26-1-6(5)

JOURNAL OF PARTIAL DIFFERENTIAL EQUATIONS J. Part. Diff. Eq., Vol. 26, No. 1, pp. 76-98 doi: 10.4208/jpde.v26.n1.6 March 2013 Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems with L1 Data AKDIM Y.1 , BENNOUNA J.1 , MEKKOUR M.1, ∗ and REDWANE H.2 1 Department of Mathematics, Laboratory LAMA, University of Fez, Faculty of Sciences Dhar El Mahraz, B.P. 1796. Atlas Fez, Morocco. 2 Faculté des Sciences Juridiques, Économiques et Sociales, Université Hassan 1, B.P. 784. Settat, Morocco. Received 7 August 2012; Accepted 14 December 2012 Abstract. We study the existence of renormalized solutions for a class of nonlinear degenerated parabolic problem. The Carathéodory function satisfying the coercivity condition, the growth condition and only the large monotonicity. The data belongs to L1 ( Q ) . AMS Subject Classifications: A7A15, A6A32, 47D20 Chinese Library Classifications: O175.27 Key Words: Weighted Sobolev spaces; truncations; nonlinear doubling parabolic equation; renormalized solutions. 1 Introduction Let Ω be a bounded open set of R N , p be a real number such that 2 ≤ p < ∞, Q = Ω×]0,T [ and w ={wi ( x), 0 ≤ i ≤ N } be a vector of weight functions (i.e., every component wi ( x) is a measurable function which is positive a.e. in Ω) satisfying some integrability conditions. The objective of this paper is to study the following problem in the weighted Sobolev space:  ∂b(u)    in Q,  ∂t − div( a( x,t,u,Du))+ div(φ(u)) = f , (1.1) b( x,u)(t = 0) = b( x,u0 ), in Ω,     u = 0, on ∂Ω×]0,T [. ∗ Corresponding author. Email address: mekkour.mounir@yahoo.fr (M. Mekkour) http://www.global-sci.org/jpde/ 76 Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems 77 The function b is assumed to be a strictly increasing C1 -function, the data f and b(u0 ) lie in L1 ( Q) and L1 (Ω), respectively. The functions φ is just assumed to be continuous of R with values in R N , and the Carathéodory function a satisfying only the large monotonicity (see assumption (H2 )). Let us point out, the difficulties that arise in problem (1.1) are due to the following facts: the data f and u0 only belong to L1 , a satisfies the large monotonicity that is [a( x,t,s,ξ )− a( x,t,s,η )](ξ − η ) ≥ 0, for all (ξ,η ) ∈ R N × R N , and the function φ(u) does not belong to ( L1loc ( Q)) N (because the function φ is just assumed to be continuous on R). To overcome this difficulty, we will apply Landes’s technical (see [1, 2]) and the framework of renormalized solutions. This notion was introduced by Diperna and P.-L. Lions [3] in their study of the Boltzmann equation. This notion was then adapted to an elliptic version of (1.1) by L. Boccardo et al. [4] when the right hand ′ side is in W −1,p (Ω), by J.-M. Rakotoson [5] when the right hand side is in L1 (Ω), and finally by G. Dal Maso, F. Murat, L. Orsina and A. Prignet [6] for the case of right hand side is general measure data. For the parabolic equation (1.1) the existence of weak solution has been proved by J.-M. Rakotoson [7] with the strict monotonicity and a measure data, the existence and uniqueness of a renormalized solution has been proved by D. Blanchard and F. Murat [8] in the case where a( x,t,s,ξ ) is independent of s, φ = 0, and by D.Blanchard, F. Murat and H. Redwane [9] with the large monotonicity on a. For the degenerated parabolic equations the existence of weak solutions have been proved by L. Aharouch et al. [10] in the case where a is strictly monotone, φ = 0 and ′ ′ f ∈ L p (0,T,W −1, p (Ω,w∗ )). See also the existence of renormalized solution by Y.Akdim et al [11] in the case where a( x,t,s,ξ ) is independent of s and φ = 0. Note that, this paper can be seen as a generalization of [9, 10] in weighted case and as a continuation of [11]. The plan of the paper is as follows. In Section 2 we give some preliminaries and the definition of weighted Sobolev spaces. In Section 3 we make precise all the assumptions on a, φ, f and u0 . In Section 4 we give some technical results. In Section 5 we give the definition of a renormalized solution of (1.1) and we establish the existence of such a solution (Theorem 5.1). Section 6 is devoted to an example which illustrates our abstract result. 2 Preliminaries Let Ω be a bounded open set of R N , p be a real number such that 1 < p < ∞ and w ={wi ( x), 0 ≤ i ≤ N } be a vector of weight functions, i.e., every component wi ( x) is a measurable function which is strictly positive a.e. in Ω. Further, we suppose in all our considerations 78 Y. Akdim et al. / J. Partial Diff. Eq., 26 (2013), pp. 76-98 that, there exists − r0 r0 − p r0 > max( N, p) such that wi ∈ L1loc (Ω), (2.1) wi ∈ L1loc (Ω), −1 p −1 wi (2.2) ∈ L1 ( Ω ), (2.3) for any 0 ≤ i ≤ N. We denote by W 1,p (Ω,w) the space of all real-valued functions u ∈ L p (Ω,w0 ) such that the derivatives in the sense of distributions fulfill ∂u ∈ L p (Ω,wi ), ∂xi for i = 1, ··· , N, which is a Banach space under the norm kuk1,p,w = Z |u( x)| p w0 ( x) dx + Ω N Z X i=1 Ω ∂u( x) p wi ( x) dx ∂xi 1/p . (2.4) The condition (2.2) implies that C0∞ (Ω) is a space of W 1,p (Ω,w) and consequently, we can 1,p introduce the subspace V = W0 (Ω,w) of W 1,p (Ω,w) as the closure of C0∞ (Ω) with respect 1,p to the norm (2.4). Moreover, condition (2.3) implies that W 1,p (Ω,w) as well as W0 (Ω,w) are reflexive Banach spaces. 1,p We recall that the dual space of weighted Sobolev spaces W0 (Ω,w) is equivalent to 1− p ′ ′ W −1,p (Ω,w∗ ), where w∗ = {w∗i = wi p′ = p/( p − 1) , (see [12]). , i = 0, ··· , N } and where p′ is the conjugate of p i.e. 3 Basic assumptions Assumption (H1) For 2 ≤ p < ∞, we assume that the expression k|u|kV = X N Z i=1 Ω ∂u( x) p wi ( x) dx ∂xi 1/p (3.1) is a norm defined on V which equivalent to the norm (2.4), and there exist a weight function σ on Ω such that, σ ∈ L1 (Ω) and σ−1 ∈ L1 (Ω). We assume also the Hardy inequality, Z q |u( x)| σdx Ω 1/q ≤c X N Z i=1 Ω ∂u( x) p wi ( x) dx ∂xi 1/p , (3.2) Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems 79 holds for every u ∈ V with a constant c > 0 independent of u, and moreover, the imbedding W 1, p (Ω,w) ֒→֒→ Lq (Ω,σ), (3.3) expressed by the inequality (3.2) is compact. Note that (V, k|·|kV ) is a uniformly convex (and thus reflexive) Banach space. Remark 3.1. If we assume that w0 ( x)≡1 and in addition the integrability condition: There exists ν ∈] Np , +∞ [∩[ p−1 1 , +∞[ such that N ν 1 w− i ∈ L (Ω) wiN−1 ∈ L1loc (Ω), and for all i = 1, ··· , N. (3.4) Notice that the assumptions (2.2) and (3.4) imply k|uk| = X N Z i=1 Ω ∂u p wi ( x) dx ∂xi 1/p , (3.5) 1,p which is a norm defined on W0 (Ω,w) and its equivalent to (2.4) and that, the imbedding 1,p W0 (Ω,w) ֒→ L p (Ω), (3.6) is compact for all 1 ≤ q ≤ p1∗ if pν < N (ν + 1) and for all q ≥ 1 if pν ≥ N (ν + 1) where p1 = pν/(ν + 1) and p1∗ is the Sobolev conjugate of p1 ; see [13, pp. 30-31]. Assumption (H2) We assume that b : Ω × R → R is a strictly increasing C1 − function with b(0) = 0, 1 1 q |ai ( x,t,s,ξ )| ≤ βwip ( x)[k( x,t)+ σ p′ |s| p′ + N X 1 ′ w jp ( x)|ξ j | p−1 ], for i = 1, ··· , N, (3.7) (3.8) j=1 ′ for a.e. ( x,t) ∈ Q, all (s,ξ ) ∈ R × R N , some function k( x,t) ∈ L p ( Q) and β > 0. Here σ and q are as in (H1), [a( x,t,s,ξ )− a( x,t,s,η )](ξ − η ) ≥ 0, a( x,t,s,ξ )· ξ ≥ α N X for all (ξ,η ) ∈ R N × R N , w i |ξ i | p , (3.9) (3.10) i=1 φ : R → R N is a continuous function, (3.11) f is an element of L1 ( Q), 1 (3.12) 1 u0 is an element of L (Ω) such that b(u0 ) ∈ L (Ω). (3.13) 80 Y. Akdim et al. / J. Partial Diff. Eq., 26 (2013), pp. 76-98 Where α is strictly positive constant. We recall that, for k > 1 and s in R, the truncation is defined as, ( s, if |s| ≤ k, Tk (s) = s k |s| , if |s| > k. 4 Some technical results Characterization of the time mollification of a function u In order to deal with time derivative, we introduce a time mollification of a function u belonging to a some weighted Lebesgue space. Thus we define for all µ ≥ 0 and all ( x,t) ∈ Q, uµ = µ Z t ũ( x,s) exp(µ(s − t))ds, where ũ( x,s) = u( x,s)χ(0,T ) (s). ∞ Proposition 4.1 ([10]). 1) if u ∈ L p ( Q,wi ) then uµ is measurable in Q and uµ L p ( Q,w i ) 1,p ∂u µ ∂t = µ(u − uµ ) and, ≤ kuk L p ( Q,wi ) . 1,p 2) If u ∈ W0 ( Q,w), then uµ → u in W0 ( Q,w) as µ → ∞. 1,p 1,p 3) If un → u in W0 ( Q,w), then (un )µ → uµ in W0 ( Q,w). Some weighted embedding and compactness results In this section we establish some embedding and compactness results in weighted Sobolev spaces, some trace results, Aubin’s and Simon’s results [14]. Let 1, p V = W0 ′ 1, p (Ω,w), H = L2 (Ω,σ), and V ∗ = W −1,p , with (2 ≤ p < ∞), X = L p (0,T;W0 (Ω,w)). ′ The dual space of X is X ∗ = L p (0,T,V ∗ ) where 1/p + 1/p′ = 1 and denoting the space Wp1 (0,T,V, H ) = {v ∈ X : v′ ∈ X ∗ }, endowed with the norm kukWp1 = kukX + u′ X∗ , which is a Banach space. Here u′ stands for the generalized derivative of u, i.e., Z 0 T ′ u (t) ϕ(t)dt = − Z T 0 u(t) ϕ′ (t)dt, for all ϕ ∈ C0∞ (0,T ). Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems 81 Lemma 4.1 ([15]). 1) The evolution triple V ⊆ H ⊆ V ∗ is verified. 2) The imbedding Wp1 (0,T,V, H ) ⊆ C (0,T, H ) is continuous. 3) The imbedding Wp1 (0,T,V, H ) ⊆ L p ( Q,σ) is compact. Lemma 4.2 ([10]). Let g ∈ Lr ( Q,γ) and let gn ∈ Lr ( Q,γ), with k gn k Lr ( Q,γ) ≤ C, 1 < r < ∞. If gn ( x) → g( x) a.e. in Q, then gn ⇀ g in Lr ( Q,γ). Lemma 4.3 ([10]). Assume that, ∂vn = αn + β n , ∂t in D ′ ( Q), where αn and β n are bounded respectively in X ∗ and in L1 ( Q). If 1, p vn is bounded in L p (0,T;W0 (Ω,w)), p then vn → v in Lloc ( Q,σ). Further vn → v strongly in L1 ( Q). 5 Main results Definition 5.1. Let f ∈ L1 ( Q) and b(u0 )∈ L1 (Ω). A real-valued function u defined on Ω×]0,T [ is a renormalized solution of problem (1.1) if 1, p Tk (u) ∈ L p (0,T;W0 (Ω,w)), for all (k ≥ 0) and b(u) ∈ L∞ (0,T; L1 (Ω)); Z a( x,t,u,Du)Dudxdt → 0, as m → +∞; (5.1) (5.2) { m ≤|u|≤ m +1}
∂BS (u) − div S′ (u) a(u,Du) + S′′ (u) a(u,Du) Du ∂t + div(S′ (u)φ(u))− S′′ (u)φ(u) Du = f S′ (u), in D ′ ( Q); for all functions S ∈ W 2, ∞ (R ) which compact support in R, where BS (z) = BS (u)(t = 0) = BS (u0 ), in Ω. (5.3) Rz 0 b′ (r)S′ (r)dr and (5.4) Remark 5.1. Eq. (5.3) is formally obtained through pointwise multiplication of Eq. (1.1) by S′ (u). However, while a(u,Du) and φ(u) does not in general make sense in (1.1), all the terms in (5.3) have a meaning in D ′ ( Q). Indeed, if M is such that suppS′ ⊂ [− M, M ], the following identifications are made in (5.3): • BS (u) belongs to L∞ ( Q) since S is a bounded function and DBS (u) = S′ (u)b′ ( TM (u)) DTM (u). 82 Y. Akdim et al. / J. Partial Diff. Eq., 26 (2013), pp. 76-98 • S′ (u) a(u,Du) identifies with S′ (u) a( TM (u),DTM (u)) a.e. in Q. Since |TM (u)| ≤ M a.e. in Q and S′ (u) ∈ L∞ ( Q), we obtain from (3.8) and (5.1) that S′ (u) a( TM (u),DTM (u)) ∈ N Y ′ L p ( Q,w∗i ). i=1 • S′′ (u) a(u,Du) Du identifies with S′′ (u) a( TM (u),DTM (u)) DTM (u) and S′′ (u) a( TM (u),DTM (u)) DTM (u) ∈ L1 ( Q). • S′′ (u)φ(u) Du and S′ (u)φ(u) respectively identify with S′′ (u)φ( TM (u)) DTM (u) and S′ (u)φ( TM (u)). Due to the properties of S′ and to (3.11), the functions S′ , S′′ and φoTM are bounded on R so that (5.1) implies that S′ (u)φ( TM (u)) ∈ ( L∞ ( Q)) N , and S′′ (u)φ( TM (u)) DTM (u) ∈ L p ( Q,w). • S′ (u) f belongs to L1 ( Q). The above considerations show that Eq. (5.3) holds in D ′ ( Q), ∂BS (u)/∂t belongs to ′ 1, p L (0,T;W −1, p (Ω,w∗i ))+ L1 ( Q) and BS (u) ∈ L p (0,T;W0 (Ω,w))∩ L∞ ( Q). It follows that BS (u) belongs to C0 ([0,T ]; L1 (Ω)) so that the initial condition (5.4) makes sense. p′ Theorem 5.1. Let f ∈ L1 ( Q) and u0 ∈ L1 (Ω). Assume that (H1) and (H2), there exists at least a renormalized solution u (in the sense of Definition 5.1). Proof. Step 1: The approximate problem. For n > 0, let us define the following approximation of b, a, φ, f and u0 ; 1 bn (r) = Tn (b(r))+ r, n an ( x,t,s,d) = a( x,t,Tn (s),d), for n > 0, (5.5) a.e. in Q, ∀s ∈ R, ∀d ∈ R N . (5.6) ′ In view of (5.6), an satisfy (3.10) and (3.8), there exists kn ∈ L p ( Q) and β n > 0 such that   N 1 1 q X 1 ′ |ani ( x,t,s,ξ )| ≤ β n wip ( x) kn ( x,t)+ σ p′ |s| p′ + w jp ( x)|ξ j | p−1  , ∀(s,ξ ) ∈ R × R N , (5.7) j=1 φn is a Lipschitz continuous bounded function from R into R N , (5.8) 83 Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems such that φn uniformly converges to φ on any compact subset of R as n tends to +∞, ′ f n ∈ L p ( Q) and f n → f , a.e. in Q and strongly in L1 ( Q) as n → +∞, (5.9) u0n ∈ D (Ω) : kbn (u0n )k L1 ≤ kb(u0 )k L1 , bn (u0n ) → b(u0 ), a.e. in Ω and strongly in L1 (Ω). Let us now consider the approximate problem:  ∂bn (un )    − div( an ( x,t,un ,Dun ))+ div(φn (un )) = f n ,  ∂t un = 0,     b (u (t = 0)) = b (u ), n n n 0n (5.10) in D ′ ( Q), in (0,T )× ∂Ω, (5.11) in Ω. 1, p As a consequence, proving existence of a weak solution un ∈ L p (0,T;W0 (Ω,w)) of (5.11) is an easy task (see e.g. [16, 17]). Step 2: The estimates derived in this step rely on standard techniques for problems of type (5.11). Using in (5.11) the test function Tk (un )χ(0,τ ) , we get, for every τ ∈ [0,T ].   Z ∂bn (un ) ,Tk (un )χ(0,τ ) + a( x,t,Tk (un ),DTk (un )) DTk (un )dxdt ∂t Qτ Z Z + φn (un )DTk (un )dxdt = f n Tk (un )dxdt, (5.12) Qτ Qτ which implies that, Z τZ Z n a( x,t,Tk (un ),DTk (un ))DTk (un )dxdt Bk (un (τ ))dx + 0 Ω Ω Z Z Z + φn (un ) DTk (un )dxdt = f n Tk (un )dxdt + Bkn (u0n )dx, Qτ Qτ (5.13) Ω Rr where Bkn (r) = 0 Tk (s)bn′ (s)ds. The Lipschitz character of φn and stokes’ formula together with the boundary condition 2 of problem (5.11) give Z τZ φn (un ) DTk (un )dxdt = 0. (5.14) 0 Ω Bkn Due to the definition of we have Z Z 0 ≤ Bkn (u0n )dx ≤ k |bn (u0n )| dx ≤ k k b(u0 )k L1 (Ω) . Ω (5.15) Ω Using (5.14), (5.15) and Bkn (un ) ≥ 0, it follows from (5.13) that Z τZ a( x,t,Tk (un ),DTk (un )) DTk (un )dxdt ≤ k(k f n k L1 ( Q) +kbn (u0n )k L1 (Ω) ) ≤ Ck. (5.16) 0 Ω 84 Y. Akdim et al. / J. Partial Diff. Eq., 26 (2013), pp. 76-98 Thanks to (3.10) we have α Z X N Q i=1 wi ( x) ∂Tk (un ) p dxdt ≤ Ck, ∂xi ∀k ≥ 1. (5.17) We deduce from that above inequality (5.13) and (5.15) that Z Bkn (un )dx ≤ k(k f k L1 ( Q) +kb(u0 )k L1 (Ω) ) ≡ Ck. (5.18) Ω 1, p 1, p Then, Tk (un ) is bounded in L p (0,T;W0 (Ω,w)), Tk (un ) ⇀ vk in L p (0,T;W0 by the compact imbedding (3.6) gives, (Ω,w)), and Tk (un ) → vk , strongly in L p ( Q,σ) and a.e. in Q. Let k > 0 large enough and BR be a ball of Ω, we have, k meas({| un | > k}∩ BR ×[0,T ]) = Z 0 ≤ Z 0 ≤ TZ TZ Z {|un |> k}∩ BR BR | Tk (un )| dxdt |Tk (un )| dxdt p Q ≤ Tc R | Tk (un )| σdxdt Z X N Q i=1  1p Z 0 TZ σ 1− p ′ p dxdt BR ∂Tk (un ) p dxdt wi ( x) ∂xi ! 1p ! 1′ 1 ≤ ck p , (5.19) which implies that, meas({| un | > k}∩ BR ×[0,T ]) ≤ c k 1− 1p , ∀k ≥ 1. So, we have lim (meas({| un | > k}∩ BR ×[0,T ])) = 0. k→+ ∞ Now we turn to prove the almost every convergence of un and bn (un ). Consider now a function non decreasing gk ∈ C2 (R ) such that gk (s) = s for | s| ≤ k/2 and gk (s) = k for |s| ≥ k. Multiplying the approximate equation by gk′ (bn (un )), we get ∂gk (bn (un )) − div( a( x,t,un ,Dun ) gk′ (bn (un )))+ a( x,t,un ,Dun ) gk′′ (bn (un ))bn′ (un ) Dun ∂t − div( gk′ (bn (un ))φn (un ))+ gk′′ (bn (un ))bn′ (un )φn (un ) Dun = f n gk′ (bn (un )), (5.20) 85 Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems in the sense of distributions, which implies that 1, p gk (bn (un )) is bounded in L p (0,T;W0 (Ω,w)), (5.21) and ∂gk (bn (un )) is bounded in X ∗ + L1 ( Q), (5.22) ∂t independently of n as soon as k < n. Due to Definition (3.7) and (5.5) of bn , it is clear that {|bn (un )| ≤ k} ⊂ {| un | ≤ k∗ }, as soon as k < n and k∗ is a constant independent of n. As a first consequence we have Dgk (bn (un )) = gk′ (bn (un ))bn′ ( Tk∗ (un )) DTk∗ (un ), a.e. in Q, (5.23) as soon as k < n. Secondly, the following estimate holds true gk′ (bn (un ))bn′ ( Tk∗ (un )) L∞ ( Q) ≤ gk′ ( max (b′ (r))+ 1). L ∞ ( Q ) |r |≤ k∗ As a consequence of (5.17), (5.23) we then obtain (5.21). To show that (5.22) holds true, due to (5.20) we obtain ∂gk (bn (un )) = div( a( x,t,un ,Dun ) gk′ (bn (un )))− a( x,t,un ,Dun ) gk′′ (bn (un ))bn′ (un ) Dun ∂t + div( gk′ (bn (un ))φn (un ))− gk′′ (bn (un ))bn′ (un )φn (un ) Dun + f n gk′ (bn (un )). (5.24) Since suppgk′ and suppgk′′ are both included in [−k,k], un may be replaced by Tk∗ (un ) in each of these terms. As a consequence, each term on the right-hand side of (5.24) is ′ ′ bounded either in L p (0,T;W −1,p (Ω,w∗ )) or in L1 ( Q). Hence Lemma 4.3 allows us to p conclude that gk (bn (un )) is compact in Lloc ( Q,σ). Thus, for a subsequence, it also converges in measure and almost every where in Q, due to the choice of gk , we conclude that for each k, the sequence Tk (bn (un )) converges almost everywhere in Q (since we have, for every λ > 0,) meas({| bn (un )− bm (um )| > λ}∩ BR ×[0,T ]) ≤ meas({| bn (un )| > k}∩ BR ×[0,T ]) + meas({|bm (um )| > k}∩ BR ×[0,T ])+ meas ({| gk (bn (un ))− gk (bm (um ))| > λ}). Let ε > 0, then, there exist k(ε) > 0 such that, meas({| bn (un )− bm (um )| > λ}∩ BR ×[0,T ]) ≤ ε, for all n,m ≥ n0 (k(ε),λ,R). This proves that (bn (un )) is a Cauchy sequence in measure in BR ×[0,T ], thus converges almost everywhere to some measurable function v. Then for a subsequence denoted again un , un → u, a.e. in Q, (5.25) 86 Y. Akdim et al. / J. Partial Diff. Eq., 26 (2013), pp. 76-98 and bn (un ) → b(u), a.e. in Q, (5.26) we can deduce from (5.17) that, 1, p Tk (un ) ⇀ Tk (u), weakly in L p (0,T;W0 (Ω,w)), (5.27) and then, the compact imbedding (3.3) gives, Tk (un ) → Tk (u), strongly in Lq ( Q,σ) and a.e. in Q. Which implies, by using (3.8), for all k > 0 that there exists a function hk ∈ such that N Y ′ a( x,t,Tk (un ),DTk (un )) ⇀ hk , weakly in L p ( Q,w∗i ). QN i=1 L p′ ( Q,w ∗ ), i (5.28) i=1 We now establish that b(u) belongs to L∞ (0,T; L1 (Ω)). Using (5.25) and passing to the limit-inf in (5.18) as n tends to +∞, we obtain that Z 1 B (u)(τ )dx ≤ [k f k L1 ( Q) +ku0 k L1 (Ω) ] ≡ C, k Ω k for almost any τ in (0,T ). Due to the definition of Bk (s) and the fact that 1k Bk (u) converges pointwise to b(u), as k tends to +∞, shows that b(u) belong to L∞ (0,T; L1 (Ω)). Step 3: This step is devoted to introduce for k ≥ 0 fixed a time regularization of the function Tk (u) and to establish the following limits: a( x,t,Tk (un ),DTk (un )) ⇀ a( x,t,Tk (u),DTk (u)), weakly in N Y ′ L p ( Q,w∗i ), (5.29) i=1 as n tends to +∞. This proof is devoted to introduce for k ≥ 0 fixed, a time regularization of the function Tk (u) in order to perform the monotonicity method. Firstly we prove the following lemma: Lemma 5.1. lim lim Z m →+ ∞ n →+ ∞ { m ≤|u |≤ m +1} n for any integer m ≥ 1. a( x,t,un ,Dun ) Dun dxdt = 0, Proof. Taking T1 (un − Tm (un )) as a test function in (5.11), we obtain  Z  ∂bn (un ) a(un ,Dun ) Dun dxdt ,T1 (un − Tm (un )) + ∂t { m ≤|un |≤ m +1}  Z u n Z Z ′ f n T1 (un − Tm (un )). φ(r) T1 (r − Tm (r)) dxdt = + div Q 0 Q (5.30) (5.31) 87 Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems Ru 1, p Using the fact that 0 n φ(r) T1′ (r − Tm (r))dxdt ∈ L p (0,T;W0 (Ω,w)) and Stokes’ formula, we get Z Z m a(un ,Dun ) Dun dxdt Bn (un )( T )dx + { m ≤|un |≤ m +1} Ω Z Z (5.32) ≤ | f n T1 (un − Tm (un ))| dxdt + Bnm (u0n )dx, Q Ω Rr where Bnm (r) = 0 bn′ (s) T1 (s − Tm (s))ds. In order to pass to the limit as n tends to +∞ in (5.32), we use Bnm (un )( T )≥ 0 and (5.9), (5.10), we obtain that Z a(un ,Dun ) Dun dxdt lim m →+ ∞ { m ≤|u |≤ m +1} n Z Z (5.33) ≤ |b(u0 ( x))| dx. | f | dxdt + {|u0 ( x )|> m } {|u( x )|> m } Finally by (3.13), (3.12) and (5.33) we get Z lim lim m →+ ∞ n →+ ∞ { m ≤|u |≤ m +1} n a(un ,Dun ) Dun dxdt = 0. (5.34) The proof is complete. The very definition of the sequence ( Tk (u))µ for µ > 0 (and fixed k) we establish the following lemma. Lemma 5.2. Let k ≥ 0 be fixed. Let ( Tk (u))µ the mollification of Tk (u). Let S be an increasing C ∞ (R )-function such that S(r) = r f or |r| ≤ k and supp S′ is compact. Then,  Z T ∂bn (un ) ′ ,S (un )( Tk (un )−( Tk (u))µ ) dxdt ≥ 0, (5.35) lim lim µ →+ ∞ n →+ ∞ 0 ∂t ′ 1, p where h·, ·i denotes the duality pairing between L1 (Ω)+W −1,p (Ω,w∗ ) and L∞ (Ω)∩W0 (Ω,w). Proof. See H. Redwane [18]. We prove the following lemma, which is the key point in the monotonicity arguments. Lemma 5.3. The subsequence of un satisfies for any k ≥ 0 Z TZ tZ limsup a( Tk (un ),DTk (un )) DTk (un )dxdsdt ≤ n →+ ∞ 0 TZ tZ Z 0 0 where hk is defined in (5.28). Ω 0 Ω hk DTk (u)dxdsdt, (5.36) 88 Y. Akdim et al. / J. Partial Diff. Eq., 26 (2013), pp. 76-98 Proof. In the following we adapt the above-mentioned method to problem (1.1) and we first introduce a sequence of increasing C ∞ (R )-functions Sm such that ′ suppSm ⊂ [−(m + 1),m + 1], Sm (r) = r if |r| ≤ m, ′′ Sm L∞ ≤ 1, for any m ≥ 1. We use the sequence Tk (u)µ of approximations of Tk (u), and plug the test function ′ ( u )( T ( u )−( T ( u )) ) (for n > 0 and µ > 0) in (5.11). Through setting, for fixed k ≤ 0, Sm n µ k n k Wµn = Tk (un )−( Tk (u))µ , we obtain upon integration over (0,t) and then over (0,T):  Z TZ tZ Z TZ t ∂bn (un ) ′ n ′ ,Sm (un )Wµ dtds + Sm (un ) an (un ,Dun ) DWµn dxdsdt ∂t 0 0 Ω 0 0 Z TZ tZ Z TZ tZ ′′ ′ + Sm (un ) an (un ,Dun ) Dun Wµn dxdsdt − Sm (un )φn (un ) DWµn dxdsdt 0 0 0 Ω 0 Ω Z TZ tZ Z TZ tZ ′′ ′ − Sm (un )φn (un ) Dun Wµn dxdsdt = f n Sm (un )Wµn dxdsdt. (5.37) 0 0 0 Ω 0 Ω In the following we pass the limit in (5.37) as n tends to +∞, then µ tends to +∞ and then m 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 :  Z TZ t n ∂Bm (un ) n ,Wµ dtds ≥ 0, for any m ≥ k, liminf lim (5.38) µ →+ ∞ n →+ ∞ 0 ∂t 0 Z TZ tZ ′ lim lim Sm (un )φn (un ) DWµn dxdsdt = 0, for any m ≥ 1, (5.39) µ →+ ∞ n →+ ∞ 0 lim lim Z µ →+ ∞ n →+ ∞ 0 0 Ω TZ tZ 0 Ω ′′ Sm (un )φn (un ) Dun Wµn dxdsdt = 0, for any m ≥ 1, lim limsup limsup m →+ ∞ µ →+ ∞ lim lim Z µ →+ ∞ m →+ ∞ 0 n →+ ∞ TZ tZ 0 Z 0 Ω TZ tZ 0 Ω ′′ Sm (un ) a(un ,Dun ) Dun Wµn dxdsdt = 0, m ≥ 1, ′ f n Sm (un )Wµn dxdsdt = 0. (5.40) (5.41) (5.42) Proof of (5.38). The function Sm belongs to C ∞ (R ) and is increasing. We have for ′ is compact. m ≥ k, Sm (r) = r for|r| ≤ k while suppSm n In view of the definition of Wµ , Lemma 5.2 applies with S = Sm for fixed m ≥ k. As a consequence (5.38) holds true. Proof of (5.39). In order to avoid repetitions in the proofs of (5.42), let us summarize the properties of Wµn . For fixed µ > 0 1, p Wµn ⇀ Tk (u)−( Tk (u))µ , weakly in L p (0,T;W0 Wµn L∞ ( Q) (Ω,w)), as n → +∞, ≤ 2k, for any n > 0 and for any µ > 0, Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems 89 we deduce that for fixed µ > 0 Wµn → Tk (u)−( Tk (u))µ , a.e. in Q and in L∞ ( Q)weak −∗, as n → +∞, ′′ ⊂ [−( m + 1), − m ]∪[m,m + 1] for any fixed m ≥ 1, we have one has suppSm ′ ′ (un )φn ( Tm+1 (un )) DWµn , a.e. in Q, Sm (un )φn (un ) DWµn = Sm (5.43) ′ ⊂ [− m − 1,m + 1]. since suppSm ′ is smooth and bounded, (3.11), (5.8), and u → u a.e. in Q lead to Since Sm n ′ ′ Sm (un )φn ( Tm+1 (un )) → Sm (u)φ( Tm+1 (u)), a.e. in Q and in L∞ ( Q) weak −∗, (5.44) as n tends to +∞. As a consequence of (5.46) and (5.44), we deduce that Z lim = n →+ ∞ 0 Z TZ tZ 0 0 TZ tZ 0 Ω Ω ′ Sm (un )φn (un ) DWµn dxdsdt ′ Sm (u)φ ( Tm+1 (u))( DTk (u)− D ( Tk (u))µ )dxdsdt, (5.45) for any µ > 0. Passing to the limit as µ → +∞ in (5.45) we conclude that (5.39) holds true. Proof of (5.40). For fixed m ≥ 1, and by the same arguments that those that lead to (5.46), we have ′′ ′′ (un )φn (un ) Dun Wµn = Sm Sm (un )φn ( Tm+1 (un )) DTm+1 (un )Wµn , a.e. in Q. (5.46) From (3.11), un → u a.e. in Q and (5.27), it follows that for any µ > 0 Z TZ tZ ′′ lim Sm (un )φn (un ) Dun Wµn dxdsdt n →+ ∞ 0 0 Ω Z TZ tZ ′′ = Sm (un )φ ( Tm+1 (u))( DTk (u)− D ( Tk (u))µ )dxdsdt, 0 0 Ω for any µ > 0. Passing to the limit as µ → +∞ in (5.45) we conclude that (5.40) holds true. ′′ ⊂ [−( m + 1), − m ]∪[m,m + 1] for any m ≥ 1. As a Proof of (5.41). One has suppSm consequence Z 0 TZ tZ 0 Ω ′′ ≤ T Sm (un ) ′′ Sm (un ) a(un ,Dun ) Dun Wµn dxdsdt L∞ Wµn Z L ∞ { m ≤|u n |≤ m +1} a(un ,Dun ) Dun dxdt (5.47) 90 Y. Akdim et al. / J. Partial Diff. Eq., 26 (2013), pp. 76-98 for any m ≥ 1, any µ > 0 and any n ≥ 1, it is possible to obtain Z limsup limsup µ →+ ∞ n →+ ∞ ≤Climsup n →+ ∞ Z 0 TZ tZ 0 { m ≤|un |≤ m +1} Ω ′′ Sm (un ) a(un ,Dun ) Dun Wµn dxdsdt a(un ,Dun ) Dun dxdt, for any m ≥ 1, where C is a constant independent of m. Appealing now to (5.30) it possible to pass the limit as m tends to +∞ to establish (5.41). Proof of (5.42). Lebesgue’s convergence theorem implies that for any µ > 0 and any m≥1 lim Z n →+ ∞ 0 TZ tZ 0 Ω ′ f n Sm (un )Wµn dxdsdt = Z 0 TZ tZ 0 Ω ′ f Sm (u)( Tk (u)−( Tk (u)µ ))dxdsdt. Now, for fixed m ≥ 1, using Lemma 4.1 and passing to the limit as µ → +∞ in the above equality to obtain (5.42). We now turn back to the proof of Lemma 5.3. Due to (5.38)-(5.42), we are in a position to pass the limit-sup when n tends to +∞, then to the limit-sup when µ tends +∞ and then to the limit as m tends to +∞ in (5.37). We obtain by using the definition of Wµn that for any k ≥ 0 lim limsup limsup m →+ ∞ µ →+ ∞ n →+ ∞ Z 0 TZ tZ 0 Ω ′ Sm (un ) an (un ,Dun )( DTk (un )− D ( Tk (u))µ )dxdsdt ≤ 0. ′ ( u ) a ( u ,Du ) DT ( u ) = a( u ,Du ) DT ( u ) for k ≤ n and k ≤ m, the above inSince Sm n n n n n n k n k n equality implies that for k ≤ m, limsup n →+ ∞ Z 0 TZ tZ 0 Ω an (un ,Dun ) DTk (un )dxdsdt ≤ lim limsup limsup m →+ ∞ µ →+ ∞ n →+ ∞ Z 0 TZ tZ 0 Ω ′ Sm (un ) an (un ,Dun ) D ( Tk (u))µ dxdsdt. The right-hand side of (5.48) is computed as follows. We have for n ≥ m + 1: ′ ′ Sm (un ) an (un ,Dun ) = Sm (un ) a( Tm+1 (un ),DTm+1 (un )) a.e. in Q. Due to the weak convergence of a( DTm+1 (un )) it follows that for fixed m ≥ 1 ′ ′ Sm (un ) an (un ,Dun ) ⇀ Sm (u)hm+1 , weakly in N Y i=1 ′ L p ( Q,w∗i ), (5.48) 91 Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems 1, p when n tends to +∞. The strong convergence of ( Tk (u))µ to Tk (u) in L p (0,T;W0 as µ tends to +∞, then we conclude that Z TZ tZ ′ lim lim Sm (un ) an (un ,Dun ) D ( Tk (u))µ dxdsdt (Ω,w)) µ →+ ∞ n →+ ∞ 0 0 Ω TZ tZ ′ (u)hm+1 DTk (u)dxdsdt, = Sm 0 0 Ω Z (5.49) ′ (r ) = 1 f or |r | ≤ m. Now for k ≤ m we have, as soon as k ≤ m, Sm a( Tm+1 (un ),DTm+1 (un ))χ{|un |<k} = a( Tk (un ),DTk (un ))χ{|un |<k} , a.e. in Q, which implies that, passing to the limit as n → +∞, hm+1 χ{|un |<k} = hk χ{|u|<k} , a.e. in Q −{| u| = k}, for k ≤ m. (5.50) As a consequence of (5.50) we have for k ≤ m, hm+1 DTk (u) = hk DTk (u), a.e. in Q. (5.51) Recalling (5.48), (5.49), (5.51) we conclude that (5.36) holds true and the proof of Lemma 5.3 is complete. In this lemma we prove the following monotonicity estimate: Lemma 5.4. The subsequence of un satisfies for any k ≥ 0 Z TZ tZ lim [a( Tk (un ),DTk (un ))− a( Tk (un ),DTk (u))] n →+ ∞ 0 0 Ω ×[ DTk (un )− DTk (u)]dxdsdt = 0. (5.52) Proof. Let k ≥ 0 be fixed. The character (3.9) of a( x,t,s,d) with respect to d implies that Z TZ tZ [a( Tk (un ),DTk (un ))− a( Tk (un ),DTk (u))] lim n →+ ∞ 0 0 Ω ×[ DTk (un )− DTk (u)]dxdsdt ≥ 0. (5.53) To pass to the limit-sup as n tends to +∞ in (5.53) imply that a( Tk (un ),DTk (u)) → a( Tk (u),DTk (u)), a.e. in Q, and that, |ai ( Tk (un ),DTk (u))|  1 p 1 p′ q p′ ≤ βwi ( x) k( x,t)+ σ | Tk (un )| + N X j=1 1 p′ ∂Tk (u) w j ( x) ∂x j p −1   , a.e. in Q, 92 Y. Akdim et al. / J. Partial Diff. Eq., 26 (2013), pp. 76-98 uniformly with respect to n. It follows that when n tends to +∞ a( Tk (un ),DTk (u)) → a( Tk (u),DTk (u)), strongly in N Y ′ L p ( Q,w∗i ). (5.54) i=1 Lemma 5.3, weak convergence of DTk (un ), a( Tk (un ), DTk (un )) and (5.54) make it possible to pass to the limit-sup as n → +∞ in (5.53) and to obtain the result. In this lemma we identify the weak limit hk and we prove the weak-L1 convergence of the “truncated” energy a( T (un ),DTk (un )) DT (un ) as n tends to +∞. Lemma 5.5. For fixed k ≥ 0, we have hk = a( T (u),DTk (u)), a.e. in Q, (5.55) 1 a( T (un ),DTk (un )) DT (un ) ⇀ a( T (u),DTk (u)) DTk (u), weakly in L ( Q). (5.56) Proof. The proof is standard once we remark that for any k ≥ 0, any n > k and any d ∈ R N an ( Tk (un ),d) = a( Tk (un ),d), a.e. in Q, which together with weak convergence of ( Tk (un )), a( DTk (un )) and (5.54) we obtain from (5.52) lim Z n →+ ∞ 0 TZ tZ 0 Ω a( Tk (un ),DTk (un )) DTk (un )dxdsdt = Z 0 TZ tZ 0 Ω hk DTk (u)dxdsdt. (5.57) The usual Minty’s argument applies in view of weak convergence of ( Tk (un )), a( DTk (un )) and (5.57). It follows that (5.55) hold true. In order to prove (5.56), we observe that monotone character of a and (5.52) give that for any k ≥ 0 and any T ′ < T [a( Tk (un ),DTk (un ))− a( Tk (u),DTk (u))][ DTk (un )− DTk (u)] → 0 (5.58) strongly in L1 ((0,T ′ )× Ω) as n → +∞. Moreover, weak convergence of ( Tk (un )) and a( DTk (un )), (5.58), (5.54) and (5.55) imply that a( Tk (un ),DTk (un )) DTk (u) ⇀ a( Tk (u),DTk (u)) DTk (u), weakly in L1 ( Q), and a( Tk (un ),DTk (u)) DTk (u) → a( Tk (un ),DTk (u)) DTk (u), strongly in L1 ( Q) as n → +∞. Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems 93 Using the above convergence results in (5.58) shows that for any k ≥ 0 and any T ′ < T a( Tk (un ),DTk (un )) DTk (un )⇀ a( Tk (u),DTk (u)) DTk (u) weakly in L1 ((0,T ′ )× Ω), (5.59) as n → +∞. At the possible expense of extending the functions a( x,t,s,d), f on a time interval (0, T̄ ) with T̄ > T in such a way that assumptions with a and f hold true with T̄ in place of T, we can show that the convergence result (5.59) is still valid in L1 ( Q)-weak, namely that (5.56) holds true. Step 4: In this step we prove that u satisfies (5.2). Lemma 5.6. The limit u of the approximate solution un of (5.11) satisfies Z lim a(u,Du) Dudxdt = 0. m →+ ∞ { m ≤|u |≤ m +1} Proof. To this end, observe that for any fixed m ≥ 0, one has Z Z a(un ,Dun ) Dun dxdt = a(un ,Dun )( DTm+1 (un )− DTm (un ))dxdt Q { m ≤|un |≤ m +1} Z Z = a( Tm+1 (un ),DTm+1 (un )) DTm+1 (un )dxdt − a( Tm (un ),DTm (un )) DTm (un )dxdt. Q Q According to (5.56), one is at liberty to pass to the limit as n → +∞ for fixed m ≥ 0 and to obtain Z a(un ,Dun ) Dun dxdt lim n →+ ∞ { m ≤|u |≤ m +1} n Z Z = a( Tm+1 (u),DTm+1 (u)) DTm+1(u)dxdt − a( Tm (u),DTm (u)) DTm (u)dxdt Q Q Z a(u,Du) Dudxdt. (5.60) = { m ≤|un |≤ m +1} Taking the limit as m → +∞ in (5.60) and using the estimate (5.30) show that u satisfies (5.2) and the proof of the lemma is complete. Step 5: In this step, u is shown to satisfy (5.3) and (5.4). Let S be a function in W 1,∞ (R ) such that S has a compact support. Let M be a positive real number such that supp(S′ ) ⊂ [− M, M ]. Pointwise multiplication of the approximate equation (5.11) by S′ (un ) leads to ∂BSn (un ) − div[S′ (un ) a(un ,Dun )]+ S′′ (un ) a(un ,Dun ) Dun + div(S′ (un )φn (un )) ∂t − S′′ (un )φn (un ) Dun = f S′ (un ), in D ′ ( Q). (5.61) 94 Y. Akdim et al. / J. Partial Diff. Eq., 26 (2013), pp. 76-98 It was follows we pass to the limit as in (5.61) n tends to +∞. • Limit of ∂BSn (un )/∂t. Since S is bounded and continuous, un → u a.e. in Q implies that BSn (un ) converges to BS (u) a.e. in Q and L∞ weak∗ . Then ∂BSn (un )/∂t converges to ∂BS (u)/∂t in D ′ ( Q) as n tends to +∞. • Limit of −div[S′ (un ) an (un ,Dun )]. Since supp(S′ ) ⊂ [− M, M ], we have for n ≥ M S′ (un ) an (un ,Dun ) = S′ (un ) a( TM (un ),DTM (un )), a.e. in Q. The pointwise convergence of un to u and (5.55) as n tends to +∞ and the bounded character of S′ permit us to conclude that ′ ′ S (un ) an (un ,Dun ) ⇀ S (u) a( TM (u),DTM (u)), in N Y ′ L p ( Q,w∗i ), (5.62) i=1 as n tends to +∞. S′ (u) a( TM (u),DTM (u)) has been denoted by S′ (u) a(u,Du) in (5.3). • Limit of S′′ (un ) a(un ,Dun ) Dun . As far as the ’energy’ term S′′ (un ) a(un ,Dun ) Dun = S′′ (un ) a( TM (un ),DTM (un )) DTM (un ), a.e. in Q. The pointwise convergence of S′ (un ) to S′ (u) and (5.56) as n tends to +∞ and the bounded character of S′′ permit us to conclude that S′′ (un ) an (un ,Dun ) Dun ⇀ S′′ (u) a( TM (u),DTM (u)) DTM (u), weakly in L1 ( Q). Recall that S′′ (u) a( TM (u),DTM (u)) DTM (u) = S′′ (u) a(u,Du) Du, a.e. in Q. • Limit of S′ (un )φn (un ). Since supp(S′ ) ⊂ [− M, M ], we have S′ (un )φn (un ) = S′ (u)φn ( TM (u)), a.e. in Q. As a consequence of (5.8) and un → u, a.e. in Q, it follows that ′ ′ S (un )φn (un ) → S (u)φ( TM (u)), strongly in N Y ′ L p ( Q,w∗i ), i=1 as n tends to +∞. The term S′ (u)φ( TM (u)) is denoted by S′ (u)φ(u). • Limit of S′′ (un )φn (un ) Dun . (5.63) Existence of a Renormalised Solutions for a Class of Nonlinear Degenerated Parabolic Problems 95 Since S′ ∈ W 1,∞ (R ) with supp(S′ ) ⊂ [− M, M ], we have S′′ (un )φn (un ) Dun = φn ( TM (un )) DS′ (un ), a.e. in Q. Moreover, DS′ (un ) converges to DS′ (u) weakly in L p ( Q,w) as n tends to +∞, while φn ( TM (un )) is uniformly bounded with respect to n and converges a.e. in Q to φ( TM (u)) as n tends to +∞. Therefore S′′ (un )φn (un ) Dun ⇀ φ( TM (u)) DS′ (u), weakly in L p ( Q,w). The term φ( TM (u)) DS′ (u) = S′′ (un )φ(u) Du. • Limit of S′ (un ) f n . Due to (5.9) and un → u a.e. in Q, we have S′ (un ) f n → S′ (u) f , strongly in L1 ( Q) as n → +∞. As a consequence of the above convergence result, we are in a position to pass to the limit as n tends to +∞ in Eq. (5.61) and to conclude that u satisfies (5.3). It remains to show that BS (u) satisfies the initial condition (5.4). To this end, firstly remark that, S being bounded, BSn (un ) is bounded in L∞ ( Q). Secondly, (5.61) and the above considerations on the behavior of the terms of this equation show that ∂BSn (un )/∂t ′ ′ is bounded in L1 ( Q)+ L p (0,T;W −1,p (Ω,w∗ )). As a consequence, an Aubin’s type lemma (see, e.g., [14]) implies that BSn (un ) lies in a compact set of C0 ([0,T ], L1 (Ω)). It follows that on the one hand, BSn (un )(t = 0) = BSn (u0n ) converges to BS (u)(t = 0) strongly in L1 (Ω). On the other hand, the smoothness of S implies that BS (u)(t = 0) = BS (u0 ), in Ω. As a conclusion of step 1 to step 5, the proof of Theorem 5.1 is complete. 6 Example Let us consider the following special case: b(r) = exp( βr)− 1, φ : r ∈ R → ( φi )i=1,··· ,N ∈ R N , where φi (r) = exp(αi r), i = 1, ··· , N, αi ∈ R, φ is a continuous function. And, ai ( x,t,d) = wi ( x)|di | p−1 sgn(di ), i = 1, ··· , N, with wi ( x) a weight function (i = 1, ··· , N ). For simplicity, we suppose that wi ( x) = w( x), for i = 1, ··· , N − 1, w N ( x) ≡ 0. 96 Y. Akdim et al. / J. Partial Diff. Eq., 26 (2013), pp. 76-98 It is easy to show that the ai (t,x,d) are Caratheodory functions satisfying the growth condition (3.8) and the coercivity (3.10). On the order hand the monotonicity condition is verified. In fact, N X i=1
ai ( x,t,d)− a( x,t,d′ ) (di − d′i ) =w( x) N −1  X |di | p−1 sgn(di )− d′i i=1 p −1  sgn(d′i ) (di − d′i ) ≥ 0, for almost all x ∈ Ω and for all d,d′ ∈ R N . This last inequality can not be strict, since for d 6= d′ with d N 6= d′N and di = d′i , i = 1, ··· , N − 1, the corresponding expression is zero. In particular, let us use special weight function, w, expressed in terms of the distance to the bounded ∂Ω. Denote d( x) = dist( x,∂Ω) and set w( x) = dλ ( x), such that  p , p−1 . (6.1) λ < min N Remark 6.1. The condition (6.1) is sufficient for (3.4). Finally, the hypotheses of Theorem 5.1 are satisfied. Therefore, for all f ∈ L1 ( Q), the following problem:  u ∈ L∞ ([0,T ]; L1 (Ω));    1, p  p  Tk (u) ∈  Z L (0,T;W0 (Ω,w)),      a(u,Du) Dudxdt = 0; lim   m →+ ∞ { m ≤|u |≤ m +1}   Z  r    B ( r ) = β(expβσ)S(σ)dσ,  S   0Z    Z   N  P ∂ϕ ∂u ∂ϕ ∂u p−1   − BS (u) dxdt + S(u) wi sgn dxdt  ∂t ∂xi ∂xi ∂xi Q i=1  (6.2)  ZQ N  P ∂u ∂u ∂u p−1  ′  sgn ϕdxdt + S (u) wi   ∂xi ∂xi ∂xi  Q i = 1  Z N Z N   P P ′ ∂ϕ ∂u    S ( u ) exp ( α u ) + S (u)exp(αi u) dxdt − ϕdxdt i   ∂xi ∂xi  Q i=1 Q i=1 Z      = f S′ (u) ϕdxdt,    Q    B ( u )( t = 0) = BS (u0 ), in Ω,  S   ∀ ϕ ∈ C0∞ ( Q) and S ∈ W 1,∞ (R ) with S′ ∈ C0∞ (R ), has at least one renormalised solution. Remark 6.2. 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