An Existence Result for Quasilinear Parabolic Systems with Lower Order Terms

. In this paper we prove the existence of weak solutions for a class of quasilinear parabolic systems, which correspond to diﬀusion problems, in the form where Ω is a bounded open domain of R n , 0 < T < ∞ be given and u 0 ∈ L 2 ( Ω ; R m ). The function v belongs to L p (cid:48) (0 , T : W − 1 ,p (cid:48) ( Ω ; R m )) is in a moving and dissolving substance, the dissolution is described by f and the motion by g . We prove the existence result by using Galerkin’s approximation and the theory of Young measures.


Introduction
Let Ω be a bounded open set of R n , n ≥ 2, T is a positive real number, Q = Ω × (0, T ) and 1 < p < ∞. In this paper, we consider the following quasilinear parabolic system: ∂u ∂t − div σ(x, t, u, Du) = v(x, t) + f (x, t, u) + div g(x, t, u) in Q, (1.1) where u : Q → R m is a vector-valued function and Du its gradient which belongs to M m×n . Here M m×n stands for the real vector space of m×n matrices equipped with the inner product ξ : η = m i=1 n j=1 ξ ij η ij . The functions σ : Q × R m × M m×n → M m×n , f : Q × R m → R m and g : Q × R m → M m×n are assumed to satisfy some conditions (see below). Moreover, the function v : Q → R m is in L p (0, T ; W −1,p (Ω; R m )) the dual space of L p (0, T : W 1,p 0 (Ω; R m )), with p = p/(p − 1) the conjugate exponent of p.
Consider first the quasilinear elliptic system −div σ(x, u, Du) = f in Ω, (1.4) endowed with the Dirichlet boundary condition. The existence result is proved in [14] by Hungerbühler. The author used the tool of Young measures and weak monotonicity over σ to achieve his result. See also [2] for a generalized p-Laplacian system. We find a generalization of (1.4) in [1], where the following quasilinear elliptic system −div σ(x, u, Du) = v(x) + f (x, u) + div g(x, u) in Ω (1. 5) was considered. This system corresponds to a diffusion problem with a source v in a moving and dissolving substance, where the motion is described by g and the dissolution by f . The authors proved existence of a weak solution for this system under classical regularity, growth, and coercivity conditions for σ, but with only very mild monotonicity assumptions. See also [3,6] for more results.
For the evolutionary problems, Hungerbühler [15] considered ∂u ∂t − div σ(x, t, u, Du) = v in Q (1. 6) with the initial and boundary conditions (1.2)-(1.3), where v ∈ L p (0, T : W −1,p (Ω; R m )) for some p ∈ ( 2n n+2 , ∞) and u 0 ∈ L 2 (Ω; R m ). The existence of a weak solution under classical regularity, growth, and coercivity conditions for σ but with only very mild monotonicity assumptions is proved. We have extended in [4] the problem (1.6) to a more general strongly quasilinear parabolic system containing the lower term g(x, t, u, Du) in the following form Under mild monotonicity assumptions on σ, we have proved the existence of weak solutions. The elliptic case of (1.7) can be found in [5], where we have investigated another mild monotonicity condition, called the strict quasimonotone. Note that, in all works mentioned above, the authors used the theory of Young measures to achieve their results, since the classical monotone operator theory can not be used for some reasons (see Remark 1). See also [8,9,10,18,19] for similar problems. Inspired by the previous works (especially [4,15]), we want to study the existence result for the problem (1.1)-(1.3). This work, can be seen as an extension of [1] (i.e. of (1.5)) to a parabolic case and generalizes both works [4,15]. We will use the Young measure as a technical tool to obtain the desired result.
The paper is organized as follows: In Section 2, we specify the assumptions on σ, f , g and u 0 needed in the present study and introduce the definition of a weak solution of (1.1)-(1.3). We present in Section 3 an overview on Young measures, while Section 4 is devoted to present the main result and its proof.
2 Assumptions on the data and the definition of a weak solution Throughout this paper, we suppose that the following assumptions hold true: Ω is a bounded open set of R n (n ≥ 2), T > 0 is given and we set Q = Ω×(0, T ). Moreover, we assume: (H0)(Continuity) σ : Q×R m ×M m×n → M m×n is a Carathéodory function, i.e. measurable w.r.t (x, t) ∈ Q and continuous w.r.t other variables.
(c) σ is strictly monotone, i.e., σ is monotone and where λ = ν (x,t) , id and ν = {ν (x,t) } (x,t)∈Q is any family of Young measures generated by a bounded sequence in L p (Q) and not a Dirac measure for a.e. (x, t) ∈ Q.
(H3)(i)(Continuity) f : Q × R m → R m is a Carathéodory function in the sense of (H0).
(ii)(Growth) There exist α 4 ≥ 0, 0 < ρ < p − 1 and d 4 ∈ L p (Q) such that for a.e. (x, t) ∈ Q and all s ∈ R m . Remark 1. Assumption (H2)(b) allows to take a potential W (x, t, s, ξ), which is only convex but not strictly convex in ξ ∈ M m×n , and to consider (1.1) with σ = D ξ W . Note that if W is assumed to be strictly convex, then σ becomes strictly monotone and the standard monotone operator may apply, but it is not the case in this paper.
A prototype of our problem (1.1)-(1.3) can be given by for 0 < γ < p − 1, 0 < ρ < p − 1 and a : Q → M m×n is a measurable function and bounded. For the potential W , one can take W := 1 p |ξ| p for ξ ∈ M m×n . Now, we can define the weak solution of (1.1)-(1.3) as follows: Here ., . denotes the dual pairing of W −1,p (Ω; R m ) and W 1,p 0 (Ω; R m ).

A review on Young measures
As stated in the introduction, we use the tool of Young measures to prove the existence result. This concept of Young measures is a nice tool to understand and control difficulties that arises when weak convergence does not behave as one desires with respect to nonlinear functionals and operators. For convenience of the readers not familiar with this concept, we give an overview needed in this paper. See [7,12,13] for more details. By C 0 (R m ) we denote the Banach space of continuous functions on R m which satisfies lim |λ|→∞ ϕ(λ) = 0. Its dual is the well known space of signed Radon measures with finite mass denoted as M(R m ). The related duality is given for ν : Ω → M(R m ), by A particular case of ϕ is the identity id, thus ν, id = R m λdν(λ). Lemma 1. [12] Let {z j } j≥1 be a measurable sequence in L ∞ (Ω; R m ). Then there exists a subsequence {z k } k ⊂ {z j } j and a Borel probability measure ν x on R m for almost every x ∈ Ω, such that for ϕ ∈ C 0 (R m ) we have Lemma 2. [13] (1) If |Ω| < ∞ and ν x is the Young measure generated by the (whole) sequence z j , then there holds

Remark 2. (1) It is shown in [7] that for any Carathéodory function
, then the Young measure ν (x,t) generated by Dw k has the following properties: We conclude this section by recalling the following useful Fatou-type inequality.

Lemma 4.
[11] Let ϕ : Q × R m × M m×n → R be a Carathéodory function and w k : Q → R m a sequence of measurable functions such that w k → w in measure and such that Dw k generates the Young measure ν (x,t) , with ν (x,t) M = 1 for almost every (x, t) ∈ Q. Then provided that the negative part ϕ − (x, t, w k , Dw k ) is equiintegrable.

Existence result
In this section we present the main result and its proof. Consider the quasilinear parabolic system (1.1)-(1.3). Let p be a real number such that 1 < p < ∞. The result of this paper reads as follows: ), every f satisfying (H3) and every g satisfying (H4).

Proof.
The proof is divided into five steps. In Step 1, we present local approximating solutions by the well known Galerkin method.
Step 2 is devoted to extend these solutions to the whole interval [0, T ]. In Step 3, some a priori estimates will be presented.
Step 4 shows the div-curl inequality which is the key ingredient to pass to the limit in Step 5.
The set Λ is non-empty since it contains a local solution. Moreover, it is an open set and closed (see e.g. [15]). Therefore, Λ = [0, T ).
Step 4: From Step 3, we have that (u k ) k is bounded in L p (0, T ; W 1,p 0 (Ω; R m )). Then by Lemma 1, it follows the existence of a Young measure ν (x,t) generated by Du k in L p (Q) such that ν (x,t) satisfies the properties of Lemma 3. Now, we show the following lemma, namely a div-curl inequality, which is the key ingredient to pass to the limit in the approximating equations.

Lemma 5.
Suppose that σ, f and g satisfy (H0)-(H4). Then the Young measure ν (x,t) associated to Du k has the following property: Proof. Let consider the sequence Since u k u in L p (0, T ; W 1,p 0 (Ω; R m )), then u k → u in measure (for a subsequence). By the growth condition in (H1), (σ(x, t, u k , Du k ) : Du) − is equiintegrable. Now, let Q ⊂ Q be measurable. The coercivity condition in (H1) yields Q min σ(x, t, u k , Du k ) : Du k , 0 dx dt Hence (σ(x, t, u k , Du k ) : Du k ) − is equiintegrable. According to Lemma 4, it follows that It is now sufficient to show that I ≤ 0. By Step 3, we have the following energy equality which is the first property of χ, F and G: On the one hand, we have where we have used u k (., T ) u(., T ) in L 2 (Ω; R m ) and u k (., 0) → u 0 (.) = u(., 0) as k → ∞. The combination of (4.7) and (4.8) implies Step 5: In this step, we will pass to the limit in the Galerkin equations by considering the conditions (a) − (d) listed in (H2). Note that, as in [4], we have σ(x, t, u, λ) − σ(x, t, u, Du) : (λ − Du) = 0 on supp ν (x,t) . (4.9) Let start with the case (c): the strict monotonicity of σ together with (4.9) implies that ν (x,t) = δ Du(x,t) . By virtue of Lemma 2, it follows that Du k → Du in measure and almost everywhere in Q as k → ∞. The continuity of σ gives σ(x, t, u k , Du k ) → σ(x, t, u, Du) almost everywhere. Since σ(x, t, u k , Du k ) is bounded (see the Equation (4.4)), then σ(x, t, u k , Du k ) → σ(x, t, u, Du) in L β (Q), ∀β ∈ [0, p ) by the Vitali convergence theorem. Hence For the case (d), we suppose by contradiction that ν (x,t) is not a Dirac measure on a set (x, t) ∈ Q ⊂ Q of positive Lebesgue measure. We have by the strict p-quasimonotone of σ that By virtue of Lemma 5 (i.e. I ≤ 0), we then have which is a contradiction. Therefore ν (x,t) = δ h(x,t) . Then Hence ν (x,t) = δ Du(x,t) and by virtue of Lemma 2, Du k → Du in measure for k → ∞. The remains of the proof in this case is similar to that in case (c).
Let λ ∈ K (x,t) , then by the Equation (4.9) Using this equation and the monotonicity of σ to obtain .
The weak L 1 -limit of h k is in fact strong since h k ≥ 0. Therefore h k −→ 0 in L 1 (Q).
Since h k is bounded in L p (Q), the Vitali convergence theorem implies that σ(x, t, u k , Du k ) σ(x, t, u, Du) in L p (Q). To conclude the proof of Theorem 1, it remains to pass to the limit on f (x, t, u k ) and g(x, t, u k ). Since u k → u in measure for k → ∞, we may infer that, after extraction of a suitable subsequence, if necessary, u k → u almost everywhere for k → ∞.
Thus for arbitrary ϕ ∈ L p (0, T ; W 1,p 0 (Ω; R m )), it follows from the continuity property in (H3)(i) and (H4)(i) that f (x, t, u k )·ϕ → f (x, t, u)·ϕ and g(x, t, u k ) : Dϕ → g(x, t, u) : Dϕ almost everywhere. Since, by (4.5) and (4.6), f (x, t, u k ) and g(x, t, u k ) are equiintegrable, it follows that f (x, t, u k )·ϕ → f (x, t, u)·ϕ and g(x, t, u k ) : Dϕ → g(x, t, u) : Dϕ in L 1 (Q) by the Vitali convergence theorem. Now, we take a test function w ∈ ∪ i∈N V i and φ ∈ C ∞ 0 ([0, T ]) in (4.1) and integrate over (0, T ) and pass to the limit k → ∞. The resulting equation is for arbitrary w ∈ ∪ i∈N V i and φ ∈ C ∞ 0 ([0, T ]). By density of the linear span of these functions in L p (0, T ; W 1,p 0 (Ω; R m )), this proves that u is in fact a weak solution. Hence the proof of Theorem 1 is complete.