)

Therefore, the open ball B(0, c_{19}(τ)) is an absorbing set in H_{0}^{1}(Ω). Hence, by theorem (1.1)( [12], p.23), we conclude to the results of theorem (2.3).

Theorem 2.4. We suppose (H_{1}) − (H_{2}) and

(H3) f ∈ C^{1}(ℝ).

Then, we have

y(t) ≡ ∥u_{t}∥^{2} ≤ c_{20}(τ), for any t ≥ τ > 0.

Proof. Differentiating equation (1.1) with respect to time(the justification of the formal derivatives can be done as in [5]), we get

u_{tt} − △u_{t} = \frac{λf′(u)u_{t}}{( ∫_{Ω} f(u) dx)^{2}} − 2λf(u)\frac{∫_{Ω} f′(u)u_{t} dx}{( ∫_{Ω} f(u) dx)^{3}} .  (2.17)

Multiplying (2.17) by u_{t}, integrating over Ω and using the L^{∞} estimate of u and Hôlder’s inequality, yields

\frac{1}{2}y′(t) ≤ c_{21}(τ)y(t).  (2.18)

On the other hand, taking the scalar product of (1.1) with u_{t}, using Young’s inequality, integrating on [t, t + τ] and using estimate (2.16), then gives

∫_{t}^{t+τ} y(s)ds ≤ c_{23}(τ), for any t ≥ τ.  (2.19)

From (2.18) and the uniform Gronwall’s lemma, we have

y(t) ≤ c_{23}(τ), for any t ≥ τ.

Therefore,

u_{t} ∈ L^{∞}(τ, ∞, L^{2}(Ω)).

By (1.1), we then get

−△u ∈ L^{∞}(τ, ∞, L^{2}(Ω)),

that is, u(t) is in a bounded subset of H^{2}(Ω).

Hence the existence of an absorbing set in H^{2}(Ω) is shown.

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Abderrahmane El Hachimi UFR Mathématiques Appliquées et Industrielles Faculté des Sciences Université Chouaib Doukkali B.P 20, El Jadida - Maroc e-mail adress: elhachimi@ucd.ac.ma; aelhachi@yahoo.fr Moulay Rchid Sidi Ammi UFR Mathématiques Appliquées et Industrielles Faculté des Sciences Université Chouaib Doukkali B.P 20, El Jadida - Maroc e-mail adress: sidiammi@hotmail.com; rachidsidiammi@yahoo.fr

(Received October 27, 2003; Revised version received January 16, 2005)