Electronic Journal of Qualitative Theory of Differential Equations 2007, No. 24, 1-18; http://www.math.u-szeged.hu/ejqtde/

EXISTENCE OF A SOLUTION FOR A CLASS OF NONLINEAR PARABOLIC SYSTEMS

HICHAM REDWANE

Faculté des Sciences Juridiques, Economiques et Sociales. Université Hassan 1, B.P. 784, Settat. Morocco

Abstract. An existence result of a solution for a class of nonlinear parabolic systems is established. The data belong to L^{1} and no growth assumption is made on the nonlinearities.

1. Introduction

In the present paper we establish an existence result of a renormalized solution for a class of nonlinear parabolic systems of the type

\frac{∂u}{∂t} − div(a(x, u, ∇u) + Φ(u)) + f_{1}(x, u, v) = 0 in (0, T) × Ω ; (1.1)

\frac{∂v}{∂t} − div(a(x, v, ∇v) + Φ(v)) + f_{2}(x, u, v) = 0 in (0, T) × Ω ; (1.2)

u = v = 0 on (0, T) × ∂Ω ; (1.3)

u(t = 0) = u_{0} in Ω. (1.4)

v(t = 0) = v_{0} in Ω. (1.5)

In Problem (1.1)-(1.5) the framework is the following : Ω is a bounded domain of R^{N}, (N ≥ 1), T is a positive real number while the data u_{0} and v_{0} in L^{1}(Ω). The operator −div(a(x, u, Du)) is a Leray-Lions operator which is coercive and which grows like |Du|^{p−1} with respect to Du, but which is not restricted by any growth condition with respect to u (see assumptions (2.1), (2.2), (2.3) and (2.4) of Section 2.). The function Φ, f_{1} and f_{2} are just assumed to be continuous on ℝ.

When Problem (1.1)-(1.5) is investigated there is difficulty is due to the facts that the data u_{0} and v_{0} only belong to L^{1} and the function a(x, u, Du), Φ(u), f_{1}(x, u, v) and f_{2}(x, u, v) does not belong (L_{loc}^{1}((0, T) × Ω))^{N} in general, so that proving existence of a weak solution (i.e. in the distribution meaning) seems to be an arduous task. To overcome this difficulty we use in this paper the framework of renormalized solutions. This notion was introduced by Lions and Di Perna [22] for the study of Boltzmann equation (see also P.-L. Lions [17] for a few applications to fluid mechanics models). This notion was then adapted to elliptic vesion of (1.1), (1.2), (1.3) in Boccardo, J.-L. Diaz, D. Giachetti, F. Murat [10], in P.-L. Lions and F. Murat [19] and F. Murat [19], [20]. At the same time the equivalent notion of

EJQTDE, 2007 No. 24, p. 1 entropy solutions has been developed independently by Bénilan and al. [2] for the study of nonlinear elliptic problems.

1991 Mathematics Subject Classification. Primary 47A15; Secondary 46A32, 47D20.

Key words and phrases. Nonlinear parabolic systems, Existence. Renormalized solutions.

As far as the parabolic equation case (1.1)-(1.5), (with, f_{i}(x, u, v) = f ∈ L^{1}(Ω × (0, T))) is concerned and still in the framework of renormalized solutions, the existence and uniqueness has been proved in D. Blanchard, F. Murat and H. Redwane [5] (see also A. Porretta [21] and H. Redwane [23]) in the case where f_{i}(x, u, v) is replaced by f + div(g) (where g belong L^{p′}(Q)^{N}). In the case where a(t, x, s, ξ) is independant of s, Φ = 0 and g = 0, existence and uniqueness has been established in D. Blanchard [3] ; D. Blanchard and F. Murat [4], and in the case where a(t, x, s, ξ) is independent of s and linear with respect to ξ, existence and uniqueness has been established in D. Blanchard and H. Redwane [7].

In the case where Φ = 0 and where the operator △_{p}u = div(|∇u|^{p−2}∇u) p-Laplacian replaces a nonlinear term div(a(x, s, ξ)), existence of a solution for nonlinear parabolic systems (1.1)-(1.5) is investigated in El Ouardi, A. El Hachimi ( [14] [15]), in Marion [18] and in A. Eden and all [1] (see also L. Dung [12]), where an existence result of as (usual) weak solution is proved.

With respect to the previous ones, the originality of the present work lies on the noncontrolled growth of the function a(x, s, ξ) with respect to s, and the function Φ, f_{1} and f_{2} are just assumed to be continuous on ℝ, and u_{0}, v_{0} are just assumed belong to L^{1}(Ω).

The paper is organized as follows : Section 2 is devoted to specify the assumptions on a(x, s, ξ), Φ, f_{1}, f_{2}, u_{0} and v_{0} needed in the present study and gives the definition of a renormalized solution of (1.1)-(1.5). In Section 3 (Theorem 3.0.4) we establish the existence of such a solution.

2. Assumptions on the data and definition of a renormalized solution

Throughout the paper, we assume that the following assumptions hold true : Ω is a bounded open set on ℝ^{N} (N ≥ 1), T > 0 is given and we set Q = Ω × (0, T), for i = 1, 2

(2.1) a : Ω × ℝ × ℝ^{N} → ℝ^{N} is a Carathéodory function,

a(x, s, ξ).ξ ≥ α|ξ|^{p} (2.2)

for almost every x ∈ Ω, for every s ∈ ℝ, for every ξ ∈ ℝ^{N}, where α > 0 given real number. For any K > 0, there exists β_{K} > 0 and a function C_{K} in L^{p′}(Ω) such that

|a(x, s, ξ)| ≤ C_{K}(x) + β_{K}|ξ|^{p−1} (2.3)

for almost every x ∈ Ω, for every s such that |s| ≤ K, and for every ξ ∈ ℝ^{N}

[a(x, s, ξ) − a(x, s, ξ′)][ξ − ξ′] ≥ 0, (2.4)

for any s ∈ ℝ, for any (ξ, ξ′) ∈ ℝ^{2N} and for almost every x ∈ Ω.

(2.5) Φ : ℝ → ℝ^{N} is a continuous function EJQTDE, 2007 No. 24, p. 2 For i = 1, 2

(2.6) f_{i} : Ω × ℝ × ℝ → ℝ is a Carathéodory function,

f_{1}(x, 0, s) = f_{2}(x, s, 0) = 0 a.e. x ∈ Ω, ∀s ∈ ℝ.

For almost every x ∈ Ω, for every s_{1}, s_{2} ∈ ℝ :

sign(s_{1})f_{1}(x, s_{1}, s_{2}) ≥ 0 and sign(s_{2})f_{2}(x, s_{1}, s_{2}) ≥ 0 (2.7)

For any K > 0, there exists σ_{K} > 0 and a function F_{K} in L^{1}(Ω) such that

|f_{1}(x, s_{1}, s_{2})| ≤ F_{K}(x) + σ_{K} |s_{2}| (2.8)

for almost every x ∈ Ω, for every s_{1} such that |s_{1}| ≤ K, and for every s_{2} ∈ ℝ. For any K > 0, there exists λ_{K} > 0 and a function G_{K} in L^{1}(Ω) such that

|f_{2}(x, s_{1}, s_{2})| ≤ G_{K}(x) + λ_{K} |s_{1}| (2.9)

for almost every x ∈ Ω, for every s_{2} such that |s_{2}| ≤ K, and for every s_{1} ∈ ℝ.

(u_{0}, v_{0}) ∈ L^{1}(Ω) × L^{1}(Ω) (2.10)

Remark 2.0.1. As already mentioned in the introduction Problem (1.1)-(1.5) does not admit a weak solution under assumptions (2.1)-(2.10) (even when f_{1} = f_{2} ≡ 0) since the growths of a(u, Du) and Φ(u) are not controlled with respect to u (so that these fields are not in general defined as distributions, even when u belongs L^{p}(0, T; W_{0}^{1,p}(Ω))).

Throughout this paper and for any non negative real number K we denote by T_{K}(r) = min(K, max(r, −K)) the truncation function at height K. For any measurable subset E of Q, we denote by meas(E) the Lebesgue measure of E. For any measurable function v defined on Q and for any real number s, χ_{{v<s}} (respectively, χ_{{v=s}}, χ_{{v>s}}) is the characteristic function of the set {(x, t) ∈ Q ; v(x, t) < s} (respectively, {(x, t) ∈ Q ; v(x, t) = s}, {(x, t) ∈ Q ; v(x, t) > s}). The definition of a renormalized solution for Problem (1.1)-(1.5) can be stated as follows.

Definition 2.0.2. A couple of functions (u, v) defined on Q is called a renormalized solution of Problem (1.1)-(1.5) if u and v satisfy :

(2.11) (T_{K}(u), T_{K}(v)) ∈ L^{p}(0, T; W_{0}^{1,p}(Ω))^{2} and (u, v) ∈ L^{∞}(0, T; L^{1}(Ω))^{2} ;

for any K ≥ 0.

(2.12) ∫_{{(t,x)∈Q ; n≤|u(x,t)|≤n+1}} a(x, u, Du)Du dx dt ⟶ 0 as n → +∞ ; ;

(2.13) ∫_{{(t,x)∈Q ; n≤|v(x,t)|≤n+1}} a(x, v, Dv)Dv dx dt ⟶ 0 as n → +∞ ;

and if, for every function S in W^{2,∞}(ℝ) which is piecewise C^{1} and such that S′ has a compact support, we have

(2.14)  \frac{∂S(u)}{∂t} − div(S′(u)a(x, u, Du)) + S′′(u)a(x, u, Du)Du
− div(S′(u)Φ(u)) + S′′(u)Φ(u)Du + f_{1}(x, u, v)S′(u) = 0 in D′(Q) ;

EJQTDE, 2007 No. 24, p. 3 and

(2.15)  \frac{∂S(v)}{∂t} − div(S′(v)a(x, v, Dv)) + S′′(v)a(x, v, Dv)Dv
− div(S′(v)Φ(v)) + S′′(v)Φ(v)Dv + f_{2}(x, u, v)S′(v) = 0 in D′(Q) ;
(2.16)  S(u)(t = 0) = S(u_{0}) and S(v)(t = 0) = S(v_{0}) in Ω.

The following remarks are concerned with a few comments on definition 2.0.2.

Remark 2.0.3. Equation (2.14) (and (2.15)) is formally obtained through pointwise multiplication of equation (1.1) by S′(u) (and equation (1.2) by S′(v)). Note that in definition 2.0.2, Du is not defined even as a distribution, but that due to (2.11) each term in (2.14) (and (2.15)) has a meaning in L^{1}(Q) + L^{p′}(0, T; W^{−1,p′}(Ω)).

Indeed if K is such that suppS′ ⊂ [−K, K], the following identifications are made in (2.14) (and in (2.15)) :

⋆ S(u) belongs to L^{∞}(Q) since S is a bounded function.

⋆ S′(u)a(u, Du) identifies with S′(u)a(T_{K}(u), DT_{K}(u)) a.e. in Q. Since indeed |T_{K}(u)| ≤ K a.e. in Q, assumptions (2.1) and (2.3) imply that

|a(T_{K}(u), DT_{K}(u))| ≤ C_{K}(t, x) + β_{K}|DT_{K}(u)|^{p−1} a.e. in Q.

As a consequence of (2.11) and of S′(u) ∈ L^{∞}(Q), it follows that

S′(u)a(T_{K}(u), DT_{K}(u)) ∈ (L^{p′}(Q))^{N}.

⋆ S′′(u)a(u, Du)Du identifies with S′′(u)a(T_{K}(u), DT_{K}(u))DT_{K}(u) and in view of (2.1), (2.3) and (2.11) one has

S′′(u)a(T_{K}(u), DT_{K}(u))DT_{K}(u) ∈ L^{1}(Q).

⋆ S′(u)Φ(u) and S′′(u)Φ(u)Du respectively identify with S′(u)Φ(T_{K}(u)) and S′′(u)Φ(T_{K}(u))DT_{K}(u). Due to the properties of S and (2.5), the functions S′, S′′ and Φ∘T_{K} are bounded on ℝ so that (2.11) implies that S′(u)Φ(T_{K}(u)) ∈ (L^{∞}(Q))^{N}, and S′′(u)Φ(T_{K}(u))DT_{K}(u) ∈ L^{p}(Q).

⋆ S′(u)f_{1}(x, u, v) identifies with S′(u)f_{1}(x, T_{K}(u), v) a.e. in Q. Since indeed |T_{K}(u)| ≤ K a.e. in Q, assumptions (2.8) imply that

|f_{1}(x, T_{K}(u), v)| ≤ F_{K}(x) + σ_{K} |v| a.e. in Q.

As a consequence of (2.11) and of S′(u) ∈ L^{∞}(Q), it follows that

S′(u)f_{1}(x, T_{K}(u), v) ∈ L^{1}(Q).

The above considerations show that equation (2.14) takes place in D′(Q) and that \frac{∂S(u)}{∂t} belongs to L^{p′}(0, T; W^{−1,p′}(Ω))+L^{1}(Q), which in turn implies that \frac{∂S(u)}{∂t} belongs to L^{1}(0, T; W^{−1,s}(Ω)) for all s < inf(p′, \frac{N}{N−1}). It follows that S(u) belongs to C^{0}([0, T]; W^{−1,s}(Ω)) so that the initial condition (2.16) makes sense. The same holds also for v.

EJQTDE, 2007 No. 24, p. 4

3. Existence result

This section is devoted to establish the following existence theorem.

Theorem 3.0.4. Under assumptions (2.1)-(2.10) there exists at least a renormalized solution (u, v) of Problem (1.1)-(1.5).

Proof. of Theorem 3.0.4. The proof is divided into 9 steps. In Step1, we introduce an approximate problem. Step 2 is devoted to establish a few a priori estimates. In Step 3, the limit (u, v) of the approximate solutions (u^{ε}, v^{ε}) is introduced and is shown of (u, v) belongs to L^{∞}(0, T; L^{1}(Ω))^{2} and to satisfy (2.11). In Step 4, we define a time regularization of the field (T_{K}(u), T_{K}(v)) and we establish Lemma 3.0.5, which a allows us to control the parabolic contribution that arises in the monotonicity method when passing to the limit. Step 5 is devoted to prove that an energy estimate (Lemma 3.0.6) which is a key point for the monotonicity arguments that are developed in Step 6 and Step 7. In Step 8, we prove that u satisfies (2.12) and v satisfies (2.13). At last, Step 9 is devoted to prove that (u, v) satisfies (2.14), (2.15) and (2.16) of definition 2.0.2 □

⋆ Step 1. Let us introduce the following regularization of the data :

(3.1)  a_{ε}(x, s, ξ) = a(x, T_{\frac{1}{ε}} (s), ξ) a.e. in Ω, ∀s ∈ ℝ, ∀ξ ∈ ℝ^{N} ;
(3.2)  Φ_{ε} is a lipschitz continuous bounded function from ℝ into ℝ^{N}

such that Φ_{ε} uniformly converges to Φ on any compact subset of ℝ as ε tends to 0.

(3.3)  f_{1}^{ε}(x, s_{1}, s_{2}) = f_{1}(x, T_{\frac{1}{ε}} (s_{1}), s_{2}) a.e. in Ω, ∀s_{1}, s_{2} ∈ ℝ ;
(3.4)  f_{2}^{ε}(x, s_{1}, s_{2}) = f_{2}(x, s_{1}, T_{\frac{1}{ε}} (s_{2})) a.e. in Ω, ∀s_{1}, s_{2} ∈ ℝ ;
(3.5)  u_{0}^{ε} and v_{0}^{ε} are a sequence of C_{0}^{∞}(Ω)- functions such that
u_{0}^{ε} → u_{0} in L^{1}(Ω)  and  v_{0}^{ε} → v_{0} in L^{1}(Ω)

as ε tends to 0.

Let us now consider the following regularized problem.

\frac{∂u^{ε}}{∂t} − div(a_{ε}(x, u^{ε}, ∇u^{ε}) + Φ_{ε}(u^{ε})) + f_{1}^{ε}(x, u^{ε}, v^{ε}) = 0 in Q ; (3.6)

\frac{∂v^{ε}}{∂t} − div(a_{ε}(x, v^{ε}, ∇v^{ε}) + Φ_{ε}(v^{ε})) + f_{2}^{ε}(x, u^{ε}, v^{ε}) = 0 in Q ; (3.7)

u^{ε} = v^{ε} = 0 on (0, T) × ∂Ω ; (3.8)

u^{ε}(t = 0) = u_{0}^{ε} in Ω. (3.9)

v^{ε}(t = 0) = v_{0}^{ε} in Ω. (3.10)

In view of (2.3), (2.8) and (2.9), a_{ε}, f_{1}^{ε} and f_{1}^{ε} satisfiy : there exists C_{ε} ∈ L^{p′}(Ω), F_{ε} ∈ L^{1}(Ω), G_{ε} ∈ L^{1}(Ω) and β_{ε} > 0, σ_{ε} > 0, λ_{ε} > 0, such that

|a_{ε}(x, s, ξ)| ≤ C_{ε}(x) + β_{ε}|ξ|^{p−1} a.e. in x ∈ Ω, ∀s ∈ ℝ, ∀ξ ∈ ℝ^{N}.
|f_{1}^{ε}(x, s_{1}, s_{2})| ≤ F_{ε}(x) + σ_{ε} |s_{2}| a.e. in x ∈ Ω, ∀s_{1}, s_{2} ∈ ℝ.

EJQTDE, 2007 No. 24, p. 5 and

|f_{2}^{ε}(x, s_{1}, s_{2})| ≤ G_{ε}(x) + λ_{ε} |s_{1}| a.e. in x ∈ Ω, ∀s_{1}, s_{2} ∈ ℝ.

As a consequence, proving existence of a weak solution (u^{ε}, v^{ε}) ∈ (L^{p}(0, T; W_{0}^{1,p}(Ω)))^{2} of (3.6)-(3.10) is an easy task (see e.g. [1], [14] and [15]).

⋆ Step 2. The estimates derived in this step rely on usual techniques for problems of type (3.6)-(3.10) and we just sketch the proof of them (the reader is referred to [3], [4], [7], [9], [5], [6] or to [10], [19], [20] for elliptic versions of (3.6)-(3.10)).

Using T_{K}(u^{ε}) as a test function in (3.6) leads to

(3.11)  ∫_{Ω} \overline{T} _{K}^{ε}(u^{ε})(t) dx + ∫_{0}^{t} ∫_{Ω} a_{ε}(x, u^{ε}, Du^{ε})DT_{K}(u^{ε}) dx ds

+ ∫_{0}^{t} ∫_{Ω} Φ_{ε}(u^{ε})DT_{K}(u^{ε}) dx ds + ∫_{0}^{t} ∫_{Ω} f_{1}^{ε}(x, u^{ε}, v^{ε})T_{K}(u^{ε}) dx ds = ∫_{Ω} \overline{T} _{K}^{ε}(u_{0}^{ε}) dx

for almost every t in (0, T), and where

\overline{T} _{K}^{ε}(r) = ∫_{0}^{r} T_{K}(s) ds = \{ \begin{matrix}\frac{r^{2}}{2} & if |r| ≤ K \\ K |r| − \frac{K^{2}}{2} & if |r| ≥ K \end{matrix}

The Lipschitz character of Φ_{ε}, Stokes formula together with the boundary condition (3.8) make it possible to obtain

(3.12)  ∫_{0}^{t} ∫_{Ω} Φ_{ε}(u^{ε})DT_{K}(u^{ε}) dx ds = 0,

for almost any t ∈ (0, T).

Since a_{ε} satisfies (2.2), f_{1}^{ε} satisfies (2.7) and the properties of \overline{T} _{K}^{ε} and u_{0}^{ε}, permit to deduce from (3.11) that

(3.13)  T_{K}(u^{ε}) is bounded in L^{p}(0, T; W_{0}^{1,p}(Ω))

independently of ε for any K ≥ 0.

Proceeding as in [4], [7] [5] and [6] that for any S ∈ W^{2,∞}(ℝ) such that S′ is compact (suppS′ ⊂ [−K, K])

(3.14)  S(u^{ε}) is bounded in L^{p}(0, T; W_{0}^{1,p}(Ω))

and

(3.15)  \frac{∂S(u^{ε})}{∂t} is bounded in L^{1}(Q) + L^{p′}(0, T; W^{−1,p′}(Ω))

independently of ε, as soon as ε < \frac{1}{K}.

Now for fixed K > 0 : a_{ε}(T_{K}(u^{ε}), DT_{K}(u^{ε})) = a(T_{K}(u^{ε}), DT_{K}(u^{ε})) a.e. in Q as soon as ε < \frac{1}{K}, while assumption (2.3) gives

|a_{ε}(T_{K}(u^{ε}), DT_{K}(u^{ε}))| ≤ C_{K}(x) + β_{K}|DT_{K}(u^{ε})|^{p−1}

where β_{K} > 0 and C_{K} ∈ L^{p′}(Q). In view (3.13), we deduce that,

(3.16)  a(T_{K}(u^{ε}), DT_{K}(u^{ε})) is bounded in (L^{p′}(Q))^{N}.

independently of ε for ε < \frac{1}{K}.

EJQTDE, 2007 No. 24, p. 6

For any integer n ≥ 1, consider the Lipschitz continuous function θ_{n} defined through

θ_{n}(r) = T_{n+1}(r) − T_{n}(r)

Remark that ∥θ_{n}∥_{L^{∞}(ℝ)} ≤ 1 for any n ≥ 1 and that θ_{n}(r) → 0 for any r when n tends to infinity.

Using the admissible test function θ_{n}(u^{ε}) in (3.6) leads to

(3.17)  ∫_{Ω} \overline{θ_{n}}(u^{ε})(t) dx + ∫_{0}^{t} ∫_{Ω} a_{ε}(u^{ε}, Du^{ε})Dθ_{n}(u^{ε}) dx ds

+ ∫_{0}^{t} ∫_{Ω} Φ_{ε}(u^{ε})Dθ_{n}(u^{ε}) dx ds + ∫_{0}^{t} ∫_{Ω} f_{1}^{ε}(x, u^{ε}, v^{ε})θ_{n}(u^{ε}) dx ds = ∫_{Ω} \overline{θ_{n}}(u_{0}^{ε}) dx,

for almost any t in (0, T) and where \overline{θ_{n}}(r) = ∫_{0}^{r} θ_{n}(s) ds.

The Lipschitz character of Φ_{ε}, Stokes formula together with boundary condition (3.8) allow to obtain

(3.18)  ∫_{0}^{t} ∫_{Ω} Φ_{ε}(u^{ε})Dθ_{n}(u^{ε}) dx ds = 0.

Since \overline{θ_{n}}(.) ≥ 0, f_{1}^{ε} satisfies (2.7), we have

(3.19)  ∫_{0}^{t} ∫_{Ω} a(u^{ε}, Du^{ε})Dθ_{n}(u^{ε}) dx ds ≤ ∫_{Ω} \overline{θ_{n}}(u_{0}^{ε}) dx,

for almost t ∈ (0, T) and for ε < \frac{1}{n+1}. ⋆ Step 3. Arguing again as in [4], [7] [5] and [6] estimates (3.14), (3.15) imply that, for a subsequence still indexed by ε,

(3.20)  u^{ε} converges almost every where to u in Q,

and with the help of (3.13),

(3.21)  T_{K}(u^{ε}) converges weakly to T_{K}(u) in L^{p}(0, T; W_{0}^{1,p}(Ω)),
(3.22)  θ_{n}(u^{ε}) ⇀ θ_{n}(u) weakly in L^{p}(0, T; W_{0}^{1,p}(Ω))
(3.23)  a_{ε}(T_{K}(u^{ε}), DT_{K}(u^{ε})) ⇀ X_{K} weakly in (L^{p′}(Q))^{N}.

The same holds for v^{ε} :

(3.24)  v^{ε} converges almost every where to v in Q,
(3.25)  T_{K}(v^{ε}) converges weakly to T_{K}(v) in L^{p}(0, T; W_{0}^{1,p}(Ω)),
(3.26)  θ_{n}(v^{ε}) ⇀ θ_{n}(v) weakly in L^{p}(0, T; W_{0}^{1,p}(Ω))
(3.27)  a_{ε}(T_{K}(v^{ε}), DT_{K}(v^{ε})) ⇀ Y_{K} weakly in (L^{p′}(Q))^{N}

as ε tends to 0 for any K > 0 and any n ≥ 1 and where for any K > 0, X_{K}, Y_{K} belongs to (L^{p′}(Q))^{N}.

EJQTDE, 2007 No. 24, p. 7

We now establish that u and v belongs to L^{∞}(0, T; L^{1}(Ω)). To this end, recalling (2.7), (3.5), (3.12) and (3.20) allows to pass to the limit-inf in (3.11) as ε tends to 0 and to obtain

∫_{Ω} \overline{T_{K}}(u)(t) dx ≤ K ∥u_{0}∥_{L^{1}(Ω)}.

Due to the definition of \overline{T_{K}}, we deduce from the above inequality that

K ∫_{Ω} |u(x, t)| dx ≤ \frac{3K^{2}}{2} mes(Ω) + K ∥u_{0}∥_{L^{1}(Ω)}

for almost any t ∈ (0, T), which shows that u belongs to L^{∞}(0, T; L^{1}(Ω)). The same holds for v belongs to L^{∞}(0, T; L^{1}(Ω)). We are now in a position to exploit (3.19). Due to the definition of θ_{n} we have

a(u^{ε}, Du^{ε})Dθ_{n}(u^{ε}) = a(u^{ε}, Du^{ε})Du^{ε}χ_{{n≤|u^{ε}|≤n+1}} ≥ α |Dθ_{n}(u^{ε})|^{p} a.e. in Q

Inequality (3.19), the weak convergence (3.22) and the pointwise convergence of u_{0}^{ε} to u_{0} then imply that

α ∫_{Q} |Dθ_{n}(u)|^{p} dx dt ≤ ∫_{Ω} \overline{θ}_{n}(u_{0}) dx.

Since θ_{n} and \overline{θ_{n}} both converge to zero everywhere as n goes to zero while

|θ_{n}(u)| ≤ 1 and |\overline{θ}_{n}(u)| ≤ |u_{0}| ∈ L^{1}(Ω)

the Lebesgue’s convergence theorem permits to conclude that

(3.28)  lim_{n→+∞} ∫_{{n≤|u|≤n+1}} |Du|^{p} dx dt = 0.

and

(3.29)  lim_{n→+∞} \overline{lim_{ε→0}} ∫_{\{n≤|u^{ε}|≤n+1\}} a_{ε}(u^{ε}, Du^{ε})Du^{ε} dx dt = 0.

⋆ Step 4. This step is devoted to introduce for K ≥ 0 fixed, a time regularization of the function T_{K}(u) in order to perform the monotonicity method which will be developed in Step 5 and Step 6. This kind of regularization has been first introduced by R. Landes (see Lemma 6 and Proposition 3, p. 230 and Proposition 4, p. 231 in [16]). More recently, it has been exploited in [8] and [11] to solve a few nonlinear evolution problems with L^{1} or measure data.

This specific time regularization of T_{K}(u) (for fixed K ≥ 0) is defined as follows. Let (v_{0}^{μ})_{μ} be a sequence of functions defined on Ω such that

(3.30)  v_{0}^{μ} ∈ L^{∞}(Ω) ∩ W_{0}^{1,p}(Ω) for all μ > 0,
(3.31)  ∥v_{0}^{μ}∥_{L^{∞}(Ω)} ≤ K ∀μ > 0,

(3.32)  v_{0}^{μ} → T_{K}(u_{0}) a.e. in Ω and \frac{1}{μ}∥Dv_{0}^{μ}∥_{L^{p}(Ω)}^{p} → 0, as μ → +∞.

EJQTDE, 2007 No. 24, p. 8 Existence of such a subsequence (v_{0}^{μ})_{μ} is easy to establish (see e.g. [13]). For fixed K ≥ 0 and μ > 0, let us consider the unique solution T_{K}(u)_{μ} ∈ L^{∞}(Q) ∩ L^{p}(0, T; W_{0}^{1,p}(Ω)) of the monotone problem :

(3.33)  \frac{∂T_{K}(u)_{μ}}{∂t} + μ(T_{K}(u)_{μ} − T_{K}(u)) = 0 in D′(Q).
(3.34)  T_{K}(u)_{μ}(t = 0) = v_{0}^{μ} in Ω.

Remark that due to (3.33), we have for μ > 0 and K ≥ 0,

(3.35)  \frac{∂T_{K}(u)_{μ}}{∂t} ∈ L^{p}(0, T; W_{0}^{1,p}(Ω)).

The behavior of T_{K}(u)_{μ} as μ → +∞ is investigated in [16] (see also [11] and [13]) and we just recall here that (3.30)-(3.34) imply that

(3.36)  T_{K}(u)_{μ} → T_{K}(u) a.e. in Q ;

and in L^{∞}(Q) weak ⋆ and strongly in L^{p}(0, T; W_{0}^{1,p}(Ω)) as μ → +∞.

(3.37)  ∥T_{K}(u)_{μ}∥_{L^{∞}(Q)} ≤ max(∥T_{K}(u)∥_{L^{∞}(Q)} ; ∥v_{0}^{μ}∥_{L^{∞}(Ω)}) ≤ K

for any μ and any K ≥ 0.

The very definition of the sequence T_{K}(u)_{μ} for μ > 0 (and fixed K) allow to establish the following lemma

Lemma 3.0.5. Let K ≥ 0 be fixed. Let S be an increasing C^{∞}(ℝ)-function such that S(r) = r for |r| ≤ K and supp(S′) is compact. Then

\underline{lim_{μ→+∞}} lim_{ε→0} ∫_{0}^{T} ∫_{0}^{s} ⟨\frac{∂S(u^{ε})}{∂t} , (T_{K}(u^{ε}) − (T_{K}(u))_{μ})⟩ dt ds ≥ 0

where ⟨ , ⟩ denotes the duality pairing between L^{1}(Ω) + W^{−1,p′}(Ω) and L^{∞}(Ω) ∩ W_{0}^{1,p}(Ω).

Proof of Lemma 3.0.5 : The Lemma is proved in [5] (see Lemma 1, p.341).

⋆ Step 5. In this step we prove the following lemma which is the key point in the monotonocity arguments that will be developed in Step 6.

Lemma 3.0.6. The subsequence of u^{ε} defined is Step 3 satisfies for any K ≥ 0 (3.38)

\overline{lim_{ε→0}} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} a(u^{ε}, DT_{K}(u^{ε}))DT_{K}(u^{ε}) dx ds dt ≤ ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} X_{K}DT_{K}(u) dx ds dt

Proof of Lemma 3.0.6 : We first introduce a sequence of increasing C^{∞}(ℝ)-functions S_{n} such that, for any n ≥ 1

(3.39)  S_{n}(r) = r for |r| ≤ n,
(3.40)  suppS′_{n} ⊂ [−(n + 1), (n + 1)],
(3.41)  ∥S′′_{n}∥_{L^{∞}(ℝ)} ≤ 1.

Pointwise multiplication of (3.6) by S′_{n}(u^{ε}) (which is licit) leads to EJQTDE, 2007 No. 24, p. 9

(3.42)  \frac{∂S_{n}(u^{ε})}{∂t} − div(S_{n}(u^{ε})a_{ε}(x, u^{ε}, Du^{ε})) + S′′_{n}(u^{ε})a_{ε}(x, u^{ε}, Du^{ε})Du^{ε}

− div(Φ_{ε}(u^{ε})S′_{n}(u^{ε})) + S′′_{n}(u^{ε})Φ_{ε}(u^{ε}) + f_{1}^{ε}(x, u^{ε}, v^{ε})S′_{n}(u^{ε}) = 0 in D′(Q).

We use the sequence T_{K}(u)_{μ} of approximations of T_{K}(u) defined by (3.33), (3.34) of Step 4 and plug the test function T_{K}(u^{ε}) − T_{K}(u)_{μ} (for ε > 0 and μ > 0) in (3.42). Through setting, for fixed K ≥ 0,

(3.43)  W_{μ}^{ε} = T_{K}(u^{ε}) − T_{K}(u)_{μ}

we obtain upon integration over (0, t) and then over (0, T) :

(3.44)

∫_{0}^{T} ∫_{0}^{t} ⟨\frac{∂S_{n}(u^{ε})}{∂t} , W_{μ}^{ε}⟩ ds dt + ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′_{n}(u^{ε})a_{ε}(x, u^{ε}, Du^{ε})DW_{μ}^{ε} dx ds dt

+ ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′′_{n}(u^{ε})W_{μ}^{ε}a_{ε}(x, u^{ε}, Du^{ε})Du^{ε} dx ds dt
+ ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} Φ_{ε}(u^{ε})S′_{n}(u^{ε})DW_{μ}^{ε} dx ds dt
+ ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′′_{n}(u^{ε})W_{μ}^{ε}Φ_{ε}(u^{ε})Du^{ε} dx ds dt
+ ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} f_{1}^{ε}(x, u^{ε}, v^{ε})S′_{n}(u^{ε})W_{μ}^{ε} dx ds dt = 0

In the following we pass to the limit in (3.44) as ε tends to 0, then μ tends to +∞ and then n tends to +∞, the real number K ≥ 0 being kept fixed. In order to perform this task we prove below the following results for fixed K ≥ 0 :

(3.45)  \underline{lim_{μ→+∞}} lim_{ε→0} ∫_{0}^{T} ∫_{0}^{t} ⟨\frac{∂S_{n}(u^{ε})}{∂t} , W_{μ}^{ε}⟩ ds dt ≥ 0 for any n ≥ K,

(3.46) lim_{μ→+∞} lim_{ε→0} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′_{n}(u^{ε})Φ_{ε}(u^{ε})DW_{μ}^{ε} dx ds dt = 0 for any n ≥ 1,

(3.47)  lim_{μ→+∞} lim_{ε→0} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′′_{n}(u^{ε})W_{μ}^{ε}Φ_{ε}(u^{ε})Du^{ε} dx ds dt = 0 for any n,

(3.48) lim_{n→+∞} \overline{lim_{μ→+∞}} \overline{lim_{ε→0}}| ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′′_{n}(u^{ε})W_{μ}^{ε}a_{ε}(u^{ε}, Du^{ε})Du^{ε} dx ds dt| = 0,

and

(3.49) lim_{μ→+∞} lim_{ε→0} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} f_{1}^{ε}(x, u^{ε}, v^{ε})S′_{n}(u^{ε})W_{μ}^{ε} dx ds dt = 0 for any n ≥ 1.

\underline{Proof of (3.45)}. In view of the definition (3.43) of W_{μ}^{ε}, lemma 3.0.5 applies with S = S_{n} for fixed n ≥ K. As a consequence (3.45) holds true.

EJQTDE, 2007 No. 24, p. 10 \underline{Proof of (3.46)}. For fixed n ≥ 1, we have

(3.50)  S′_{n}(u^{ε})Φ_{ε}(u^{ε})DW_{μ}^{ε} = S′_{n}(u^{ε})Φ_{ε}(T_{n+1}(u^{ε}))DW_{μ}^{ε}

a.e. in Q, and for all ε ≤ \frac{1}{n+1}, and where suppS′_{n} ⊂ [−(n + 1), n + 1].

Since S′_{n} is smooth and bounded, (2.5), (3.2) and (3.20) lead to

(3.51)  S′_{n}(u^{ε})Φ_{ε}(T_{n+1}(u^{ε})) → S′_{n}(u)Φ(T_{n+1}(u))

a.e. in Q and in L^{∞}(Q) weak ⋆, as ε tends to 0.

For fixed μ > 0, we have

(3.52)  W_{μ}^{ε} ⇀ T_{K}(u) − T_{K}(u)_{μ} weakly in L^{p}(0, T; W_{0}^{1,p}(Ω))

and a.e. in Q and in L^{∞}(Q) weak ⋆, as ε tends to 0.

As a consequence of (3.50), (3.51) and (3.52) we deduce that

(3.53)  lim_{ε→0} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′_{n}(u^{ε})Φ_{ε}(u^{ε})DW_{μ}^{ε} dx ds dt

= ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′_{n}(u)Φ(u)[DT_{K}(u) − DT_{K}(u)_{μ}] dx ds dt

for any μ > 0.

Appealing now to (3.36) and passing to the limit as μ → +∞ in (3.53) allows to conclude that (3.46) holds true.

\underline{Proof of (3.47)}. For fixed n ≥ 1, and by the same arguments that those that lead to (3.50), we have

S′′_{n}(u^{ε})Φ_{ε}(u^{ε})Du^{ε}W_{μ}^{ε} = S′′_{n}(u^{ε})Φ_{ε}(T_{n+1}(u^{ε}))DT_{n+1}(u^{ε})W_{μ}^{ε} a.e. in Q.

From (2.5), (3.2) and (3.20), it follows that for any μ > 0

lim_{ε→0} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′′_{n}(u^{ε})Φ_{ε}(u^{ε})W_{μ}^{ε} dx ds dt

= ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′′_{n}(u)Φ(u)[DT_{K}(u) − DT_{K}(u)_{μ}] dx ds dt

with the help of (3.36) passing to the limit, as μ tends to +∞, in the above equality leads to (3.47).

\underline{Proof of (3.48)}. For any n ≥ 1 fixed, we have suppS′′_{n} ⊂ [−(n + 1), −n] ∪ [n, n + 1]. As a consequence

| ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′′_{n}(u^{ε})a_{ε}(u^{ε}, Du^{ε})Du^{ε}W_{μ}^{ε} dx ds dt|
≤ T ∥S′′_{n}∥_{L^{∞}(ℝ)}∥W_{μ}^{ε}∥_{L^{∞}(Q)} ∫_{{n≤|u^{ε}|≤n+1}} a_{ε}(u^{ε}, Du^{ε})Du^{ε} dx dt,

for any n ≥ 1, and any μ > 0. The above inequality together with (3.37) and (3.41) make it possible to obtain

(3.54)  \overline{lim_{μ→+∞}} \overline{lim_{ε→0}}| ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′′_{n}(u^{ε})a_{ε}(u^{ε}, Du^{ε})Du^{ε}W_{μ}^{ε} dx ds dt|

EJQTDE, 2007 No. 24, p. 11

≤ C\overline{lim_{ε→0}} ∫_{\{n≤|u^{ε}|≤n+1\}} a_{ε}(u^{ε}, Du^{ε})Du^{ε} dx dt,

for any n ≥ 1, where C is a constant independent of n.

Appealing now to (3.29) permits to pass to the limit as n tends to +∞ in (3.54) and to establish (3.48).

\underline{Proof of (3.49)}. For fixed n ≥ 1, we have

f_{1}^{ε}(x, u^{ε}, v^{ε})S′_{n}(u^{ε}) = f_{1}(x, T_{n+1}(u^{ε}), v^{ε})

a.e. in Q, and for all ε ≤ \frac{1}{n+1}.

In view (2.8), (3.20) and (3.24), Lebesgue’s convergence theorem implies that for any μ > 0 and any n ≥ 1

lim_{ε→0} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} f_{1}^{ε}(x, u^{ε}, v^{ε})S′_{n}(u^{ε})W_{μ}^{ε} dx ds dt

= ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} f_{1}(x, u, v)S′_{n}(u)(T_{K}(u) − T_{K}(u)_{μ}) dx ds dt.

Now for fixed n ≥ 1, using (3.36) permits to pass to the limit as μ tends to +∞ in the above equality to obtain (3.49).

We now turn back to the proof of lemma 3.0.6, due to (3.44), (3.45), (3.46), (3.47), (3.48) and (3.49), we are in a position to pass to the lim-sup when ε tends to zero, then to the limit-sup when μ tends to +∞ and then to the limit as n tends to +∞ in (3.44). We obtain using the definition of W_{μ}^{ε} that for any K ≥ 0

lim_{n→+∞} \overline{lim_{μ→+∞}}\overline{lim_{ε→0}} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′_{n}(u^{ε})a_{ε}(u^{ε}, Du^{ε})(DT_{K}(u^{ε}) − DT_{K}(u)_{μ}) dx ds dt ≤ 0.

Since S′_{n}(u^{ε})a_{ε}(u^{ε}, Du^{ε})DT_{K}(u^{ε}) = a(u^{ε}, Du^{ε})DT_{K}(u^{ε}) for ε ≤ \frac{1}{K} and K ≤ n. The above inequality implies that for K ≤ n

(3.55)  \overline{lim_{ε→0}} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} a_{ε}(u^{ε}, Du^{ε})DT_{K}(u^{ε}) dx ds dt

≤ lim_{n→+∞} \overline{lim_{μ→+∞}} \overline{lim_{ε→0}} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′_{n}(u^{ε})a_{ε}(u^{ε}, Du^{ε})DT_{K}(u)_{μ} dx ds dt

The right hand side of (3.55) is computed as follows. In view (3.1) and (3.40), we have for ε ≤ \frac{1}{n+1}.

S′_{n}(u^{ε})a_{ε}(u^{ε}, Du^{ε}) = S′_{n}(u^{ε})a(T_{n+1}(u^{ε}), DT_{n+1}(u^{ε})) a.e. in Q.

Due to (3.23) it follows that for fixed n ≥ 1

S′_{n}(u^{ε})a_{ε}(u^{ε}, Du^{ε}) ⇀ S′_{n}(u)X_{n+1} weakly in L^{p′}(Q),

when ε tends to 0. The strong convergence of T_{K}(u)_{μ} to T_{K}(u) in L^{p}(0, T; W_{0}^{1,p}(Ω)) as μ tends to +∞, then allows to conclude that

(3.56)  lim_{μ→+∞} lim_{ε→0} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′_{n}(u^{ε})a_{ε}(u^{ε}, Du^{ε})DT_{K}(u)_{μ} dx ds dt

EJQTDE, 2007 No. 24, p. 12

= ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} S′_{n}(u)X_{n+1}DT_{K}(u) dx ds dt = ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} X_{n+1}DT_{K}(u) dx ds dt

as soon as K ≤ n, since S′_{n}(r) = 1 for |r| ≤ n. Now for K ≤ n we have

a(T_{n+1}(u^{ε}), DT_{n+1}(u^{ε}))χ_{{|u^{ε}|<K}} = a(T_{K}(u^{ε}), DT_{K}(u^{ε}))χ_{{|u^{ε}|<K}} a.e. in Q

Passing to the limit as ε tends to 0, we obtain

(3.57)  X_{n+1}χ_{{|u|<K}} = X_{K}χ_{{|u|<K}} a.e. in Q − {|u| = K} for K ≤ n.

As a consequence of (3.57) we have for K ≤ n

(3.58)  X_{n+1}DT_{K}(u) = X_{K}DT_{K}(u) a.e. in Q.

Recalling (3.55), (3.56) and (3.58) allows to conclude (3.38) holds true and the proof of lemma 3.0.6 is complete.

⋆ Step 6. In this step we prove the following monotonicity estimate :

Lemma 3.0.7. The subsequence of u^{ε} defined in step 3 satisfies for any K ≥ 0

(3.59)  lim_{ε→0} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} [a(T_{K}(u^{ε}), DT_{K}(u^{ε})) − a(T_{K}(u^{ε}), DT_{K}(u))]
[DT_{K}(u^{ε}) − DT_{K}(u)] dx ds dt = 0

Proof of Lemma 3.0.7. Let K ≥ 0 be fixed. The monotone character (2.4) of a(s, ξ) with respect to ξ implies that

(3.60)  ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} [a(T_{K}(u^{ε}), DT_{K}(u^{ε})) − a(T_{K}(u^{ε}), DT_{K}(u))]
[DT_{K}(u^{ε}) − DT_{K}(u)] dx ds dt ≥ 0,

To pass to the limit-sup as ε tends to 0 in (3.60), let us remark that (2.1), (2.3) and (3.20) imply that

a(T_{K}(u^{ε}), DT_{K}(u)) → a(T_{K}(u), DT_{K}(u)) a.e. in Q,

as ε tends to 0, and that

|a(T_{K}(u^{ε}), DT_{K}(u))| ≤ C_{K}(t, x) + β_{K}|DT_{K}(u)|^{p−1}

a.e. in Q, uniformly with respect to ε.

It follows that when ε tends to 0

(3.61)  a(T_{K}(u^{ε}), DT_{K}(u)) → a(T_{K}(u), DT_{K}(u)) strongly in (L^{p′}(Q))^{N}.

Using (3.38) of lemma (3.0.6), (3.21), (3.23) and (3.61) allow to pass to the lim-sup as ε tends to zero in (3.60) and to obtain (3.59) of lemma 3.0.7.

⋆ Step 7. In this step we identify the weak limit X_{K} and we prove the weak L^{1} convergence of the ”truncated” energy a(T_{K}(u^{ε}), DT_{K}(u^{ε}))DT_{K}(u^{ε}) as ε tends to 0.

EJQTDE, 2007 No. 24, p. 13 Lemma 3.0.8. For fixed K ≥ 0, we have as ε tends to 0

(3.62)  X_{K} = a(T_{K}(u^{ε}), DT_{K}(u^{ε})) a.e. in Q.

And as ε tends to 0 (3.63) a(T_{K}(u^{ε}), DT_{K}(u^{ε}))DT_{K}(u^{ε}) ⇀ a(T_{K}(u), DT_{K}(u))DT_{K}(u) weakly in L^{1}(Q).

Proof of Lemma (3.0.8). The proof is standard once we remark that for any K ≥ 0, any 0 < ε < \frac{1}{K} and any ξ ∈ ℝ^{N}

a_{ε}(T_{K}(u^{ε}), ξ) = a(T_{K}(u^{ε}), ξ) = a_{\frac{1}{K}} (T_{K}(u^{ε}), ξ) a.e. in Q

which together with (3.21), (3.61) makes it possible to obtain from (3.59) of lemma 3.0.7

(3.64)  lim_{ε→0} ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} a_{\frac{1}{K}} (T_{K}(u^{ε}), DT_{K}(u^{ε}))DT_{K}(u^{ε}) dx ds dt

= ∫_{0}^{T} ∫_{0}^{t} ∫_{Ω} σ_{K}DT_{K}(u) dx ds dt.

Since, for fixed K > 0, the function a_{\frac{1}{K}} (s, ξ) is continuous and bounded with respect to s, the usual Minty’s argument applies in view (3.21), (3.23), and (3.64). It follows that (3.62) holds true (the case K = 0 being trivial). In order to prove (3.63), we observe that the monotone character of a (with respect to ξ) and (3.59) give that for any K ≥ 0 and any T′ < T

(3.65) [a(T_{K}(u^{ε}), DT_{K}(u^{ε})) − a(T_{K}(u^{ε}), DT_{K}(u))][DT_{K}(u^{ε}) − DT_{K}(u)] → 0

strongly in L^{1}((0, T′) × Ω) as ε tends to 0.

Moreover (3.21), (3.23), (3.61) and (3.62) imply that

a(T_{K}(u^{ε}), DT_{K}(u^{ε}))DT_{K}(u) ⇀ a(T_{K}(u), DT_{K}(u))DT_{K}(u) weakly in L^{1}(Q),

a(T_{K}(u^{ε}), DT_{K}(u))DT_{K}(u^{ε}) ⇀ a(T_{K}(u), DT_{K}(u))DT_{K}(u) weakly in L^{1}(Q),

and

a(T_{K}(u^{ε}), DT_{K}(u))DT_{K}(u) ⟶ a(T_{K}(u), DT_{K}(u))DT_{K}(u) strongly in L^{1}(Q),

as tends to 0. Using the above convergence results in (3.65) shows that for any K ≥ 0 and any T′ < T

(3.66)  a(T_{K}(u^{ε}), DT_{K}(u^{ε}))DT_{K}(u^{ε}) ⇀ a(T_{K}(u), DT_{K}(u))DT_{K}(u)

weakly in L^{1}((0, T′) × Ω) as tends to 0.

Remark that for \overline{T} > T, we have (2.1), (2.2), (2.3), (2.4), (2.5), (2.6), (2.7), (2.8) and (2.9) hold true with \overline{T} in place of T, we can show that the convergence result (3.66) is still in L^{1}(Q) weak, namely that (3.63) holds true.

EJQTDE, 2007 No. 24, p. 14 ⋆ Step 8. In this step we prove that u satisfies (2.12) (and (2.13)). To this end, remark that for any fixed n ≥ 0 one has

∫_{{(t,x)/ n≤|u^{ε}|≤n+1}} a(u^{ε}, Du^{ε})Du^{ε} dx dt
= ∫_{Q} a_{ε}(u^{ε}, Du^{ε})[DT_{n+1}(u^{ε}) − DT_{n}(u^{ε})] dx dt
= ∫_{Q} a_{ε}(T_{n+1}(u^{ε}), DT_{n+1}(u^{ε}))DT_{n+1}(u^{ε}) dx dt
− ∫_{Q} a_{ε}(T_{n}(u^{ε}), DT_{n}(u^{ε}))DT_{n}(u^{ε}) dx dt

for ε < \frac{1}{n+1}.

According to (3.63), one is at liberty to pass to the limit as ε tends to 0 for fixed n ≥ 0 and to obtain

(3.67)  lim_{ε→0} ∫_{{(t,x)/ n≤|u^{ε}|≤n+1}} a_{ε}(u^{ε}, Du^{ε})Du^{ε} dx dt
= ∫_{Q} a(T_{n+1}(u), DT_{n+1}(u))DT_{n+1}(u) dx dt
− ∫_{Q} a(T_{n}(u), DT_{n}(u))DT_{n}(u) dx dt
= ∫_{{(t,x)/ n≤|u|≤n+1}} a(u, Du)Du dx dt

Taking the limit as n tends to +∞ in (3.68) and using the estimate (3.67) show that u satisfies (2.12), (and v satisfies (2.13)).

⋆ Step 9. In this step, u is shown to satisfies (2.14) and (2.16 for u) (and v is shown to satisfies (2.15) and (2.16) for v). Let S be a function in W^{2,∞}(ℝ) such that S′ has a compact support. Let K be a positive real number such that suppS′ ⊂ [−K, K]. Pointwise multiplication of the approximate equation (3.6) by S′(u^{ε}) (and (3.7) by S′(v^{ε})) leads to

(3.68)  \frac{∂S(u^{ε})}{∂t} − div(S′(u^{ε})a_{ε}(u^{ε}, Du^{ε})) + S′′(u^{ε})a_{ε}(u^{ε}, Du^{ε})Du^{ε}

− div(S′(u^{ε})Φ_{ε}(u^{ε})) + S′′(u^{ε})Φ_{ε}(u^{ε})Du^{ε} + f_{1}^{ε}(x, u^{ε}, v^{ε})S′(u^{ε}) = 0 in D′(Q).

In what follows we pass to the limit as ε tends to 0 in each term of (3.68).

⋆ Limit of \frac{∂S(u^{ε})}{∂t}

Since S is bounded and continuous, and S(u^{ε}) converges to S(u) a.e. in Q and in L^{∞}(Q) weak ⋆. Then \frac{∂S(u^{ε})}{∂t} converges to \frac{∂S(u)}{∂t} in D′(Q) as ε tends to 0.

⋆ Limit of − div(S′(u^{ε})a_{ε}(u^{ε}, Du^{ε}))

Since suppS′ ⊂ [−K, K], we have for ε < \frac{1}{K}, EJQTDE, 2007 No. 24, p. 15

S′(u^{ε})a_{ε}(u^{ε}, Du^{ε}) = S′(u^{ε})a_{ε}(T_{K}(u^{ε}), DT_{K}(u^{ε})) a.e. in Q.

The pointwise convergence of u^{ε} to u as ε tends to 0, the bounded character of S, (3.21) and (3.62) of Lemma (3.0.8) imply that

S′(u^{ε})a_{ε}(T_{K}(u^{ε}), DT_{K}(u^{ε})) ⇀ S′(u)a(T_{K}(u), DT_{K}(u)) weakly in L^{p′}(Q),

as ε tends to 0, because S′(u) = 0 for |u| ≥ K a.e. in Q. And the term

S′(u)a(T_{K}(u), DT_{K}(u)) = S′(u)a(u, Du) a.e. in Q.

⋆ Limit of S′′(u^{ε})a_{ε}(u^{ε}, Du^{ε})Du^{ε}

Since suppS′′ ⊂ [−K, K], we have for ε ≤ \frac{1}{K^{⋆}}

S′′(u^{ε})a_{ε}(u^{ε}, Du^{ε})Du^{ε} = S′′(u^{ε})a_{ε}(T_{K}(u^{ε}), DT_{K}(u^{ε}))DT_{K}(u^{ε}) a.e. in Q.

The pointwise convergence of S′′(u^{ε}) to S′′(u) as ε tends to 0, the bounded character of S′′, T_{K} and (3.63) of lemma (3.0.8) allow to conclude that

S′′(u^{ε})a_{ε}(u^{ε}, Du^{ε})Du^{ε} ⇀ S′′(u)a(T_{K}(u), DT_{K}(u))DT_{K}(u)

weakly in L^{1}(Q), as ε tends to 0. And

S′′(u)a(T_{K}(u), DT_{K}(u))DT_{K}(u) = S′′(u)a(u, u)Du a.e. in Q.

⋆ Limit of S′(u^{ε})Φ_{ε}(u^{ε})

Since suppS′ ⊂ [−K, K], we have for ε ≤ \frac{1}{K^{⋆}}

S′(u^{ε})Φ_{ε}(u^{ε}) = S′(u^{ε})Φ_{ε}(T_{K}(u^{ε})) a.e. in Q.

As a consequence of (2.5), (3.2) and (3.20), it follows that for any 1 ≤ q < +∞

S′(u^{ε})Φ_{ε}(u^{ε}) → S′(u)Φ(T_{K}(u)) strongly in L^{q}(Q),

as ε tends to 0. The term S′(u)Φ(T_{K}(u)) is denoted by S′(u)Φ(u).

⋆ Limit of S′′(u^{ε})Φ_{ε}(u^{ε})Du^{ε} Since S′ ∈ W^{1,∞}(ℝ) with suppS′ ⊂ [−K, K], we have

S′′(u^{ε})Φ_{ε}(u^{ε})Du^{ε} = Φ_{ε}(T_{K}(u^{ε}))DS′(u^{ε}) a.e. in Q,

we have, DS′(u^{ε}) converges to DS′(u) weakly in L^{p}(Q)^{N} as ε tends to 0, while Φ_{ε}(T_{K}(u^{ε})) is uniformly bounded with respect to ε and converges a.e. in Q to Φ(T_{K}(u)) as ε tends to 0. Therefore

S′′(u^{ε})Φ_{ε}(u^{ε})Du^{ε} ⇀ Φ_{ε}(T_{K}(u^{ε}))DS′(u^{ε}) weakly in L^{p}(Q).

⋆ Limit of f^{ε}(x, u^{ε}, v^{ε})S′(u^{ε})

Due to (2.6), (2.8), (3.3), (3.20) and (3.24), we have

f^{ε}(x, u^{ε}, v^{ε})S′(u^{ε}) → f(x, u, v)S′(u) strongly in L^{1}(Q),

as ε tends to 0.

As a consequence of the above convergence result, we are in a position to pass to the limit as ε tends to 0 in equation (3.68) and to conclude that u satisfies (2.14), (and v satisfies (2.15))

It remains to show that S(u) (and S(v)) satisfies the initial condition (2.16 for u) (and (2.16 for v). To this end, firstly remark that, S being bounded, S(u^{ε}) is bounded in L^{∞}(Q). Secondly, (3.68) and the above considerations on the behavior of the terms of this equation show that \frac{∂S(u^{ε})}{∂t} is bounded in L^{1}(Q) + L^{p′}(0, T; W^{−1,p′}(Ω)). As a consequence, an Aubin’s type lemma (see, e.g, [24], Corollary 4) implies that S(u^{ε}) lies in a compact set of C^{0}([0, T]; W^{−1,s}(Ω)) for any s < inf(p′, \frac{N}{N−1}). It follows that, on one hand, S(u^{ε})(t = 0) = S(u_{0}^{ε}) con-

verges to S(u)(t = 0) strongly in W^{−1,s}(Ω). On the order hand, (3.5) and the smoothness of S imply that S(u_{0}^{ε}) converges to S(u)(t = 0) strongly in L^{q}(Ω) for all q < +∞. Then we conclude that

S(u)(t = 0) = S(u_{0}) in Ω.

The same holds also for v

S(v)(t = 0) = S(v_{0}) in Ω.

As a conclusion of step 3, step 8 and step 9, the proof of theorem 3.0.4 is complete.

References

1. J. Rakotoson A. Eden, B. Michaux, Doubly nonlinear parabolic-type equations as dynamical systems, Journal of Dynamics and Differential Equations 3 (1991).

2. P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, and J.-L. Vazquez, An l^{1}-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa 22 (1995), 241–273.

3. D. Blanchard, Truncation and monotonicity methods for parabolic equations equations, Nonlinear Anal. 21 (1993), 725–743.

4. D. Blanchard and F. Murat, Renormalized solutions of nonlinear parabolic problems with l^{1} data, existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 1137–1152.

5. D. Blanchard, F. Murat, and H. Redwane, Existence et unicité de la solution reormalisée d’un problème parabolique assez général, C. R. Acad. Sci. Paris Sér I329 (1999), 575–580.

6. , Existence and Uniqueness of a Renormalized Solution for a Fairly General Class of Nonlinear Parabolic Problems, J. Differential Equations 177 (2001), 331–374.

7. D. Blanchard and H. Redwane, Renormalized solutions of nonlinear parabolic evolution problems, J. Math. Pure Appl. 77 (1998), 117–151.

8. L. Boccardo, A. Dall’Aglio, T. Gallouët, and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal. 87 (1989), 149–169.

9. L. Boccardo and T. Gallouët, On some nonlinear elliptic equations with right-hand side measures, Commun. Partial Differential Equations 17 (1992), 641–655.

10. L. Boccardo, D. Giachetti, J.-I. Diaz, and F. Murat, Existence and regularity of renormalized solutions for some elliptic problems involving derivation of nonlinear terms, J. Differential Equations 106 (1993), 215–237.

11. A. Dall’Aglio and L. Orsina, Nonlinear parabolic equations with natural growth conditions

and l^{1} data, Nonlinear Anal. 27 (1996), 59–73.

EJQTDE, 2007 No. 24, p. 17

12. L. Dung, Ultimately uniform boundedness of solutions and gradients for degenerate parabolic systems, Nonlinear Analysis T.M.A. in press.

13. N. Grenon, Résultats d’existence et comportement asymptotique pour des équations paraboliques quasi-linéaire, Thèse Université d’Orléans, France, 1990.

14. A. EL Hachimi and H. EL Ouardi, Existence and regularity of a global attractor for doubly nonlinear parabolic equtions, Electron. J. Diff. Eqns 45 (2002), 1–15.

15. , Attractors for a class of doubly nonlinear parabolic systems, Electron. J. Diff. Eqns 1 (2006), 1–15.

16. R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh Sect A89 (1981), 217–237.

17. J.-P. Lions, Mathematical topics in fluid mechanics, vol. 1 : Incompressible models, Oxford Univ. Press, Oxford, 1996.

18. M. Marion, Attractors for reaction-diffusion eqution : existence of their dimension, Applicable Analysis 25 (1987), 101–147.

19. F. Murat, Soluciones renormalizadas de edp elipticas non lineales, cours à l’université de séville, Publication R93023, Laboratoire d’Analyse Numérique, Paris VI, 1993.

20. , Equations elliptiques non linéaires avec second membre l^{1}s ou mesure, Comptes Rendus du 26ème Congrès National d’Analyse Numérique Les Karellis (1994), A12–A24.

21. A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of trauncations, Ann. Mat. Pura ed Applicata 177 (1999), 143–172.

22. R.-J.DiPerna and P.-L. Lions, On the cauchy problem for boltzmann equations : Global existence and weak stability, Ann. Math. 130 (1989), 321–366.

23. H. Redwane, Existence of a solution for a class of parabolic equations with three unbounded nonlinearities, Abstract and Applied Analysis ID 24126 (2007), Accepted.

24. J. Simon, Compact sets in l^{p}(0, t; B), Ann. Mat. Pura Appl. 146 (1987), 65–96.

(Received May 3, 2006)

Hicham REDWANE : Faculté des Sciences Juridiques, Economiques et Sociales. Université Hassan 1, B.P. 784, Settat. Morocco

   E-mail address: redwane hicham@yahoo.fr

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