EXISTENCE OF WEAK SOLUTIONS FOR QUASILINEAR PARABOLIC SYSTEMS IN DIVERGENCE FORM WITH VARIABLE GROWTH

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1 Electronic Journal of Differential Equations, Vol , No. 113, pp ISSN: URL: or EXISTENCE OF WEAK SOLUTIONS FOR UASILINEAR PARABOLIC SYSTEMS IN DIVERGENCE FORM WITH VARIABLE GROWTH MIAOMIAO YANG, YONGIANG FU Communicated by Vicentiu D. Radulescu Abstract. In this article we study the existence of weak solutions for quasilinear parabolic system in divergence form with variable growth. By means of Young measures, Galerkin s approximation method and the theory of variable exponents spaces, we obtain the existence of weak solutions. 1. Introduction and statement of main result The spaces L px and W m,px were first discussed by Kováčik and Rákosník in [24]. Lately, a lot of attention has been paid to the study of various mathematical problems with variable exponent growth conditions; see [9, 11, 12, 14, 16] for the properties of such spaces, and [6, 15, 2] for applications of variable exponent spaces on partial differential equations. The theory of variable exponent spaces has been driven by various problems in nonlinear elastic mechanics, imaging processing, electrorheological fluids and other physics phenomena; see for example [1, 2, 3, 7, 27, 39]. When px is a constant function, Norbert Hungerbühler studied the following problem in [22]: div σ x, ux, Dux = f, x 1.1 ux =, x The classical monotone operator methods developed by [5, 25, 29, 36] cannot be applied here. Norbert Hungerbühler obtain the existence of weak solutions for 1.1 by Young measures which were proposed by Young in [38]. Many applications and developments of Young measures to the calculus of variations, optimal control theory and nonlinear partial differential equations are presented by MacShane, Gamkrelidze and Tarter in [21, 26, 35, 37]. Inspired by the works mentioned above, results from [22] were extended in [18] to the case that σ satisfies variable growth conditions, by Young measures generated by sequences in variable exponent spaces; see [18, 19] for the basic theorems and properties. 21 Mathematics Subject Classification. 35D3, 35K59, 46E35. Key words and phrases. Variable exponent; Young measures; weak solutions; quasilinear parabolic system; Galerkin s approximation. c 218 Texas State University. Submitted November 16, 217. Published May 1,

2 2 M. YANG, Y. FU EJDE-218/113 In this article, we consider the initial and boundary value problem for the quasilinear parabolic system: u t div σ x, t, Dux, t = div f, x, t ux, t =, x, t, T ux, = u x, x, 1.2 where R N N 2 is a bounded open domain, u :, T R m, < T <, =, T, px is Lipschitz continuous and 1 < p := inf x px px p + := sup x px <, f L p x ; M m N, u L 2 ; R m, σ satisfies the conditions H1 H3 below. Inspired by [8], we consider that px only depends x in this problem. The related definition and properties will be given in section 2. In our paper, we denote by M m n the real vector space of m n matrices equipped with the inner product M N := M ij N ij. 1 i m, 1 j n Now we give conditions required for σ in 1.2. H1 Continuity σ :, T M m N M m N is a Carathéodory function, i.e. x, t σx, t, ξ is measurable for every ξ M m N and ξ σx, t, ξ is continuous for almost every x, t. H2 Growth and coercivity There exist c 1, c 2 >, < a L p x, b L 1, such that σx, t, ξ ax, t + c 1 ξ px 1, σx, t, ξ ξ bx, t + c 2 ξ px. H3 Monotonicity σ satisfies one of the following conditions: i For all x, t, ξ σx, t, ξ is a C 1 -function and is monotone, i.e. for all x, t and ξ, η M m N, we have σx, t, ξ σx, t, η ξ η. ii There exists a function W :, T M m N R such that σx, t, ξ = D ξ W x, t, ξ, and ξ W x, t, ξ is convex and C 1 for all x, t. iii σ is strictly monotone, i.e. σ is monotone and σx, t, ξ σx, t, η ξ η = implies ξ = η. iv σx, t, λ σx, t, λ λ λ dνx,t λ dx dt > M m n where λ = ν x,t, I, ν = {ν x,t } x,t is any family of Young measures generated by a bounded sequence in L px and not a Dirac measure for a.e. x, t. Our main result is as follows: Theorem 1.1. If σ satisfies conditions H1 H3, then problem 1.2 has a weak solution for every f L px ; M m N and every u L 2 ; R m.

3 EJDE-218/113 PARABOLIC SYSTEMS IN DIVERGENCE FORM 3 Condition H2 states the variable growth and coercivity condition. H3iv is weaker than typical strictly monotone condition, even than the p-quasimonotone condition introduced by Norbert Hungerbühler in [22] when px is a constant. This article is organized as the following: In Section 2, several important properties on variable exponent spaces and the theory of Young measures will be presented. In Section 3, we will give the Galerkin approximation and necessary priori estimates. In section 4, the existence of weak solutions for problem 1.2 will be proved; the conclusions will be given in section Preliminaries In this section, we recall some facts on variable exponent spaces L px and W k,px. Let P be the set of all Lebesgue measurable functions p : [1, +, where R n n 2 is a nonempty open subset. Denote ρ px u = ux px dx, 2.1 u px = inf{t > : ρ px u 1}. 2.2 t The variable exponent Lebesgue space L px is the class of all functions u such that ρ px t u < for some t >. L px is a Banach space endowed with the norm is called the modular of u in L px. For a given px P, we define the conjugate function p x as {, if x 1 = {x : px = 1}; p x = px px 1, for other x. Lemma 2.1 [9]. Let p P, then ux vx dx 2 u px v p x for every u L px and every v L p x. In the rest of this section, for every p P, we assume that 1 p px p + <. Lemma 2.2 [16]. For every u L px, we have: 1 If u px 1, then u p px ρ pxu u p+ px. 2 If u px < 1, then u p+ px ρ pxu u p px. Lemma 2.3 [16]. If p > 1, L px is reflexive, and the dual space of L px is L p x. Lemma 2.4 [24]. Let <, where denotes the Lebesgue measure of, p 1 x, p 2 x P, then a necessary and sufficient condition for L p2x L p1x is that p 1 x p 2 x for almost every x, and in this case the embedding is continuous. We assume that R d is a bounded domain,, T R, T is a fixed real number, =, T, p is a Lipschitz function. We define { } X := u L 2 d : u L p, d d, uτ, V τ a.e. τ, T

4 4 M. YANG, Y. FU EJDE-218/113 The norm on X is given by where for all τ, T, we have u X := u L2 d + u L p, d d V τ := { u L 2 d W 1,1 d : u L pτ, d d} The norm on V τ is defined by u Vτ := u L 2 d + u L pτ, d d Lemma 2.5 [1]. The space X is a Banach space under the norm X ; and C is density in X. Lemma 2.6 [1]. The space X is reflexible. Lemma 2.7 [1]. The dual space X is isomorphic to the subspace of D consisting of distributions T of the form } T := inf { g L 2 + G d L p, : T = g div G, d d where g L 2 d. Lemma 2.8 [4]. Let R n be Lebesgue measurable not necessarily bounded and z j : R m, j = 1, 2,..., be a sequence of Lebesgue measurable functions. Then there exists a subsequence z k and a family {ν x } x of nonnegative Radon measures on R n, such that i ν x := dν x 1 for almost every x. ii ϕz k ϕ weakly* in L for any ϕ C R m, where ϕx = ν x, ϕ and C R m = {ϕ CR m : lim z ϕz = }. iii If for any R > lim sup meas{x B, R : z k x L} =, L k N then ν x = 1 for almost every x, and for any measurable A there holds ϕz k ϕ = ν x, ϕ weakly in L 1 A for continuous ϕ provided the sequence ϕz k is weakly precompact in L 1 A. Lemma 2.9 [4]. If meas < and ν x is a Young measure generated by the sequence {u j }, then u j converges by measures to u if and only if for a.e. x we have ν x = δ ux. Lemma 2.1 [5]. Let {f j } be a uniformly boundedness in L 1, sup f j L 1 = C <. j There exists a subsequence, not relabled, a nonincreasing sequence of measurable sets n, n+1 n, and f L 1 such that for all n. f j f in L 1 \ n Lemma 2.11 [5]. If {z j } is a sequence of measurable functions with associated Young measure ν = {ν x } x, lim inf ψ x, z j x dx ψx, λ dν x λ dx, j E E R m for every nonnegative, Carathéodory function ψ and every measurable subset E.

5 EJDE-218/113 PARABOLIC SYSTEMS IN DIVERGENCE FORM 5 The above theorem is obtained in [31] by proving a complicated lemma. We will give a much easier proof by using the contradiction method. Proof. Assume that lim inf ψ x, z j x dx <. j E Then ψx, z j x is a bounded sequence in L 1 E. Let ψx = ψx, λ dν x λ. R m By Lemmas 2.8 and 2.1, there exists E n, E n+1 E n, mease n as n, such that ψ x, z j x dx ψ dx 2.3 E\E n E\E n as j for all n. On the other hand, it is apparent that ψ dx ψ dx E\E n as n. Now we show the proof of our conclusion by contradiction. Assume that lim inf ψ x, z j x dx < ψx, λ dν x λ dx. j E E R m Let a := ψx, λ dν x λ dx lim inf ψ x, z j x dx >. E R m j E Since lim n E\E n ψ dx = E ψ dx, for a >, there exist n which is large enough, such that ψ dx ψ dx < a = ψx, λ dν x λ dx lim inf ψ x, z j x dx. E E\E n E R m j E Therefore E\E n ψ dx > lim inf ψ x, z j x dx. j E Combining this with 2.3 leads to a contradiction. Lemma 2.12 [18]. If {u j } is bounded in L px, R m, then {u j } can generate Young measure ν x satisfied that ν x = 1 and there is a subsequence of {u j } weakly convergent to R m λ dν x λ in L 1, R m. Let X := 3. Galerkin approximation and a priori estimates { } u L 2 ; R m : Du L px ; M m N, u, τ V τ a.e. τ, T, where for τ, T, { } V τ := u L 2 ; R m W 1,1 ; R m : Du, τ L px ; M m N. The norm on X is defined by u X := u L 2 ;R m + Du L px ;M m N. E

6 6 M. YANG, Y. FU EJDE-218/113 According to Lemmas , it is easy to show that X is a Banach space and C ; R m is dense in X. X denotes the dual space of X. For all g X, u X, there exists g L 2 ; R m, g 1 L p x ; M m N, such that g, u = g u dx dt + g 1 Du dx dt. Based on the above notes, we will show the definition of weak solutions for 1.2. Definition 3.1. A function u L, T ; L 2 X is called as the weak solution of problem 1.2, if u ϕ dx dt+ ux, tϕx, t dx T + σx, t, Du Dϕ dx dt = f Dϕ dx dt t holds for all ϕ C 1, T ; C. We choose an L 2 ; R m -orthonormal base {ω j } j=1, such that {ω j } j=1 C ; R m, C ; R m n=1 V nc 1 ;R m. Here V n = span{ω 1, ω 2,..., ω n }. Since f L p x ; M m N and C ; M m N is identity in L p x ; M m N, there exists a sequence {f n } C ; M m N such that f n f in L px ; M m N as n. For every u L 2 ; R m, there is a sequence {ψ n } n=1, such that ψ n n=1v n and ψ n u in L 2 ; R m as n. Definition 3.2. u n C 1, T ; V n is called by the Galerkin solution of problem 1.2, if u n φ dx dt + σx, t, Du n Dφ dx dt = f n Dφ dx dt t τ τ τ holds for all τ, T ] and φ C 1, T ; V k k n, where τ =, τ. Now we construct the Galerkin solution of problem 1.2. Define P n t, η : [, T ] R n R n n Pnt, ηi = σx, t, η j Dω j Dω i dx where η = η 1,, η n. Since σ is a Carathéodory function, P n t, η is continuous in t, η. Consider the ordinary differential equation j=1 where η t + P n t, ηt = Fn η = U n F n i = f n Dω i dx, U n i = ψ n xω i dx. 3.1

7 EJDE-218/113 PARABOLIC SYSTEMS IN DIVERGENCE FORM 7 From 3.1 we have η η + P n t, ηη = F n η. Furthermore, n n P n t, ηη = σx, t, η j Dω j η i Dω i dx dt It is apparent that j=1 bx, t dx dt + c 2 i=1 n px η i Dω i dx dt C. i=1 η η + C F n η 1 2 F n ηt 2. Consequently, 1 ηt t 2 F n ηt 2 + C After integrating the both sides of this inequality, we obtain ηt 2 C n + T ηs 2 ds Then by Gronwall s inequality, ηt C n T. Let M n = max F n P n t, η, t,η [,T ] Bη,2C nt T n = min { T, 2C nt M n } where B η, 2C n T is a ball of radius 2C n T with the center at the point η in R n. By Peano s theorem, 3.1 has a C 1 solution on [, T n ]. Let t 1 = T n and ηt 1 be a initial value, then we can repeat the above process and get a C 1 solution on [t 1, t 2 ], where t 2 = t 1 + T n. Thus there is a interval [t i 1, t i 2 ] [, T ], such that 3.1 admits a solution on [t i 1, t i 2 ], where t i = t i 1 +T n, i = 1, 2,..., l 1, t l = T. Moreover we can get a solution η n t C 1 [, T ]. From the definition of P n, it is easy to know that u n x, t = n j=1 ηn t j ω jx is the Galerkin solution of 1.2. Now we study the boundedness and convergence of some function sequences. Lemma 3.3. The sequence {u n } is bounded in X, and {σx, t, Du n } is bounded in L p x ; M m N. Proof. Let φ = u n. By Definition 3.2, for every τ [, T ], one has u n t u n dx dt + σx, t, Du n Du n dx dt = f n Du n dx dt, τ τ τ which is denoted as I + II = III. By integration and H2, and I = 1 2 u n, τ 2 L u n, 2 L 2 II bx, t dx dt + c 2 τ Du n px dx dt τ Since f n L p x ; M m N, we have III C f n L px τ ;M m N Du n L px τ ;M m N

8 8 M. YANG, Y. FU EJDE-218/113 We know that u n x, = ψ n x u in L 2. As a result, u 2 nx, dx = ψ n x 2 dx C forall n. Consequently, 1 2 u n, τ 2 L 2 + c 2 Du n px dx dt τ 1 2 u n, 2 L 2 + b L 1 τ C f n L p x τ ;M m N Du n L px τ ;M m N C + C Du n L px τ ;M m N By Lemma 2.2, it follows that { 1/p, Du n L px τ max Du n px dx dt Du n px 1/p+ } dx dt. τ τ If Du n L px τ ;M m N is unbounded, then τ Du n px dx dt is unbounded. This contradict 3.2. Thus Du n L px ;M m N C. Moreover u n, τ 2 L 2 C 3.3 Then we can get the conclusion that {u n } is bounded in X. By Lemma 2.6, there is a subsequence of {u n } also denoted by {u n } satisfying u n u in X, as n. Owing to H2, we obtain σx, t, Du n p x dx dt C ax, t p x dx dt + c 1 Du n px dx dt. Since a L p x and Du n L px ;M m N C, it follows that σx, t, Du n p x dx dt C. From Lemma 2.2 we have σx, t, Du n L p x ;M m N C. 3.4 Then σx, t, Du n χ in L p x ; M m N as n we can choose a proper subsequence if necessary. Lemma 3.4. For function sequences {u n } constructed above, we have u n, T u, T u, = u. inl 2, Proof. Thanks to 3.3, the sequence {u n } is bounded in L, T ; L 2. Thus there exists a subsequence also denoted by {u n } such that u n, T z in L 2 as n. We will prove that z = u, T, and u, = u. We denote u, T as ut, and denote u, as u.

9 EJDE-218/113 PARABOLIC SYSTEMS IN DIVERGENCE FORM 9 For every ψ C [, T ], v V k, k n, we have T t u n vψ dx dt + σx, t, Du n Dvψ dx dt = T After integrating, one gets u n T ψt v dx = + T T If n, then zψt v dx T = u n ψv dx σx, t, Du n Dvψ dx dt + u n vψ dx dt. u ψv dx f Dvψ dx dt Let ψ = ψt =. Then T T f Dvψ dx dt T Thus by 3.5, we can obtain zψt v dx u ψv dx = = = T T f n Dvψ dx dt f n Dvψ dx dt. T χ Dψv dx dt + ψ vu dx dt. 3.5 T T χ Dψv dx = ψ vu dx = ψvu dx. T T ψvu dx + uψv dx T ut ψt v dx ψ vu dx uψv dx Let k, if we take ψt = and ψ = 1, then we have u = u ; if we take ψt = 1 and ψ =, then we have ut = z. 4. Existence of weak solutions The proof of Lemma 3.3 implies that {Du n } is bounded in L px ; M m N. By Lemma 2.12, {Du n } can generate a family of Young measures ν x,t, and ν x,t, I = Dux, t. By Lemmas 2.3 and 2.6, we can choose a proper subsequence if necessary such that u n u in X, n, Du n Du in L px ; M m N. Lemma 4.1. Suppose that σ satisfies H1 H3, then the Young measures ν x,t generated by {Du n }, which is the gradient of Galerkin sequence {u n } constructed before, satisfy σx, t, λ λ dν x,t λ dx dt M m N 4.1 σx, t, λ Du dν x,t λ dx dt M m N

10 1 M. YANG, Y. FU EJDE-218/113 Proof. Consider the sequence I n := σx, t, Du n σx, t, Du Du n Du = σx, t, Du n Du n Du σx, t, Du Du n Du =: I n,1 + I n,2 Assumption H2 implies σx, t, Du p x dx dt C ax, t p x dx dt + c 1 Du px dx dt. Since Du L px ; M m N, it follow that σ L p x ; M m N. Because of the weak convergence of {Du n }, we obtain I n,2 as n. It follows from Lemma 2.11 that I := lim inf I n dx dt n = lim inf I n,1 dx dt n 4.2 = lim inf σx, t, Du n Du n Du dx dt n σx, t, λ λ Dudν x,t λ dx dt M m N Now we prove that I. Using u n t u n dx dt + σx, t, Du n Du n dx dt = f n Du n dx dt, we find that I = lim inf σx, t, Du n Du n Du dx dt n = lim inf σx, t, Du n Du n dx n = lim inf n Obviously, f n Du n dx dt f n Du n dx dt u n t u n dx dt σx, t, Du n Du dx dt σx, t, Du n Du dx dt. f Du dx dt = f n Du n dx dt f Du n dx dt f Du n dx dt + f Du dx dt. Since f n f L p x ;M m N, as n, it is easy to see that f n Du n dx dt f Du n dx dt C f n f L p x ;M m N Du n L px ;M m N, as n.

11 EJDE-218/113 PARABOLIC SYSTEMS IN DIVERGENCE FORM 11 Because of the weak convergence of Du n, f Du n dx dt f Du dx dt, as n. Consequently, f n Du n dx dt f Du dx dt, as n. From the weak convergence of σx, t, Du n, σx, t, Du n Du dx dt χ Du dx dt, as n. For every ψ C 1, T ; V k, k n, ψ t u n dx dt σx, t, Du n Dψ dx dt = f n Dψ dx dt. After integrating, we have u n, T ψt v dx u n, ψv dx u n t ψ dx dt + σx, t, Du n Dψ dx dt = f n Dψ dx dt. Letting n, we have u, T ψt v dx = f Dψ dx dt. u, ψv dx u t ψ dx dt + χ Dψ dx dt Let k, for all ψ C 1, T ; C 1. The the above equality is valid. Then for all ψ C, the above equality also holds. Thus u t ψ dx dt = χ Dψ dx dt + f Dψ dx dt = divχ f, ψ. Obviously t u = divχ f. For u X, we can derive that u t u dx dt = χ Du dx dt + f Du dx dt. On the other hand, u t u dx dt = 1 2 u, T 2 L u, 2 L 2, u n t u n dx dt = 1 2 u n, T 2 L u n, 2 L 2. From the structure of u n, we obtain u n, L 2 u, L 2.

12 12 M. YANG, Y. FU EJDE-218/113 Using Lemma 3.4, we have u n, T u, T in L 2. Owing to the weakly lower semicontinuity of the norm, Clearly, u, T L2 lim inf n u n, T L2. lim inf u n t u n dx dt 1 n 2 u, T 2 L u, 2 L 2. Thus we arrived at the conclusion that I. Lemma 4.2. For a.e. x, t, we have σx, t, λ σx, t, Du λ Du = on supp νx,t. Proof. Since λ dν x,t λ = ν x,t, I = Dux, t, M m N and ν x,t is a family of probability measures, 1dν M m N x,t = 1. Consequently σx, t, Du λ Du dν x,t λ dx dt M m N = σx, t, Du λ dν x,t λ dx dt M m N σx, t, Du Du dν x,t λ dx dt M m N = σx, t, Du λ dν x,t λ dx dt M m N σx, t, Du Du 1 dν x,t λ dx dt M m N = σx, t, Du Du dx dt σx, t, Du Du 1 dν x,t λ dx dt =. M m N From Lemma 4.1, we obtain Thus M m N σx, t, λ λ Du dν x,t λ dx dt. M m N σx, t, λ σx, t, Du λ Dudν x,t λ dx dt. By the monotonicity of σ, the integrand in the above inequality is nonnegative. Then for a.e. x, t, we can obtain that σx, t, λ σx, t, Du λ Du = in supp ν x. We are now in a position to show the existence of solutions of 1.2. Proof of Theorem 1.1. We consider 4 cases which correspond to the 4 cases in H3.

13 EJDE-218/113 PARABOLIC SYSTEMS IN DIVERGENCE FORM 13 Case i. We prove that for a.e. x, t and every µ M m N the following equation holds on supp ν x, σx, t, λ µ = σx, t, Du µ + σx, t, Duµ Du λ, 4.3 where is the derivative with respect to the third variable of σ. Actually, by the monotonicity of σ, for all α R, we have σx, t, λ σx, t, Du + αµ λ Du αµ. From Lemma 4.2 on supp ν x we obtain σx, t, λ σx, t, Du + αµ λ Du αµ = σx, t, λ λ Du σx, t, λ αµ σx, t, Du + αµ λ Du αµ = σx, t, Du λ Du σx, t, λ αµ σx, t, Du + αµ λ Du αµ. It can be easily seen that σx, t, λ αµ σx, t, Du λ Du + σx, t, Du + αµ λ Du αµ, and Then we infer that σx, t, Du + αµ = σx, t, Du + σx, t, Duαµ + oα. σx, t, Du + αµ λ Du αµ = σx, t, Du + αµ λ Du σx, t, Du + αµ αµ = σx, t, Du λ Du + σx, t, Duαµ λ Du σx, t, Du αµ + σx, t, Duαµ αµ + oα = σx, t, Du λ Du + α σx, t, Duµ λ Du σx, t, Du µ + oα. Moreover σx, σx, t, λ αµ α t, Duµ λ Du σx, t, Du µ + oα Since the sign of α is arbitrary, the above equation implies 4.2. Set µ = E ij, where E ij is the matrix whose entry in the ith row and jth column is 1 and others are. Then by 4.2, σx, t, λ ij = σx, t, Du ij + σx, t, DuE ij Du λ. Furthermore, σx, t, λ ij dν x,t λ supp ν x,t = σx, t, Du ij dν x,t λ supp ν x,t + σx, t, DuE ij Du λ dν x,t λ. supp ν x,t Note that Du λ dν x,t λ = Dux, t supp ν x,t λ dν x,t λ =. supp ν x,t

14 14 M. YANG, Y. FU EJDE-218/113 Thus we can derived that σx, u, λ dν x,t λ = σx, t, Du dν x,t λ supp ν x,t supp ν x,t = σx, t, Du dν x,t λ = σx, t, Du. supp ν x,t Since {σx, t, Du n } is weakly convergent in L p x ; M m N. By Dunford-Pettis criterion and Lemma 2.9, {σx, t, Du n } has a L 1 -weak limit: σ := σx, t, λ dν x,t λ = σx, t, Du. supp ν x,t Evidently, σx, t, Du n σx, t, Du inl p x ; M m N. For all φ C 1, T ; V k, k n, one has φ t u n dx dt + σx, t, Du n Dφ dx dt = f n Dφ dx dt, where φ t u n dx dt = Letting n we obtain u, T φt dx u, φv dx = f Dφ dx dt. u n, T φt dx u n, φv dx u n t φ dx dt. u t φ dx dt + σx, t, Du Dφ dx dt Let k, then for φ C 1, T ; C, we are led to the conclusion that u φ dx dt+ ux, tφx, t dx T + σx, t, Du Dφ dx dt = f Dφ dx dt. t Case ii. We prove that for all x, t we have supp ν x,t K x,t = { λ M m N : W x, t, λ = W x, t, Du + σx, t, Du λ Du }. If λ supp ν x,t, by Lemma 4.2, for every β [, 1], 1 β σx, t, λ σx, t, Du λ Du =. By monotonicity, for β [, 1], we have 1 β σ x, t, Du + βλ Du σx, t, λ Du λ. Thus for all β [, 1], 1 β σ x, t, Du + tλ Du σx, t, Du Du λ. In view of the monotonicity condition, σ x, t, Du + βλ Du σx, t, Du β λ Du.

15 EJDE-218/113 PARABOLIC SYSTEMS IN DIVERGENCE FORM 15 Since β [, 1], we have σ x, t, Du + βλ Du σx, t, Du 1 β λ Du. For all β [, 1], if λ supp ν x,t, then σ x, t, Du + βλ Du σx, t, Du λ Du =. 4.4 It follows that W x, t, λ = W x, t, Du + 1 = W x, t, Du + σx, t, Du λ Du. σ x, t, Du + βλ Du λ Du dβ So we can get λ K x,t, i.e. supp ν x,t K x,t. On account of the convexity of W, for all ξ M m N, For all λ K x,t, put W x, t, ξ W x, u, Du + σx, t, Du ξ Du. P λ = W x, t, λ, λ = W x, t, Du + σx, t, Du λ Du. As λ W x, u, λ is continuous and differentiable, for every ϕ M m N, γ R, Thus DP = D, and Consequently, P λ + γϕ P λ λ + γϕ λ γ γ P λ + γϕ P λ λ + γϕ λ γ γ γ >, γ <. σx, t, λ = σx, t, Du λ K x,t supp ν x,t. 4.5 σx, t := σx, t, λ dν x,t λ M m N = σx, t, λ dν x,t λ = σx, t, Du. supp ν x,t Now we consider the Carathéodory function gx, t, λ = σx, t, λ σx, t, λ M m N. 4.6 Since σx, t, Du n is weakly convergent in L p x ; M m N, then σx, t, Du n is equi-integrable. Thus g n x, t = gx, t, Du n is equi-integrable, and g n g in L 1. Taking 4.4 and 4.5 into consideration, we obtain gx, t = σx, t, λ σx, t dνx,t λ M m N = σx, t, λ σx, t dνx,t λ supp ν x,t = σx, t, λ σx, t, Dux, t dνx,t λ =. supp ν x,t

16 16 M. YANG, Y. FU EJDE-218/113 It turns out that σx, t, Dun σx, t, Du dx dt. The remainder of the argument is similar to that in case i and so is omitted. Case iii. By the strict monotonicity and Lemma 4.2, we have supp ν x,t = {Dux, t}. Thus for a.e. x, t, ν x,t = δ Dux,t. Using Lemma 2.9, we find Du n Du in measure. For a proper subsequence, we assert that Du n Du a.e. in. It follows that σx, t, Du n σx, t, Du a.e. in. Moreover σx, t, Du n σx, t, Du in measure. From a similar analysis in case i, we obtain the existence of 1.2 for case ii. Case iv. Suppose that ν x,t is not a Dirac measure, for a.e. x, t, then we have < σx, t, λ σx, t, λ λ λ dν x,t λ dx dt M m N = σx, t, λ λ σx, t, λ λ Since M m N σx, t, λ λ + σx, t, λ λ dν x,t λ dx dt. 1 dν x,t λ = 1 M m N we obtain and M m N λ dν x,t λ = λ = Dux, t, σx, t, λ λ dν x,t λ dx dt M m N > σx, t, λ λ + σx, t, λ λ σx, t, λ λ dν x,t λ dx dt M m N = σx, t, λ dν x,t λ λ + σx, t, λ M m N λ dν x,t λ M m N σx, t, λ λ 1 dν x,t λ dx dt M m N = σx, t, λ dν x,t λ λ dx dt M m N = σx, t, λ dν x,t λ Dux, t dx dt. M m N By Lemma 4.1, σx, t, λ Du dν x,t λ dx dt M m N σx, t, λ λ dν x,t λ dx dt M m N > σx, t, λ dν x,t λ Du dx dt. M m N

17 EJDE-218/113 PARABOLIC SYSTEMS IN DIVERGENCE FORM 17 This is a contradiction. Hence ν x,t is a Dirac measure. Assume that ν x,t = δ hx,t. Then hx, t = λ dδ hx,t λ = λ dν x,t λ = Dux, t M m N M m N Thus ν x,t = δ Dux,t. Lemma 2.1 implies that Du n Du as n. Moreover σx, t, Du n σx, t, Du in measure as n. An argument similar to the one in case iii shows the conclusion we want. The proof is complete. Conclusions. In this article, we study the existence of weak solutions for quasilinear parabolic system in divergence form with variable growth by means of Young measures generated by sequences in variable exponent spaces. We can conclude that problem 1.2 has a weak solution under four kinds of monotonicity conditions in H3. We need notice that H3iii requires σ is strictly monotone. Actually classical monotonicity operator method can get our result under H3iii. We give the other method to obtain the main theorem by Young measures in our paper under H3iii. But conventional method can not prove the main result under the other monotonicity conditions. And in H3iv, we define a new monotonicity condition. If σ is strictly monotone, then H3iv holds. Obviously, H3iv is weaker than typical strictly monotone condition. Currently, the research on Young measures generated by sequences in variable exponent Lebesgue and Sobolev spaces is still in exploration. Our results enrich and perfect the theory of variable exponent spaces and Young measures. For related results on nonlinear problems with variable growth we refer to the monograph by Rădulescu and Repovš [33] and the survey paper by Rădulescu [32]. Recent contributions to this field may be found in the papers [28, 29, 3, 34]. Acknowledgements. This work was supported by the National Natural Science Foundation of China No , No , and by the Natural Science Foundation of Shandong Province ZR216AM13, ZR218PA3. References [1] Acerbi, E.; Mingione, G.; Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal., , [2] Acerbi, E.; Mingione, G.; Seregin, G.A.; Regularity results for parabolic systems related to a class of non-newtonian fluids. Ann. Inst. H. Poincare Anal. Non Lineaire, 21 24, [3] Antontsev, S.; Shmarev, S.; A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal., 6 25, [4] Ball, J. M.; A Version of the fundamental theorem for young measures. PDEs and Continuum Models of Phase Transitions: Proceedings of an NSF-CNRS Joint Seminar, France.1989, [5] Brézis, H.; Operateurs Maximaux Monotones et Semigroups de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam, [6] Chabrowski, J.; Fu, Y.; Existent of solutions for px-laplacian problems on a bounded domain. J. Math. Anal. Appl., 36 25, Corrigendum: J. Math. Anal. Appl., , [7] Chen, Y. M.; Levine, S. E.; Rao, M: Variable exponent linear growth functionals in image restoration. SIAM J. Appl. Math., 66 26, [8] Chen, Y. M.; Levine, S. E.; Rao, M.; Variable exponent, linear growth functionals in image restoration. SIMA Journal on Applied Mathematics , [9] Diening, L.; Harjulehto, P.; Hästö, P.; R užička, M.; Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics. 217, Springer, Berlin, 211.

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19 EJDE-218/113 PARABOLIC SYSTEMS IN DIVERGENCE FORM 19 [37] Warga, J.; Relaxed variational problems. Journal of Mathematical Analysis and Applications, , [38] Young, L. C.; Generalized curves and the existence of an attained absolute minimum in the calculus of variations. Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, , [39] Zhikov, V.; Averaging of functionals in the calculus of variations and elasticity. Math. USSR, Izv., , Miaomiao Yang School of Science, ilu Institute of Technology, Jinan 251, China address: 3d7@163.com Yongqiang Fu Department of Mathematics, Harbin Institute of Technology, Harbin 151, China address: fuyongqiang@hit.edu.cn