Journal of Differential Equations

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1 J. Differential Equations Contents lists available at ScienceDirect Journal of Differential Equations Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L 1 data Chao Zhang, Shulin Zhou LMAM, School of Mathematical Sciences, Peking University, Beijing 1871, PR China article info abstract Article history: Received 3 March 29 Revised 28 October 29 Available online 3 December 29 MSC: primary 35D5 secondary 35D1, 46E35 In this paper we prove the existence and uniqueness of both renormalized solutions and entropy solutions for nonlinear parabolic equations with variable exponents and L 1 data. And moreover, we obtain the equivalence of renormalized solutions and entropy solutions. 29 Elsevier Inc. All rights reserved. Keywords: Variable exponents Renormalized solutions Entropy solutions Existence Uniqueness 1. Introduction Suppose that is a bounded open domain of R N with Lipschitz boundary, T is a positive number. In this paper we study the following nonlinear parabolic problem u t div u px 2 u = f in Q, T, 1.1 u = onγ, T, ux, = u x on, where the variable exponent p : 1, is a continuous function, f L 1 Q and u L 1. This work was supported in part by the NBRPC under Grant 26CB757 and the NSFC under Grant * Corresponding author. addresses: czhang@math.pku.edu.cn C. Zhang, szhou@math.pku.edu.cn S. Zhou /$ see front matter 29 Elsevier Inc. All rights reserved. doi:1.116/j.jde

2 C. Zhang, S. Zhou / J. Differential Equations The study of differential equations and variational problems with nonstandard growth conditions arouses much interest with the development of elastic mechanics, electro-rheological fluid dynamics and image processing, etc. We refer the readers to [31,32,36,15] and references therein. px-growth conditions can be regarded as a very important class of nonstandard p, q-growth conditions. There are already numerous results for such kind of problems see [1 3,19,2,18,5]. The functional spaces to deal with these problems are the generalized Lebesgue spaces L px and the generalized Lebesgue Sobolev spaces W k,px. Under our assumptions, it is reasonable to work with entropy solutions or renormalized solutions, which need less regularity than the usual weak solutions. The notion of renormalized solutions was first introduced by DiPerna and Lions [17] for the study of Boltzmann equation. It was then adapted to the study of some nonlinear elliptic or parabolic problems and evolution problems in fluid mechanics. We refer to [14,16,8,1,9,26] for details. At the same time the notion of entropy solutions has been proposed by Bénilan et al. in [7] for the nonlinear elliptic problems. This framework was extended to related problems with constant p in [13,3,11,4,28]. Recently, Sanchón and Urbano in [33] studied a Dirichlet problem of px-laplace equation and obtained the existence and uniqueness of entropy solutions for L 1 data, as well as integrability results for the solution and its gradient. The proofs rely crucially on a priori estimates in Marcinkiewicz spaces with variable exponents. Besides, Bendahmane and Wittbold in [6] proved the existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and L 1 data. The aim of this paper is to extend the results in [33,6] to the case of parabolic equations. As far as we know, there are no papers concerned with the nonlinear parabolic equations involving variable exponents and L 1 data. Inspired by [29] and [3], we develop a refined method. The advantage of our method is that we cannot only obtain the existence and uniqueness of renormalized solutions for problem 1.1, but also find that the renormalized solution is equivalent to the entropy solution for problem 1.1. We first employ the difference and variation methods to prove the existence and uniqueness of weak solutions for the approximate problem of 1.1 under appropriate assumptions. Then we construct an approximate solution sequence and establish some a priori estimates. Next, we draw a subsequence to obtain a limit function, and prove this function is a renormalized solution. Based on the strong convergence of the truncations of approximate solutions, we obtain that the renormalized solution of problem 1.1 is also an entropy solution, which leads to an equality in the entropy formulation. By choosing suitable test functions, we prove the uniqueness of renormalized solutions and entropy solutions, and thus the equivalence of renormalized solutions and entropy solutions. For the convenience of the readers, we recall some definitions and basic properties of the generalized Lebesgue spaces L px and generalized Lebesgue Sobolev spaces W k,px. Set C ={h C : min x hx>1}. Foranyh C we define h = sup hx and h = inf hx. x x For any p C, we introduce the variable exponent Lebesgue space L p to consist of all measurable functions such that ux px dx <, endowed with the Luxemburg norm { u p = inf λ>: ux λ px dx 1 },

3 1378 C. Zhang, S. Zhou / J. Differential Equations which is a separable and reflexive Banach space. The dual space of L px is L p x, where 1/px 1/p x = 1. If px is a constant function, then the variable exponent Lebesgue space coincides with the classical Lebesgue space. The variable exponent Lebesgue spaces is a special case of Orlicz Musielak spaces treated by Musielak in [27]. For any positive integer k, denote W k,px = { u L px : D α u L px, α k }, where the norm is defined as u W k,px = α k D α u px. W k,px is called generalized Lebesgue Sobolev space, which is a special generalized Orlicz Sobolev space. An interesting feature of a generalized Lebesgue Sobolev space is that smooth functions are not dense in it without additional assumptions on the exponent px. This was observed by Zhikov [35] in connection with Lavrentiev phenomenon. However, when the exponent px is log- Hölder continuous, i.e., there is a constant C such that px py C log x y 1.2 for every x, y with x y 1, then smooth functions are dense in variable exponent Sobolev 2 spaces and there is no confusion in defining the Sobolev space with zero boundary values, W 1,p, as the completion of C with respect to the norm u W 1,p see [21]. Throughout this paper we assume that px C satisfies the log-hölder continuity condition 1.2. Let T k denote the truncation function at height k : and its primitive Θ k : R R by T k r = { min k, max{r, k} } k if r k, = r if r < k, k if r k, r Θ k r = T k s ds = It is obvious that Θ k r and Θ k r k r. We denote { r 2 2 if r k, k r k2 2 if r k. T 1,p Q = { u:, T ] R is measurable Tk u L p, T ; W 1,p with T k u L p Q N, for every k > }. Next we define the very weak gradient of a measurable function u T 1,p Q. The proof follows from Lemma 2.1 of [7] due to the fact that W 1,p W 1,p.

4 C. Zhang, S. Zhou / J. Differential Equations Proposition 1.1. For every measurable function u T 1,p Q, there exists a unique measurable function v : Q R N, which we call the very weak gradient of u and denote v = u, such that T k u = vχ { u <k}, almost everywhere in Q and for every k >, where χ E denotes the characteristic function of a measurable set E. Moreover, if u belongs to L 1, T ; W 1,1, then v coincides with the weak gradient of u. The notion of the very weak gradient allows us to give the following definitions of renormalized solutions and entropy solutions for problem 1.1. Definition 1.1. Afunctionu T 1,p Q C[, T ]; L 1 is a renormalized solution to problem 1.1 if the following conditions are satisfied: i lim n {x,t Q : n ux,t n1} u px dxdt = ; ii for every function ϕ C 1 Q with ϕ, T = and S in W 2, R which is piecewise C 1 satisfying that S has a compact support, Su ϕx, dx T T Su ϕ t dxdt [ S u u px 2 u ϕ S u u px ϕ ] dxdt = T fs uϕ dxdt 1.3 holds. Definition 1.2. Afunctionu T 1,p Q C[, T ]; L 1 is an entropy solution to problem 1.1 if Θ k u φt dx Θ k u φ dx u px 2 u T k u φdxdt = T φt, T k u φ dt ft k u φdxdt, 1.4 Q Q for all k > and φ C 1 Q with φ Γ =. Now we state our main results. Theorem 1.1. Assume that condition 1.2 holds. Then there exists a unique renormalized solution for problem 1.1. Theorem 1.2. Assume that condition 1.2 holds. Then the renormalized solution u in Theorem 1.1 is also an entropy solution for problem 1.1. And the entropy solution is unique. Remark 1.1. The renormalized solution for problem 1.1 is equivalent to the entropy solution for problem 1.1.

5 138 C. Zhang, S. Zhou / J. Differential Equations The rest of this paper is organized as follows. In Section 2, we state some basic results that will be used later. We will prove the main results in Section 3. In the following sections C will represent a generic constant that may change from line to line even if in the same inequality. 2. Preliminaries In this section, we first state some elementary results for the generalized Lebesgue spaces L px and the generalized Lebesgue Sobolev spaces W k,px. The basic properties of these spaces can be found from [23], and many of these properties were independently established in [2]. Lemma 2.1. See [2,23]. 1 The space L p is a separable, uniform convex Banach space, and its conjugate space is L p where 1/px 1/p x = 1. For any u L p and v L p, wehave uvdx 1 1 p p u px v p x 2 u px v p x; 2 If p 1, p 2 C,p 1 x p 2 x for any x, then there exists the continuous embedding L p 2x L p 1x, whose norm does not exceed 1. Lemma 2.2. See [2]. If we denote ρu = u px dx, u L px, then { min u p px, } { u p p px ρu max u px, u p px}. Lemma 2.3. See [2]. W k,px is a separable and reflexive Banach space. Lemma 2.4. See [22,23]. Let p C satisfy the log-hölder continuity condition 1.2. Then,foru W 1,p, thep -Poincaré inequality u px C u px holds, where the positive constant C depends on p and. Lemma 2.5. Assume that u L 2 and f L p, T ; L p x. Then the following problem u t div u px 2 u = f in Q, u = on Γ, ux, = u on, admits a unique weak solution u L p, T ; W 1,p C[, T ]; L 2 with u L p Q N such that for any ϕ C 1 Q with ϕ, T =,

6 C. Zhang, S. Zhou / J. Differential Equations u xϕx, dx T [ uϕt u px 2 u ϕ ] dxdt = T f ϕ dxdt holds. Proof. By employing the difference and variation methods see [34], we give a sketched proof. Let n be a positive integer. Denote h = T /n. We first consider the following time-discrete problem { uk u k 1 div u k px 2 u k =[f ]h k 1h, h u k =, k = 1, 2,...,n, 2.1 where [ f ] h denotes the Steklov average of f defined by [ f ] h x, t = 1 h th t f x,τ dτ. It is easy to see that [ f ] h L p. For k = 1, we introduce the variational problem min { Ju u W }, where W = { u W 1,px L 2 } and functional J is Ju = 1 2h u 2 dx 1 px u px dx 1 h u udx [ f ] h udx. We will establish that Ju has a minimizer u 1 x in W. By Lemmas 2.1, 2.4, Young s inequality and Lemma 2.2, we have [ f ] h udx 2 [ f ]h p x u px C [ f ]h p x u px ε u p px Cε [ f ] h p p x β p ε u px dx Cε [ f ] h p p x ε u px dx 1 Cε [ f ] h p p x,

7 1382 C. Zhang, S. Zhou / J. Differential Equations where ε is a small positive number and β = { 1 p if u p 1, 1 p if u p 1. Choosing ε sufficiently small and using Young s inequality, we obtain Ju 1 u px dx 1 u 2 dx C 2p 4h u 2 dx [ f ]h p p x 1, and thus Ju is lower bounded and coercive on W. On the other hand, Ju is weakly lower semicontinuous on W. Therefore, there exists a function u 1 W such that Ju 1 = inf Ju. u W Thus the function u 1 is a weak solution of the corresponding Euler Lagrange equation of Ju, which is 2.1 in the case k = 1. And it is unique. Following the same procedures, we find weak solutions u k of 2.1 for k = 2,...,n. It follows that, for every ϕ W, u k u k 1 h ϕ dx u k px 2 u k ϕdx = For every h = T /n, we define the approximate solutions u x, t =, u 1 x, < t h,...,..., u h x, t = u j x, j 1h < t jh,...,..., u n x, n 1h < t nh = T. Taking ϕ = u k in 2.2, we can obtain an a priori estimate [ f ] h k 1h ϕ dx. 2.2 T u 2 h x, t dx u h x, t px dxdt u 2 dx C T f p p x dt, which implies from Lemma 2.2 that T { T min u h p px, u h p } px dt u h px dxdt C and u h L,T ;L 2 u h px,q u h L p,t ;W 1,px C.

8 C. Zhang, S. Zhou / J. Differential Equations Thus we may choose a subsequence we also denote it by the original sequence for simplicity such that u h u, weakly- in L, T ; L 2, u h u, weakly in L p, T ; W 1,px, u h px 2 u h ξ, weakly in L p x Q N. Following the arguments in [34] with necessary changes in detail, we use the monotonicity method to show that ξ = u px 2 u a.e. in Q. Recalling the fact that u L p, T ; W 1,px L, T ; L 2 and u t L p, T ; W 1,p x from the equation, we conclude that u belongs to C[, T ]; L 2. Therefore, we obtain the existence of weak solutions. For uniqueness, suppose there exist two weak solutions u and v of problem 1.1. Then w = u v satisfies the following problem w t div u px 2 u v px 2 v = in Q, w = onγ, wx, = on. Choosing w as a test function in the above problem, we have, for almost every t, T, 1 2 w 2 t dx t [ u px 2 u v px 2 v ] u v dxds =. Since the two terms on the left-hand side are nonnegative, we have u = v a.e. in Q. This finishes the proof. 3. The proofs of main results Now we are ready to prove the main results. Some of the reasoning is based on the ideas developed in [29] and [3] for the constant exponent case. First we prove the existence and uniqueness of renormalized solutions for problem 1.1. Proof of Theorem Existence of renormalized solutions. We first introduce the approximate problems. Find two sequences of functions { f n } C Q and {u n } C strongly converging respectively to f in L1 Q and to u in L 1 such that f n L1 Q f L 1 Q, u n L1 u L Then we consider the approximate problem of 1.1 u n div u n px 2 u n = fn in Q, t u n = onγ, u n x, = u n on. 3.2 By Lemma 2.5, we can find a weak solution u n L p, T ; W 1,p with u n L p Q N for problem 3.2. Our aim is to prove that a subsequence of these approximate solutions {u n } converges to a measurable function u, which is a renormalized solution of problem 1.1. We will divide the

9 1384 C. Zhang, S. Zhou / J. Differential Equations proof into several steps. Although some of the arguments are not new, we present a self-contained proof for the sake of clarity and readability. Step 1. Prove the convergence of {u n } in C[, T ]; L 1 and find its subsequence which is almost everywhere convergent in Q. Let m and n be two integers, then from 3.2 we can write the weak form as T un u m t,φ dt T [ un px 2 u n u m px 2 u m ] φ dxdt = T f n f m φ dxdt, for all φ L p, T ; W 1,p L Q with φ L p Q N. Choosing φ = T 1 u n u m χ,t with t T and discarding the positive term, we get Θ 1 u n u m t dx Therefore, we conclude that Θ 1 u n u m dx f n f m L1 Q u n u m L 1 f n f m L 1 Q := a n,m. { u n u m <1} u n u m 2 t dx 2 { u n u m 1} [ Θ1 u n u m ] t dx a n,m. u n u m t dx 2 It follows that u n u m t dx = { u n u m <1} { u n u m <1} u n u m t dx u n u m 2 t dx { u n u m 1} 1 2 u n u m t dx meas 1 2 2an,m 2meas an,m 2a n,m. Since { f n } and {u n } are convergent in L 1,wehavea n,m forn,m.thus{u n } is a Cauchy sequence in C[, T ]; L 1 and u n converges to u in C[, T ]; L 1. Then we find an a.e. convergent subsequence still denoted by {u n }inq such that u n u a.e. in Q. 3.3 Step 2. Prove T k u n strongly converges to T k u in L p Q N,foreveryk >.

10 C. Zhang, S. Zhou / J. Differential Equations Choosing T k u n as a test function in 3.2, we have Θ k u n T dx Θ k u n dx T T k u n px dxdt = T f n T k u n dxdt. It follows from the definition of Θ k r and 3.1 that T T k u n px dxdt k f n L 1 Q u n L 1 k f L1 Q u L Combining 3.4 with Lemma 2.2, we deduce that T { min Tk u n p px, Tk u n T p } px dt ρ T k u n dt C, that is T k u n is bounded in L p, T ; W 1,px. For every k, h >, using the boundedness of T k u n and T 2k u n T h u n in L p Q N,we draw a subsequence still denoted by {u n }from{u n } such that T k u n T k u weakly in L p Q N, 3.5 T 2k un T h u n T 2k u Th u weakly in L p Q N. 3.6 In order to deal with the time derivative of truncations, we will use the regularization method of Landes [24] and use the sequence T k u μ as approximation of T k u. Forμ >, we define the regularization in time of the function T k u given by Tk u t μ x, t := μ e μs t T k ux, s ds, extending T k u by for s <. Observe that T k u μ L p, T ; W 1,p L Q with T k u μ L p Q N, it is differentiable for a.e. t, T with After computation, we can get T k u μ x, t k 1 e μt < k a.e. in Q, T k u μ = μ T k u T k u t μ. T k u μ T ku strongly in L p Q N. Let us take now a sequence {ψ j } of C functions that strongly converge to u in L 1, and set

11 1386 C. Zhang, S. Zhou / J. Differential Equations η μ, j u T k u μ e μt T k ψ j. The definition of η μ, j, which is a smooth approximation of T k u, is needed to deal with a nonzero initial datum see also [29]. Note that this function has the following properties: ημ, j u = μ T t k u η μ, j u, η μ, j u = T k ψ j, η μ, j u k, η μ, j u T k u strongly in L p Q N, as μ. 3.7 Fix a positive number k. Leth > k. We choose w n = T 2k un T h u n T k u n η μ, j u as a test function in 3.2. The use of w n as a test function to prove the strong convergence of truncations was first introduced in the elliptic case in [25], then adapted to parabolic equations in [29]. If we set M = 4k h, then it is easy to see that w n = where u n > M. Therefore, we may write the weak form of 3.2 as T un t, w n dt T T M u n px 2 T M u n w n dxdt = T f n w n dxdt. In the following, denote wn, μ, j, h all quantities such that lim lim lim lim wn,μ, j, h =. h j μ n First as far as the first term is concerned, that is Since η μ, j u k, w n can be written as T un t, w n dt. w n = T hk un η μ, j u T h k un T k u n. Applying Lemma 2.1 in [29], we can obtain that From the above estimate, we have T un t, w n dt wn, j, h. T T M u n px 2 T M u n w n dxdt T f n w n dxdt wn, j, h.

12 C. Zhang, S. Zhou / J. Differential Equations Splitting the integral in the left-hand side on the sets where u n k and where u n > k and discarding some nonnegative terms, we find T T T M u n px 2 T M u n T 2k un T h u n T k u n η μ, j u dxdt T k u n px 2 T k u n T k u n η μ, j u dxdt { u n >k} It follows from the above inequality that T T M u n px 2 T M u n η μ, j u dxdt. T k u n px 2 T k u n T k u n η μ, j u dxdt T M u n px 2 T M u n η μ, j u T dxdt { u n >k} f n w n dxdt wn,μ, j, h. Using the fact that η μ, j u T k u strongly in L p Q N as μ, we conclude that T Tk u n px 2 T k u n T k u n T k u dxdt T M u n px 2 T M u n η μ, j u T dxdt { u n >k} f n w n dxdt wn,μ, j, h. Furthermore, we have T Tk u n px 2 T k u n Tk u px 2 T k u T k u n T k u dxdt { u n >k} T T T M u n px 2 T M u n η μ, j u dxdt f n T 2k un T h u n T k u n η μ, j u dxdt T k u px 2 T k u T k u n T k u dxdt wn,μ, j, h = I 1 I 2 I 3 wn,μ, j, h. 3.8

13 1388 C. Zhang, S. Zhou / J. Differential Equations Now we show the limits of I 1, I 2 and I 3 are zeros when n, μ and then h tend to infinity respectively. Limit of I 1. We observe that T M u n px 2 T M u n is bounded in L p x Q, and by the dominated convergence theorem χ { un >k} η μ, j u converges strongly in L px Q to χ { u >k} T k u, which is zero, as n and μ tends to infinity. Thus we obtain lim lim I 1 = lim lim T M u n px 2 T M u n T k u dxdt =. 3.9 μ n μ n { u n >k} Limit of I 2. Notice that I 2 T 2k T T f n f T2k un T h u n T k u n η μ, j u dxdt ft2k un T h u n T k u n η μ, j u dxdt f n f dx T ft2k un T h u n T k u n η μ, j u dxdt. Since f n is strongly compact in L 1 Q, using 3.3, the definition of η μ, j and the Lebesgue dominated convergence theorem, we have lim h lim lim I 2 lim μ n h Limit of I 3. Recalling 3.5, we have T ft2k u Th u dxdt =. 3.1 lim I 3 = n Therefore, passing to the limits in 3.8 as n, μ, j, and then h tend to infinity, by means of 3.9, 3.1 and 3.11, we deduce that where lim En =, n En = T T k u n px 2 T k u n T k u px 2 T k u T k u n T k u dxdt. We recall the following well-known inequalities: for any two real vectors a, b R N, a a p 2 b b p 2 a b cp a b p, if p 2

14 C. Zhang, S. Zhou / J. Differential Equations and for every ε, 1], a b p cpε p 2/p a a p 2 b b p 2 a b ε b p, if 1 < p < 2, where cp = 21 p 32 p when p 2 and cp = p 1 p 1 Therefore, we have {x,t Q : px 2} when 1 < p < 2. T k u n T k u px dxdt 2 p 1 p 1En 3.12 and {x,t Q :1<px<2} T k u n T k u px dxdt 32 p p 1 εp 2/p En ε T T k u px dxdt Since En as n, then using the arbitrariness of ε and T k u is bounded in L p Q N, we conclude that lim n which implies that, for every k >, T T k u n T k u px dxdt =, T k u n T k u strongly in L p Q N 3.14 and T k u n px 2 T k u n T k u px 2 T k u in L p Q N Thanks to Lemma 2.2, we know that T k u n T k u strongly in L p, T ; W 1,p. Step 3. Show that u is a renormalized solution. For given a, k >, define the function T k,a s = T a s T k s as s k signs if k s < k a, T k,a s = a signs if s k a, if s k. Using T k,a u n as a test function in 3.2, we find

15 139 C. Zhang, S. Zhou / J. Differential Equations Θ a u n kt dx Θ a u n k dx u n px 2 u n u n dxdt { u n >k} { u n >k} {k u n ka} f n T k,a u n dxdt, which yields that u n px dxdt a f n dxdt u n dx. {k u n ka} { u n >k} { u n >k} Recalling the convergence of {u n } in C[, T ]; L 1, wehave lim meas{ x, t Q : u n > } k = uniformly with respect to n. k Therefore, passing to the limit first in n then in k, we conclude that lim k {x,t Q : k ux,t ka} Choosing a = 1, we obtain the renormalized condition, i.e., lim k {x,t Q : k ux,t k1} u px dxdt =. u px dxdt =. Let S W 2, R be such that supp S [ M, M] for some M >. For every ϕ C Q with ϕx, T =, S u n ϕ is a test function in 3.2. It yields T = Su n ϕ dxdt t T T [ S u n u n px 2 u n ϕ S u n u n px ϕ ] dxdt f n S u n ϕ dxdt First we consider the first term on the left-hand side of Since S is bounded and continuous, 3.3 implies that Su n converges to Su a.e. in Q and weakly- in L Q. Then Su n converges t to Su in D Q as n, that is t T Su n ϕ dxdt t T Su ϕ dxdt. t For the other terms on the left-hand side of 3.16, because of supp S [ M, M] we know S u n u n px 2 u n = S u n T M u n px 2 T M u n

16 C. Zhang, S. Zhou / J. Differential Equations and Using 3.3, 3.14 and 3.15, we have S u n u n px = S u n T M u n px. S u n T M u n px 2 T M u n S u T M u px 2 T M u in L p Q N and S u n T M u n px S u T M u px in L 1 Q. Noting that S u T M u px 2 T M u = S u u px 2 u, S u T M u px = S u u px, we deduce S u n u n px 2 u n S u u px 2 u in L p Q N and S u n u n px S u u px in L 1 Q. For the right-hand side of 3.16, thanks to the strong convergence of f n, it is easy to pass to the limits. Therefore, we obtain = Su ϕx, dx T fs uϕ dxdt. T Su ϕ t dxdt T [ S u u px 2 u ϕ S u u px ϕ ] dxdt This completes the proof of the existence of renormalized solutions. 2 Uniqueness of renormalized solutions. Now we prove the uniqueness of renormalized solutions for problem 1.1 by choosing an appropriate test function motivated by [9] and [6]. Let u and v be two renormalized solutions for problem 1.1. Fix a positive number k. Forσ >, let S σ be the function defined by S σ r = r if r < σ, S σ r = σ r σ 1 if σ ±r σ 1, 2 2 S σ r =± σ 1 if ±r > σ It is obvious that

17 1392 C. Zhang, S. Zhou / J. Differential Equations S σ r = 1 if r < σ, S σ r = σ 1 r if σ r σ 1, S σ r = if r > σ 1. It is easy to check S σ W 2, R with supp S σ [ σ 1, σ 1] and supp S σ [σ, σ 1] [ σ 1, σ ]. Therefore, we may take S = S σ in 1.3 to have T = S σ u T t ϕ dxdt T fs σ uϕ dxdt [ S σ u u px 2 u ϕ S σ u u px ϕ ] dxdt and T = S σ v T t ϕ dxdt T fs σ vϕ dxdt. [ S σ v v px 2 v ϕ S σ v v px ϕ ] dxdt We plug ϕ = T k S σ u S σ v as a test function in the above equalities and subtract them to obtain that J J 1 J 2 = J 3, 3.18 where J = J 1 = J 2 = J 3 = T Sσ u S σ v T T t, T k Sσ u S σ v dt, S σ u u px 2 u S σ v v px 2 v T k Sσ u S σ v dxdt, T [ S σ u u px S σ v v px] T k Sσ u S σ v dxdt, f S σ u S σ v T k Sσ u S σ v dxdt. We estimate J, J 1, J 2 and J 3 one by one. Recalling the definition of Θ k r, J can be written as

18 J = C. Zhang, S. Zhou / J. Differential Equations Θ k Sσ u S σ v T dx Θ k Sσ u S σ v dx. Due to the same initial condition for u and v, and the properties of Θ k,weget Writing J 1 = T J = Θ k Sσ u S σ v T dx. [ S σ u px 2 S σ u S σ v px 2 S σ v ] T k Sσ u S σ v dxdt T [ S σ u S σ u S σ u px 2 ] u px 2 u T k Sσ u S σ v dxdt T [ S σ v S σ v S σ v px 2 ] v px 2 v T k Sσ u S σ v dxdt := J 1 1 J 2 1 J 3 1, and setting σ k, wehave J 1 1 { u v k} { u, v k} u px 2 u v px 2 v u v dxdt Recalling supp S σ [ σ 1, σ 1] and supp S σ [σ, σ 1] [ σ 1, σ ], weobtain J C {σ u σ 1} u px dxdt {σ u σ 1} { v σ 1} { Sσ u Sσ v k} {σ u σ 1} {σ u σ 1} u px dxdt u px dxdt u px 1 v dxdt {σ u σ 1} {σ k v σ 1} {σ k v σ 1} v px dxdt And we may get the similar estimate for J 3 1.Furthermore,wehave J 2 C {σ u σ 1} u px dxdt {σ v σ 1} u px 1 v dxdt. v px dxdt.

19 1394 C. Zhang, S. Zhou / J. Differential Equations From the above estimates and i in Definition 1.1, we obtain Observing lim σ J 2 J 3 J 2 =. 1 1 f S σ u S σ v strongly in L 1 Q as σ and using the Lebesgue dominated convergence theorem, we deduce that lim σ J 3 =. Therefore, sending σ in 3.18 and recalling 3.19, we have u px 2 u v px 2 v u v dxdt =, { u k 2, v k 2 } which implies u = v a.e. on the set { u k 2, v k }. Since k is arbitrary, we conclude that 2 u = v a.e. in Q. Then, set ξ n = T 1 T n1 u T n1 v. Wehaveξ n L p, T ; W 1,px and { on u n 1, v n } 1, ξ n = uχ { { u Tn1 v 1} on u n 1, v > n } 1, vχ { { Tn1 u v 1} on u > n 1, v n } 1, such that Q ξ n px dxdt u px dxdt v px dxdt. {n u n1} {n v n1} Thanks to Lemma 2.2 and i in Definition 1.1, we deduce that ξ n strongly in L p, T ; W 1,px. Sinceξ n T 1 u v a.e. in Q, we conclude that T 1 u v =, hence u = v a.e. in Q. Therefore we obtain the uniqueness of renormalized solutions. This completes the proof of Theorem 1.1. Next, we prove that the renormalized solution u is also an entropy solution of problem 1.1 and the entropy solution of problem 1.1 is unique. Proof of Theorem The renormalized solution is an entropy solution. Now we choose v n = T k u n φ asatestfunctionin3.2fork > and φ C 1 Q with φ Γ =. We note that, if L = k φ L Q, then T = u n px 2 u n T k u n φdxdt T T L u n px 2 T L u n T k T L u n φ dxdt

20 C. Zhang, S. Zhou / J. Differential Equations and T un t, T k u n φ dt T T L u n px 2 T L u n T k T L u n φ dxdt = T f n T k u n φdxdt. Since u n t = u n φ t φ t,wehave T un t, T k u n φ dt = Θ k u n φt dx Θ k u n φ dx T φt, T k u n φ dt, which yields that Θ k u n φt dx = T T Θ k u n φ dx T φt, T k T L u n φ dt T L u n px 2 T L u n T k T L u n φ dxdt f n T k u n φdxdt. 3.2 Recalling u n converges to u in C[, T ]; L 1, hence t T, u n t ut in L 1. SinceΘ k is Lipschitz continuous, we get Θ k u n φt dx Θ k u φt dx and Θ k u n φ dx Θ k u φ dx, as n. Using the strong convergence of f n, 3.5 and 3.15, we can pass to the limits as n tends to infinity for the other terms to conclude

21 1396 C. Zhang, S. Zhou / J. Differential Equations Θ k u φt dx T Θ k u φ dx u px 2 u T k u φdx = T φt, T k u φ dt ft k u φdx, for all k > and φ C 1 Q with φ Γ =. Therefore, we finish the proof of the existence of entropy solutions. 2 Uniqueness of entropy solutions. Suppose that u and v are two entropy solutions of problem 1.1. Let {u n } be a sequence constructed in 3.2, which satisfies T k u n strongly converges to T k u in L p Q N,foreveryk >. Choosing S σ u n as a test function in 1.4 for entropy solution v, wehave Θ k v Sσ u n T dx = T T Θ k u S σ u n dx v px 2 v T k v Sσ u n dxdt T un t, S σ u nt k v S σ u n dt ft k v Sσ u n dxdt In order to deal with the third term on the left-hand side of 3.21, we take S σ u nψ with Ψ = T k v S σ u n as a test function for problem 3.2 to obtain T un t, S σ u nψ dt T T = f n S σ u nψ dxdt. S σ u nψ u n px dxdt T S σ u n u n px 2 u n Ψ dxdt 3.22 Thus we deduce from 3.21 and 3.22 that Θ k v Sσ u n T dx T T Θ k u S σ u n dx S σ u nt k v Sσ u n u n px dxdt S σ u n u n px 2 u n T k v Sσ u n dxdt

22 = T T C. Zhang, S. Zhou / J. Differential Equations v px 2 v T k v Sσ u n dxdt ft k v Sσ u n dxdt T f n S σ u nt k v Sσ u n dxdt. We will pass to the limit as n and σ successively. Let us denote A 3 for the third term on the left-hand side of the above equality for simplicity. Recalling supp S σ [σ, σ 1] [ σ 1, σ ], wehave A 3 k {σ u n σ 1} u n px dxdt. Observe that S σ u n u n px 2 u n = S σ u n T σ 1 u n px 2 T σ 1 u n,thenweget Θ k v Sσ u n T dx T T Θ k u S σ u n dx v px 2 v S σ u n Tσ 1 u n px 2 T σ 1 u n T k v Sσ u n dxdt f fn S σ u n T k v Sσ u n dxdt k {σ u n σ 1} u n px dxdt. Thanks to the fact that T k u n T k u strongly in L p Q N convergence theorem, letting n, weobtain and the Lebesgue dominated Θ k v Sσ u T dx T T Θ k u S σ u dx v px 2 v S σ u Tσ 1 u px 2 T σ 1 u T k v Sσ u dxdt f 1 S σ u T k v Sσ u dxdt k {σ u σ 1} u px dxdt Let us denote A 3 for the third term on the left-hand side of Then we can write A 3 as T A 3 = v px 2 v S σ u u px 2 u T k v Sσ u dxdt

23 1398 C. Zhang, S. Zhou / J. Differential Equations = T v px 2 v S σ u px 2 S σ u T k v Sσ u dxdt T [ S σ u px 2 S σ u S σ u] u px 2 u T k v Sσ u dxdt = I 1 I 2. Recalling the definition of S σ,wehave I C {σ u σ 1} {σ u σ 1} {σ u σ 1} Now we let σ.since u px dxdt u px dxdt u px dxdt {σ u σ 1} { v Sσ u k} {σ u σ 1} {σ k v σ k1} {σ k v σ k1} v px dxdt u px 1 vdxdt u px 1 vdxdt Θ k v Sσ u T k vt ut, Θ k u S σ u k u, by the Lebesgue dominated convergence theorem, we have Θ k u S σ u dx, According to the fact that lim k {x,t Q : k ux,t k1} and Fatou s lemma, we deduce from 3.23 and 3.24 that Θ k v ut dx { u k 2, v k 2 } Θ k v Sσ u T dx u px dxdt = Θ k v ut dx. v px 2 v u px 2 u v u dxdt. Using the positivity of Θ k,wehave u = v a.e. in Q,forallk. Similar to the case of renormalized solutions, we conclude that u = v a.e. in Q. Therefore we obtain the uniqueness of entropy solutions. This completes the proof of Theorem 1.2. Remark 3.1. Furthermore, we may improve the integrability of the renormalized solution or entropy solution u for problem 1.1 by assuming that p > 2 1. Then we can prove that N1 u L q,t ;W 1,q C,

24 C. Zhang, S. Zhou / J. Differential Equations with 1 q < p N 1 N. N 1 Recalling i in Definition 1.1, L p Q L p Q and Lemma 2.2, we get u p,b m = u p,b m C u p,bm { 1/p C max u px dxdt, u px dxdt 1/p } C, B m B m where B m = { x, t Q : m ux, t < m 1 }. Following the arguments in [12] and u C[, T ]; L 1, we can conclude that Acknowledgment u Lq,T ;W 1,q C. The authors wish to thank the referee for careful reading of the early version of this manuscript and providing valuable suggestions and comments. References [1] E. Acerbi, G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal [2] E. Acerbi, G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal [3] E. Acerbi, G. Mingione, G.A. Seregin, Regularity results for parabolic systems related to a class of non-newtonian fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire [4] A. Alvino, L. Boccardo, V. Ferone, L. Orsina, G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura Appl [5] S.N. Antontsev, S.I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions, Nonlinear Anal [6] M. Bendahmane, P. Wittbold, Renormalized solutions for nonlinear elliptic equations with variable exponents and L 1 data, Nonlinear Anal [7] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J.L. Vazquez, An L 1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci [8] D. Blanchard, F. Murat, Renormalised solutions of nonlinear parabolic problems with L 1 data: Existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A [9] D. Blanchard, F. Murat, H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differential Equations [1] D. Blanchard, H. Redwane, Renormalized solutions for a class of nonlinear evolution problems, J. Math. Pure Appl [11] L. Boccardo, G.R. Cirmi, Existence and uniqueness of solution of unilateral problems with L 1 data, J. Convex. Anal [12] L. Boccardo, T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal [13] L. Boccardo, T. Gallouët, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire

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