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PARABOLIC INEQUALITIES WITH NONSTANDARD GROWTHS AND L 1 DATA R. ABOULAICH, B. ACHCHAB, D. MESKINE, AND A. SOUISSI Received 25 July 2005; Revised 13 December 2005; Accepted 19 December 2005 We prove an existence result for solutions of nonlinear parabolic inequalities with L 1 data in Orlicz spaces. Copyright © 2006 R. Aboulaich et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let Ω be an open bounded subset of R N , N ≥ 2, let Q be the cylinder Ω × (0,T)with some given T>0. Consider the following nonlinear parabolic problem: ∂u ∂t + A(u) = χ in Q, u(x,t) = 0on∂Ω × (0,T), u(x,0) = u 0 in Ω, (1.1) where A(u) =−div(a(x,t,u,∇u)) is a Leray-Lions operator defined on D(A)⊂ W 1,x 0 L M (Ω), with M is an N-function, and χ is a given data. In the variational case (i.e., where χ ∈ W −1,x E M (Ω)), it is well known that the solvabil- ity of (1.1) is done by Donaldson [2]andRobert[11] when the operator A is monotone, t 2 M(t), and M satisfies a Δ 2 condition, and by finally the recent work [3]forthegen- eral case. In the L 1 case, an existence theorem is given in [4]. However, the techniques used in [4] do not allow us to adapt it for parabolic inequalities. It is our purpose in this paper to solve the obstacle problem associated to (1.1) in the case where χ ∈ L 1 (Q)+W −1,x E M (Q) and without assuming any growth restriction on M. The existence of solutions is proved via a sequence of penalized problems, with solutions u n . A priori estimates of the tr un- cation of u n are obtained in some suitable Orlicz space. For the passage to the limit, the Hindawi Publishing Corporation Boundary Value Problems Volume 2006, Article ID 29286, Pages 1–18 DOI 10.1155/BVP/2006/29286 2 Parabolic inequalities in L 1 almost everywhere convergence of ∇u n is proved via new techniques. As operators mod- els, we can consider slow or fast growth: A(u) =−div 1+|u| 2 ∇u log 1+|∇u| |∇u| , A(u) =−div(∇uexp |∇ u| . (1.2) For some classical and recent results in the setting of Orlicz spaces dealing with elliptic and parabolic equations, the reader can be referred to [8, 10, 12–14]. 2. Preliminaries 2.1. Let M : R + → R + be an N-function, that is, M is continous, convex, with M(t) > 0 for t>0, M(t)/t → 0ast → 0, and M(t)/t →∞as t →∞. Equivalently, M admits the representation M(t) = t 0 a(s)ds,wherea : R + → R + is non- decreasing, right continuous, with a(0) = 0, a(t) > 0fort>0, and a(t)tendsto∞ as t →∞. The N-function M conjugate to M is defined by M(t) = t 0 ¯ a(s)ds,wherea : R + → R + is given by ¯ a(t) = sup{s : a(s) ≤ t} (see [1]). The N-function is said to satisfy the Δ 2 condion if, for some k>0, M(2t) ≤ kM(t), ∀t ≥ 0, (2.1) when (2.1)holdsonlyfort ≥ some t 0 > 0, then M is said to satisfy the Δ 2 condition near infinity. We will extend these N-functions into even functions on all R. Let P and Q be two N-functions. P Q means that P grows essentially less rapidly than Q, that is, for each > 0, P(t)/Q(t) → 0ast →∞. This is the case if and only if lim t→∞ (Q −1 (t))/(P −1 (t)) = 0. 2.2. Let Ω be an open subset of R N .TheOrliczclassK M (Ω) (resp., the Orlicz space L M (Ω)) is defined as the set of (equivalence classes of) real-valued measurable functions u on Ω such that Ω M u(x) dx < +∞ resp ., Ω M u(x) λ dx < +∞ for some λ>0 . (2.2) L M (Ω) is a Banach space under the norm u M,Ω = inf λ>0: Ω M u(x) λ dx ≤ 1 (2.3) and K M (Ω)isaconvexsubsetofL M (Ω). The closure in L M (Ω) of the set of bounded measurable functions with compact sup- port in Ω is denoted by E M (Ω). The equality E M (Ω) = L M (Ω) holds if and only if M satisfies the Δ 2 condition, for all t or for t large, according to whether Ω has infinite measure or not. R. Aboulaich et al. 3 The dual of E M (Ω) can be identified with L M (Ω) by means of the pairing Ω uv dx,and the dual norm of L M (Ω)isequivalentto· M,Ω . The space L M (Ω)isreflexiveifandonlyifM and M satisfy the Δ 2 condition, for all t or for t large, according to whether Ω has infinite measure or not. 2.3. We now turn to the Orlicz-Sobolev space, W 1 L M (Ω)(resp.,W 1 E M (Ω)) is the space of all functions u such that u and its distributional derivatives up to order 1 lie in L M (Ω) (resp., E M (Ω)). It is a Banach space under the norm u 1,M = |α|≤1 D α u M . (2.4) Thus, W 1 L M (Ω)andW 1 E M (Ω) can be identified with subspaces of product of N +1 copies of L M (Ω). Denoting this product by L M , we will use the weak topologies σ( L M , E M )andσ( L M , L M ). The space W 1 0 E M (Ω) is defined as the (norm) closure of the Schwartz space D(Ω)in W 1 E M (Ω) and the space W 1 0 L M (Ω)astheσ( L M , E M )closureofD(Ω)inW 1 L M (Ω). We say that u n converges to u for the modular convergence in W 1 L M (Ω)ifforsomeλ>0, Ω M D α u n − D α u λ dx −→ 0, ∀|α|≤1. (2.5) This implies convergence for σ( L M , L M ). If M satisfies the Δ 2 condition on R + ,then modular convergence coincides with norm convergence. 2.4. Let W −1 L M (Ω)(resp.,W −1 E M (Ω)) denote the space of distributions on Ω which can be written as sums of derivatives of order ≤ 1 of functions in L M (resp., E M (Ω)). It is a Banach space under the usual quotient norm. If the open set Ω has the segment property, then the space D(Ω) is dense in W 1 0 L M (Ω) for the modular convergence and thus for the topology σ( L M , L M )(cf.[6, 7]). Con- sequently, the action of a distribution in W −1 L M (Ω) on an element of W 1 0 L M (Ω)iswell defined. 2.5. Let Ω be a bounded open subset of R N , T>0, and set Q = Ω × (0,T). Let M be an N-function. For each α ∈ N N , denote by D α x the distributional derivatives on Q of order α withrespecttothevariablex ∈ R N . The inhomogeneous Orlicz-Sobolev spaces of order 1 are defined as follows: W 1,x L M (Q) = u ∈ L M (Q):D α x u ∈ L M (Q), ∀|α|≤1 , W 1,x E M (Q) = u ∈ E M (Q):D α x u ∈ E M (Q), ∀|α|≤1 . (2.6) The latest space is a subset of the first one. They are Banach spaces under the norm u= |α|=1 D α x u M,Q . (2.7) We can easily show that they form a complementary system when Ω satisfies the seg- ment property.These spaces are considered as subspaces of the product spaces L M (Q) 4 Parabolic inequalities in L 1 which has N + 1 copies. We will also consider the weak topologies σ( L M , E M )and σ( L M , L M ). If u ∈ W 1,x L M (Q), then the function t → u(t) = u(·,t)isdefinedon (0,T) with values in W 1 L M (Ω). If, further, u ∈ W 1,x E M (Q), then u(t)isW 1 E M (Ω)- valued and is strongly measurable. Furthermore, the following continuous imbedding holds: W 1,x E M (Q) ⊂ L 1 (0,T;W 1 E M (Ω)). The space W 1,x L M (Q) is not in general sepa- rable, if u ∈ W 1,x L M (Q), we cannot conclude that the function u(t)ismeasurablefrom (0,T)intoW 1 L M (Ω). However, the scalar function t →D α x u(t) M,Ω is in L 1 (0,T)forall |α|≤1. 2.6. The space W 1,x 0 E M (Q)isdefinedasthe(norm)closureinW 1,x E M (Q)ofD(Q). We can easily show as in [7] that when Ω has the segment property, then for all u ∈ D( Q) σ( L M , E M ) there exist some λ>0 and a sequence (u n ) ⊂ D(Q) such that for all |α|≤1, Ω M((D α x u n − D α x u)/λ)dx → 0whenn →∞. Consequently, D(Q) σ( L M , E M ) = D( Q) σ( L M , L M ) , this space w ill b e denoted by W 1,x 0 L M (Q). Furthermore, W 1,x 0 E M (Q) = W 1,x 0 L M (Q) ∩ E M .Poincar ´ e’s inequality also holds in W 1,x 0 L M (Q) and then there is a constant C>0 such that for all u ∈ W 1,x 0 L M (Q), one has |α|≤1 D α x u M,Q ≤ C |α|=1 D α x u M,Q , (2.8) thus both sides of the last inequality are equivalent norms on W 1,x 0 L M (Q). We have then the following complementary system: W 1,x 0 L M (Q) F W 1,x 0 E M (Q) F 0 , (2.9) F being the dual space of W 1,x 0 E M (Q). It is also, up to an isomorphism, the quotient of L M by the polar set W 1,x 0 E M (Q) ⊥ , and will be denoted by F = W −1,x L M (Q) and it is shown that W −1,x L M (Q) = f = |α|≤1 D α x f α : f α ∈ L M (Q) . (2.10) This space will be equipped with the usual quotient norm: f =inf |α|≤1 f α M,Q , (2.11) where the inf is taken on all possible decompositions f = |α|≤1 D α x f α , f α ∈ L M (Q). The space F 0 is then given by F 0 ={f = |α|≤1 D α x f α : f α ∈ E M (Q)} and is denoted by F 0 = W −1,x E M (Q). Defint ion 2.1. We say that u n → u in W −1,x L M (Q)+L 1 (Q) for the modular convergence if we can write u n = |α|≤1 D α x u α n + u 0 n , u = |α|≤1 D α x u α + u 0 (2.12) R. Aboulaich et al. 5 with u α n → u α in L M (Q) for the modular convergence for all |α|≤1andu 0 n → u 0 strongly in L 1 (Q). We will give the following approximation theorem which plays a crucial role when proving the existence result of solutions for parabolic inequalities. Theorem 2.2. Let φ ∈ W 1,x 0 E M (Q) ∩ L ∞ (Q) and consider the convex s et φ ={v ∈ W 1,x 0 L M (Q):v ≥ φ a.e. in Q}. Then for every u ∈ φ ∩ L ∞ (Q) such that ∂u/∂t ∈ W −1,x L M (Q)+L 1 (Q),thereexistsv j ∈ φ ∩ D(Q) such that v j −→ u in W 1,x L M (Q), ∂v j ∂t −→ ∂u ∂t in W −1,x L M (Q)+L 1 (Q) (2.13) for the modular convergence. Proof. It is easily adapted from that g iven in [4, Theorem 3] and the approximation tech- niques of [9]. Remark 2.3. The result is still true for φ ∈ W 1,x E M (Q) ∩ L ∞ (Q), when Ω is more regular (see [9]). In order to deal with the time derivative, we introduce a time mollification of a func- tion v ∈ L M (Q). Thus, we define, for all μ>0andall(x, t) ∈ Q, v μ (x, t) = μ t −∞ v(x,s)exp μ(s − t) ds, (2.14) where v(x,s) = v(x,s)χ (0,T) (s) is the zero extension of v. The following proposition is fun- damental in the sequel. Proposition 2.4 [5]. If v ∈ L M (Q), then v μ is measurable in Q, ∂v μ /∂t = μ(v − v μ ) and Q M v μ dxdt ≤ Q M(v)dxdt. (2.15) Recall now the following compactness result which is proved in [5]. Proposition 2.5. Assume that (u n ) n is a bounded sequence in W 1 0 L M (Q) such that ∂u n /∂t is bounded in W −1,x L M (Q)+L 1 (Q), then u n is relatively compact in L 1 (Q). 3. The main result Let Ω be an open bounded subset of R N , N ≥ 2, w ith the segment propert y. Let P and M be two N-functions such that P M. Consider now the operator A : D(A) ⊂ W 1,x 0 L M (Q) → W −1 L M (Q)indivergenceformA(u) =−div(a(x,t,u,∇u)), where a : Ω × R × R × R N → R N is a Carath ´ eodory function satisfy ing for a.e. x ∈ Ω and for all ζ,ζ ∈ R N , 6 Parabolic inequalities in L 1 (ζ = ζ )andalls,t ∈ R: a(x, t,s,ζ) ≤ h(x,t)+k 1 P −1 M k 2 |s| + k 3 M −1 M k 4 |ζ| , a(x, t,s,ζ) − a x, t,s,ζ ζ − ζ > 0, a(x, t,s,ζ)ζ ≥ αM | ζ| − d(x, t), (3.1) with d ∈ L 1 (Q), α,k 1 ,k 2 ,k 3 ,k 4 > 0, and h ∈ E M (Q). Let ψ ∈ W 1 0 E M (Ω) ∩ L ∞ (Ω). (3.2) Finally, consider f ∈ L 1 (Q). (3.3) We define for all t ∈ R, k ≥ 0, T k (t) = max(−k,min(k,t)), and S k (t) = t 0 T k (η)dη. We will prove the following existence theorem. Theorem 3.1. Let u 0 ∈ L 1 (Ω) such that u 0 ≥ 0. Assume that (3.1)–(3.3)holdtrue.Then thereexistsatleastonesolutionu ∈ C([0,T]; L 1 (Ω)) such that u(x,0) = u 0 a.e. and for all τ ∈]0,T], u ≥ ψ a.e. in Q, T k (u) ∈ W 1,x 0 L M (Q), Ω S k u(τ) − v(τ) dx + ∂v ∂t ,T k (u − v) Q τ + Q τ a(x, t,u,∇u)∇T k (u − v)dxdt ≤ Q τ fT k (u − v)dxdt + Ω S k u 0 − v(x,0) dx, ∀k>0 and ∀v ∈ ψ ∩ L ∞ (Q) such that ∂v ∂t ∈ W −1,x L M (Q)+L 1 (Q), (p ψ ) where Q τ = Ω×]0,τ[. Remark 3.2. Since {v ∈ ψ ∩ L ∞ (Q):∂v/∂t ∈ W −1,x L M (Q)+L 1 (Q)}⊂C([0,T],L 1 (Ω)), (see [4]), the first and the latest terms of problem (p ψ )arewelldefined. Proof Step 1. Aprioriestimates. For the sake of simplicity, we assume that d(x,t) = 0. Consider the approximate equations ∂u n ∂t − div a x, t,u n ,∇u n − nT n u n − ψ) − = f n , u n ∈ W 1,x 0 L M (Q), u n (x,0)= u n 0 , (P n ) R. Aboulaich et al. 7 where f n → f strongly in L 1 (Q)andu n 0 → u 0 strongly in L 1 (Ω). Thanks to [3,Theo- rem 3.1], there exists at least one solution u n of problem (P n ). By choosing T k (u n − T h (u n )),h ≥ψ ∞ as test function in (P n ), we get ∂u n ∂t ,T k u n − T h u n + h≤|u n |≤h+k a u n ,∇u n ∇ u n dxdt − Q nT n u n − ψ − T k u n − T h u n dxdt = Q f n T k u n − T h u n dxdt. (3.4) On the one hand, we have ∂u n ∂t ,T k u n − T h u n = Ω S h k u n (T) dx − Ω S h k u 0 n dx, (3.5) where S h k (s) = s 0 T k (q − T h (q))dq, and by using the fact that Ω S h k (u n (T))dx ≥ 0and| Ω S h k (u 0 n )|≤ku 0 n 1 ,weget α h≤|u n |≤h+k M ∇ u n dxdt − Q nT n u n − ψ) − T k u n − T h u n dxdt ≤ Ck, ∀n ∈ N, (3.6) so that − Q nT n u n − ψ − T k u n − T h u n k dxdt ≤ C. (3.7) Since −nT n (u n − ψ) − T k (u n − T h (u n )) ≥ 0, for every h ≥ψ ∞ , we deduce by Fatou’s lemma as k → 0that Q nT n u n − ψ − ≤ C. (3.8) Using in (P n ) the test function T k (u n )χ(0,τ), we get for every τ ∈ (0,T), Ω S k u n (τ) dx + Q τ a x, t,T k u n ,∇T k u n ∇ T k u n dxdt + Q τ nT n u n − ψ − T k u n dxdt ≤ Ck (3.9) which gives thanks to (3.8) Ω S k u n (τ) dx + Q τ a x, t,T k u n ,∇T k u n ∇ T k u n dxdt ≤ Ck, (3.10) Q M ∇ T k u n dxdt ≤ Ck. (3.11) 8 Parabolic inequalities in L 1 On the other hand, by using [6, Lemma 5.7], there exist two positive constants μ 1 and μ 2 such that Q M T k u n μ 1 dxdt ≤ μ 2 Q M ∇ T k u n dxdt (3.12) which implies, by using (3.11), that meas u n >k ≤ μ 2 Ck M k/μ 1 (3.13) so that lim k→∞ meas u n >k = 0 uniformly with respect to n. (3.14) Take now a nondecreasing function θ k ∈ C 2 (R)suchthatθ k (s) = s for |s|≤k/2and θ k (s) = ksign(s)for|s| >k. By multiplying the approximate equation by θ k (u n ), we get ∂θ k u n ∂t − div a x, t,u n ,∇u n θ u n + a x, t,u n ,∇u n ∇ u n θ u n − nT n u n − ψ − θ k u n = f n θ k u n , (3.15) which implies that ∂θ k (u n )/∂t is bounded in W −1,x L M (Q)+L 1 (Q). Since θ k (u n )isbound- ed in W 1,x 0 L M (Q), we have by Proposition 2.5 that θ k (u n ) is relatively compact in L 1 (Q) and so that u n → u a.e. in Q,andfrom(3.8) by using Fatou’s lemma, we get u ≥ ψ a.e. in Q. Consequently, T k u n −→ T k (u) weakly in W 1,x 0 L M (Q) (3.16) for the topology σ( L M , E M ). Step 2. Almost everywhere convergence of the gradients. Since T k (u) ∈ W 1,x 0 L M (Q), then there exists a sequence (α k j ) ⊂ D(Q)suchthatα k j → T k (u) for the modular convergence in W 1,x 0 L M (Q). In the sequel and throughout the pa- per, χ j,s and χ s will denote, respectively, the characteristic functions of the sets Q j,s = { (x, t) ∈ Ω : |∇T k (α k j )|≤s} and Q s ={(x,t) ∈ Ω : |∇T k (u)|≤s}. For the sake of sim- plicity, we will write only (n, j,μ,s) to mean all quantities (possibly different) such that lim s→∞ lim μ→∞ lim j→∞ lim n→∞ (n, j,μ,s) = 0. Taking now T η (u n − T k (α k j ) μ ), η>0 as test function in (P n ), we get ∂u n ∂t ,T η u n − T k α k j μ + Q a x, u n ,∇u n ∇ T η u n − T k α k j μ − Q nT n u n − ψ − T η u n − T k α k j μ dxdt ≤ Cη, (3.17) and by using (3.8), we get ∂u n ∂t ,T η u n − T k α k j μ + Q a u n ,∇u n ∇ T η u n − T k α k j μ ≤ Cη. (3.18) R. Aboulaich et al. 9 The first term of the left-hand side of the last equality reads as ∂u n ∂t ,T η u n − T k α k j μ = ∂u n ∂t − ∂T k α k j μ ∂t ,T η u n − T k α k j μ + ∂T k α k j μ ∂t ,T η u n − T k α k j μ . (3.19) The second term of the last equality can be written as ∂u n ∂t − ∂T k α k j μ ∂t ,T η u n − T k α k j μ = Ω S η u n (T) − T k α k j μ (T) dx − Ω S η u n 0 dx ≥−η Ω u n 0 dx ≥−ηC, (3.20) the third term can be written as ∂T k α k j μ ∂t ,T η u n − T k α k j μ = μ Q T k α k j − T k α k j μ T η u n − T k α k j μ , (3.21) thus by letting n, j →∞and since α k j → T k (u)a.e.inQ and by using Lebesgue theorem, Q T k α k j − T k α k j μ T η u n − T k α k j μ dxdt = Q T k (u) − T k (u) μ T η u − T k (u) μ dxdt + (n, j). (3.22) Consequently, ∂u n ∂t ,T η T k u n − T k α k j μ ≥ (n, j) − ηC. (3.23) On the other hand, Q a u n ,∇u n ∇ T η u n − T k α k j μ dxdt = {|T k (u n )−T k (α k j ) μ |<η} a T k u n ,∇T k u n ∇ T k u n −∇ T k α k j μ χ j,s dxdt + {k<|u n |}∩{|u n −T k (α k j ) μ |<η} a u n ,∇u n ∇ u n dxdt − {k<|u n |}∩{|u n −T k (α k j ) μ |<η} a u n ,∇u n ∇ T k α k j μ χ {|∇T k (α k j )|>s} dxdt (3.24) 10 Parabolic inequalities in L 1 which implies, by using the fact that {k<|u n |}∩{|u n −T k (α k j ) μ |<η} a(u n ,∇u n )∇u n dxdt ≥ 0, that {|T k (u n )−T k (α k j ) μ |<η} a u n ,∇u n ∇ T k u n −∇ T k α k j μ χ j,s dxdt ≤ Cη + {k<|u n |}∩{|u n −T k (α k j ) μ |<η} a u n ,∇u n ∇ T k α k j μ χ {|∇T k (α k j )|>s} dxdt. (3.25) Since a(T k+η (u n ),∇T k+η (u n )) is bounded in (L M (Q)) N , there exists some h k+η ∈(L M (Q)) N such that a T k+η u n ,∇T k+η u n h k+η weakly in L M (Q) N for σ L M , E M . (3.26) Consequently, {k<|u n |}∩{|u n −T k (α k j ) μ |<η} a u n ,∇u n ∇ T k α k j μ χ {|∇T k (α k j )|>s} dxdt = {k<|u|}∩{|u−T k (α k j ) μ |<η} h k+η ∇T k α k j μ χ {|∇T k (α k j )|>s} dxdt + (n), (3.27) where we have used the fact that ∇T k (α k j ) μ χ {k<|u n |}∩{|u n −T k (α k j ) μ |<η} tends strongly to ∇T k (α k j ) μ χ {k<|u|}∩{|u−T k (α k j ) μ |<η} in (E M (Q)) N . Letting j →∞,weobtain {k<|u n |}∩{|u n −T k (α k j ) μ |<η} a u n ,∇u n ∇ T k α k j μ χ {|∇T k (α k j )|>s} dxdt = {k<|u|}∩{|u−T k (u) μ |<η} h k+η ∇T k (u) μ χ {|∇T k (u)|>s} dxdt + (n, j). (3.28) Thanks to Proposition 2.4, one easily has {k<|u|}∩{|u−T k (u) μ |<η} h k+η ∇T k (u) μ χ {|∇T k (u)|>s} dxdt = {k<|u|}∩{|u−T k (u)|<η} h k+η ∇T k (u)χ {|∇T k (u)|>s} dxdt + (μ) = (μ,s). (3.29) Hence {|T k (u n )−T k (α k j ) μ |<η} a T k u n ,∇T k u n ∇ T k u n −∇ T k α k j μ χ j,s dxdt ≤ Cη + (n, j,μ,s). (3.30) [...]... [8] Analysis and Its Applications, Part 1 (Berkeley, Calif, 1983), Proc Sympos Pure Math., vol 45, American Mathematical Society, Rhode Island, 1986, pp 455–462 [9] J.-P Gossez and V Mustonen, Variational inequalities in Orlicz-Sobolev spaces, Nonlinear Analysis Theory, Methods & Applications 11 (1987), no 3, 379–392 [10] V K Le and K Schmitt, Quasilinear elliptic equations and inequalities with rapidly... evolution in Banach spaces, Ph.D thesis, University of Utah, Utah, 2003 18 Parabolic inequalities in L1 [13] , Weak and strong solvability of parabolic variational inequalities in Banach spaces, Journal of Evolution Equations 4 (2004), no 4, 497–517 [14] M Rudd and K Schmitt, Variational inequalities of elliptic and parabolic type, Taiwanese Journal of Mathematics 6 (2002), no 3, 287–322 ´ R Aboulaich:... Tl ηi dx (3.63) and using (3.55) and (3.59), we see that Ω Sk un (τ) − Φi,l dx ≤ (n, j,μ,i,l) j,μ (3.64) Parabolic inequalities in L1 16 which implies, by writing Ω un (τ) − um (τ) 1 dx ≤ 2 2 Sk Ω Sk un (τ) − Φi,l dx + j,μ Ω Sk um (τ) − Φi,l dx , j,μ (3.65) that Ω Sk un (τ) − um (τ) dx ≤ 2 1 (n,m), (3.66) we deduce then that Ω un (τ) − um (τ) dx ≤ not depending on τ, 2 (n,m), (3.67) and thus (un ) is... Donaldson, Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems, Journal of Differential Equations 16 (1974), no 2, 201–256 [3] A Elmahi and D Meskine, Parabolic equations in Orlicz spaces, Journal of the London Mathematical Society Second Series 72 (2005), no 2, 410–428 , Strongly nonlinear parabolic equations with natural growth terms and L1 data in Orlicz [4] spaces, Portugaliae... (3.70) ∂v j ∂v − → ∂t ∂t for the modular in W −1,x LM (Q) + L1 (Q), we deduce then that Ω Sk u(τ) − v(τ) dx + ≤ Qτ ∂v ,Tk (u − v) ∂t f Tk (u − v)dx dt + which completes the proof Ω + Qτ Qτ a x,t,u, ∇u ∇Tk (u − v)dx dt Sk u(0) − v(0) dx (3.71) R Aboulaich et al 17 Remark 3.3 A similar result can be proved when dealing with the right-hand side in L1 (Q) + W −1,x EM (Q) or replacing the assumption (3.1)... Tk un , ∇Tk (u) δ 0, {|Tk (un )−Tk (αk )μ | . PARABOLIC INEQUALITIES WITH NONSTANDARD GROWTHS AND L 1 DATA R. ABOULAICH, B. ACHCHAB, D. MESKINE, AND A. SOUISSI Received 25 July 2005; Revised 13 December. Variational inequalities in Orlicz-Sobolev spaces, Nonlinear Anal- ysis. Theory, Methods & Applications 11 (1987), no. 3, 379–392. [10] V. K. Le and K. Schmitt, Quasilinear elliptic equations and inequalities. Parabolic inequalities in L 1 [13] , Weak and strong solvability of parabolic variational inequalities in Banach spaces,Journal of Evolution Equations 4 (2004), no. 4, 497–517. [14] M.RuddandK.Schmitt,Variational
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