Journal of Differential Equations
|
|
- Dayna Richards
- 5 years ago
- Views:
Transcription
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
25 14 C. Zhang, S. Zhou / J. Differential Equations [14] L. Boccardo, D. Giachetti, J.I. Diaz, F. Murat, Existence and regularity of renormalized solutions for some elliptic problems involving derivations of nonlinear terms, J. Differential Equations [15] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math [16] G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci [17] R.J. DiPerna, P.L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math [18] X. Fan, Global C 1,α regularity for variable exponent elliptic equations in divergence form, J. Differential Equations [19] X. Fan, J. Shen, D. Zhao, Sobolev embedding theorems for spaces W k,px, J. Math. Anal. Appl [2] X. Fan, D. Zhao, On the spaces L px and W m,px, J. Math. Anal. Appl [21] P. Harjulehto, Variable exponent Sobolev spaces with zero boundary values, Math. Bohem [22] P. Harjulehto, P. Hästö, M. Koskenoja, S. Varonen, The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential Anal [23] O. Kováčik, J. Rákosník, On spaces L p x and W 1,px, Czechoslovak Math. J [24] R. Landes, On the existence of weak solutions for quasilinear parabolic initial boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A [25] C. Leone, A. Porretta, Entropy solutions for nonlinear elliptic equations in L 1, Nonlinear Anal [26] P.L. Lions, Mathematical Topics in Fluid Mechanics, vol. 1: Incompressible models, Oxford Univ. Press, Oxford, [27] J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin, [28] M.C. Palmeri, Entropy subsolutions and supersolutions for nonlinear elliptic equations in L 1, Ricerche Mat [29] A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura Appl [3] A. Prignet, Existence and uniqueness of entropy solutions of parabolic problems with L 1 data, Nonlinear Anal [31] K. Rajagopal, M. R užička, Mathematical modelling of electro-rheological fluids, Contin. Mech. Thermodyn [32] M. R užička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2. [33] M. Sanchón, J.M. Urbano, Entropy solutions for the px-laplace equation, Trans. Amer. Math. Soc [34] M. Xu, S. Zhou, Existence and uniqueness of weak solutions for a generalized thin film equation, Nonlinear Anal [35] V.V. Zhikov, On some variational problems, Russ. J. Math. Phys [36] V.V. Zhikov, On the density of smooth functions in Sobolev Orlicz spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. POMI
Weak Solutions to Nonlinear Parabolic Problems with Variable Exponent
International Journal of Mathematical Analysis Vol. 1, 216, no. 12, 553-564 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.216.6223 Weak Solutions to Nonlinear Parabolic Problems with Variable
More informationLARGE TIME BEHAVIOR FOR p(x)-laplacian EQUATIONS WITH IRREGULAR DATA
Electronic Journal of Differential Equations, Vol. 215 (215), No. 61, pp. 1 16. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LARGE TIME BEHAVIOR
More informationA Caffarelli-Kohn-Nirenberg type inequality with variable exponent and applications to PDE s
A Caffarelli-Kohn-Nirenberg type ineuality with variable exponent and applications to PDE s Mihai Mihăilescu a,b Vicenţiu Rădulescu a,c Denisa Stancu-Dumitru a a Department of Mathematics, University of
More informationASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR PARABOLIC OPERATORS OF LERAY-LIONS TYPE AND MEASURE DATA
ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR PARABOLIC OPERATORS OF LERAY-LIONS TYPE AND MEASURE DATA FRANCESCO PETITTA Abstract. Let R N a bounded open set, N 2, and let p > 1; we study the asymptotic behavior
More informationRenormalized Solutions of a Nonlinear Parabolic Equation with Double Degeneracy
Electronic Journal of Qualitative Theory of Differential Equations 26, No. 5, -2; http://www.math.u-szeged.hu/ejqtde/ Renormalized Solutions of a Nonlinear Parabolic Equation with Double Degeneracy Zejia
More informationEXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS
EXISTECE AD REGULARITY RESULTS FOR SOME OLIEAR PARABOLIC EUATIOS Lucio BOCCARDO 1 Andrea DALL AGLIO 2 Thierry GALLOUËT3 Luigi ORSIA 1 Abstract We prove summability results for the solutions of nonlinear
More informationRENORMALIZED SOLUTIONS OF STEFAN DEGENERATE ELLIPTIC NONLINEAR PROBLEMS WITH VARIABLE EXPONENT
Journal of Nonlinear Evolution Equations and Applications ISSN 2161-3680 Volume 2015, Number 7, pp. 105 119 (July 2016) http://www.jneea.com RENORMALIZED SOLUTIONS OF STEFAN DEGENERATE ELLIPTIC NONLINEAR
More informationELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 76, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ELLIPTIC
More informationWell-Posedness Results for Anisotropic Nonlinear Elliptic Equations with Variable Exponent and L 1 -Data
CUBO A Mathematical Journal Vol.12, N ō 01, (133 148). March 2010 Well-Posedness Results for Anisotropic Nonlinear Elliptic Equations with Variable Exponent and L 1 -Data Stanislas OUARO Laboratoire d
More informationWELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL FLUID EQUATIONS WITH DEGENERACY ON THE BOUNDARY
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 13, pp. 1 15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu WELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL
More informationNONLINEAR PARABOLIC PROBLEMS WITH VARIABLE EXPONENT AND L 1 -DATA
Electronic Journal of Differential Equations, Vol. 217 217), No. 32, pp. 1 32. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR PARABOLIC PROBLEMS WITH VARIABLE EXPONENT
More informationJournal of Differential Equations
J. Differential Equations 249 21) 1483 1515 Contents lists available at ScienceDirect Journal of Differential Equations www.elsevier.com/locate/jde Renormalized solutions for a nonlinear parabolic equation
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied athematics http://jipam.vu.edu.au/ Volume 4, Issue 5, Article 98, 2003 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES D. ESKINE AND A. ELAHI DÉPARTEENT
More informationRELATIONSHIP BETWEEN SOLUTIONS TO A QUASILINEAR ELLIPTIC EQUATION IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 265, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu RELATIONSHIP
More informationCRITICAL POINT METHODS IN DEGENERATE ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT. We are interested in discussing the problem:
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 4, December 2010 CRITICAL POINT METHODS IN DEGENERATE ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT MARIA-MAGDALENA BOUREANU Abstract. We work on
More informationThe p(x)-laplacian and applications
The p(x)-laplacian and applications Peter A. Hästö Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland October 3, 2005 Abstract The present article is based
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
More informationStrongly nonlinear parabolic initial-boundary value problems in Orlicz spaces
2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 203 220. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationPROPERTIES OF CAPACITIES IN VARIABLE EXPONENT SOBOLEV SPACES
PROPERTIES OF CAPACITIES IN VARIABLE EXPONENT SOBOLEV SPACES PETTERI HARJULEHTO, PETER HÄSTÖ, AND MIKA KOSKENOJA Abstract. In this paper we introduce two new capacities in the variable exponent setting:
More informationHarnack Inequality and Continuity of Solutions for Quasilinear Elliptic Equations in Sobolev Spaces with Variable Exponent
Nonl. Analysis and Differential Equations, Vol. 2, 2014, no. 2, 69-81 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/nade.2014.31225 Harnack Inequality and Continuity of Solutions for Quasilinear
More informationCOMPACT EMBEDDINGS ON A SUBSPACE OF WEIGHTED VARIABLE EXPONENT SOBOLEV SPACES
Adv Oper Theory https://doiorg/05352/aot803-335 ISSN: 2538-225X electronic https://projecteuclidorg/aot COMPACT EMBEDDINGS ON A SUBSPACE OF WEIGHTED VARIABLE EXPONENT SOBOLEV SPACES CIHAN UNAL and ISMAIL
More informationA continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces
A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces Mihai Mihailescu, Vicentiu Radulescu To cite this version: Mihai Mihailescu, Vicentiu Radulescu. A continuous spectrum
More informationVariable Exponents Spaces and Their Applications to Fluid Dynamics
Variable Exponents Spaces and Their Applications to Fluid Dynamics Martin Rapp TU Darmstadt November 7, 213 Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, 213 1 / 14 Overview 1 Variable
More informationANISOTROPIC EQUATIONS: UNIQUENESS AND EXISTENCE RESULTS
ANISOTROPIC EQUATIONS: UNIQUENESS AND EXISTENCE RESULTS STANISLAV ANTONTSEV, MICHEL CHIPOT Abstract. We study uniqueness of weak solutions for elliptic equations of the following type ) xi (a i (x, u)
More informationHARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 37, 2012, 571 577 HARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH Olli Toivanen University of Eastern Finland, Department of
More informationEXISTENCE OF BOUNDED SOLUTIONS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol 2007(2007, o 54, pp 1 13 ISS: 1072-6691 URL: http://ejdemathtxstateedu or http://ejdemathuntedu ftp ejdemathtxstateedu (login: ftp EXISTECE OF BOUDED SOLUTIOS
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
More informationSome Applications to Lebesgue Points in Variable Exponent Lebesgue Spaces
Çankaya University Journal of Science and Engineering Volume 7 (200), No. 2, 05 3 Some Applications to Lebesgue Points in Variable Exponent Lebesgue Spaces Rabil A. Mashiyev Dicle University, Department
More informationUniversität des Saarlandes. Fachrichtung 6.1 Mathematik
Universität des Saarlandes U N I V E R S I T A S S A R A V I E N I S S Fachrichtung 6.1 Mathematik Preprint Nr. 155 A posteriori error estimates for stationary slow flows of power-law fluids Michael Bildhauer,
More informationResearch Article Nontrivial Solution for a Nonlocal Elliptic Transmission Problem in Variable Exponent Sobolev Spaces
Abstract and Applied Analysis Volume 21, Article ID 38548, 12 pages doi:1.1155/21/38548 Research Article Nontrivial Solution for a Nonlocal Elliptic Transmission Problem in Variable Exponent Sobolev Spaces
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationEXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
More informationExistence of periodic solutions for some quasilinear parabolic problems with variable exponents
Arab. J. Math. (27) 6:263 28 DOI.7/s465-7-78- Arabian Journal of Mathematics A. El Hachimi S. Maatouk Existence of periodic solutions for some quasilinear parabolic problems with variable exponents Received:
More informationON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM. Paweł Goncerz
Opuscula Mathematica Vol. 32 No. 3 2012 http://dx.doi.org/10.7494/opmath.2012.32.3.473 ON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM Paweł Goncerz Abstract. We consider a quasilinear
More informationSYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.
More informationGOOD RADON MEASURE FOR ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT
Electronic Journal of Differential Equations, Vol. 2016 2016, No. 221, pp. 1 19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GOOD RADON MEASURE FOR ANISOTROPIC PROBLEMS
More informationCMAT Centro de Matemática da Universidade do Minho
Universidade do Minho CMAT Centro de Matemática da Universidade do Minho Campus de Gualtar 471-57 Braga Portugal wwwcmatuminhopt Universidade do Minho Escola de Ciências Centro de Matemática Blow-up and
More informationEXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM. Saeid Shokooh and Ghasem A. Afrouzi. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69 4 (217 271 28 December 217 research paper originalni nauqni rad EXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM Saeid Shokooh and Ghasem A.
More informationarxiv: v1 [math.ap] 10 May 2013
0 SOLUTIOS I SOM BORDRLI CASS OF LLIPTIC QUATIOS WITH DGRAT CORCIVITY arxiv:1305.36v1 [math.ap] 10 May 013 LUCIO BOCCARDO, GISLLA CROC Abstract. We study a degenerate elliptic equation, proving existence
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationExistence of Solutions for a Class of p(x)-biharmonic Problems without (A-R) Type Conditions
International Journal of Mathematical Analysis Vol. 2, 208, no., 505-55 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/ijma.208.886 Existence of Solutions for a Class of p(x)-biharmonic Problems without
More informationThe Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:
Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply
More informationSTEKLOV PROBLEMS INVOLVING THE p(x)-laplacian
Electronic Journal of Differential Equations, Vol. 204 204), No. 34, pp.. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STEKLOV PROBLEMS INVOLVING
More informationResearch Article Function Spaces with a Random Variable Exponent
Abstract and Applied Analysis Volume 211, Article I 17968, 12 pages doi:1.1155/211/17968 Research Article Function Spaces with a Random Variable Exponent Boping Tian, Yongqiang Fu, and Bochi Xu epartment
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 3, Issue 3, Article 46, 2002 WEAK PERIODIC SOLUTIONS OF SOME QUASILINEAR PARABOLIC EQUATIONS WITH DATA MEASURES N.
More informationQuasilinear degenerated equations with L 1 datum and without coercivity in perturbation terms
Electronic Journal of Qualitative Theory of Differential Equations 2006, No. 19, 1-18; htt://www.math.u-szeged.hu/ejqtde/ Quasilinear degenerated equations with L 1 datum and without coercivity in erturbation
More informationVariable Lebesgue Spaces
Variable Lebesgue Trinity College Summer School and Workshop Harmonic Analysis and Related Topics Lisbon, June 21-25, 2010 Joint work with: Alberto Fiorenza José María Martell Carlos Pérez Special thanks
More informationThe variable exponent BV-Sobolev capacity
The variable exponent BV-Sobolev capacity Heikki Hakkarainen Matti Nuortio 4th April 2011 Abstract In this article we study basic properties of the mixed BV-Sobolev capacity with variable exponent p. We
More informationBLOW-UP FOR PARABOLIC AND HYPERBOLIC PROBLEMS WITH VARIABLE EXPONENTS. 1. Introduction In this paper we will study the following parabolic problem
BLOW-UP FOR PARABOLIC AND HYPERBOLIC PROBLEMS WITH VARIABLE EXPONENTS JUAN PABLO PINASCO Abstract. In this paper we study the blow up problem for positive solutions of parabolic and hyperbolic problems
More informationNON-EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION SYSTEM WITH NONLOCAL SOURCES
Electronic Journal of Differential Equations, Vol. 2016 (2016, No. 45, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NON-EXTINCTION OF
More informationhal , version 1-22 Nov 2009
Author manuscript, published in "Kinet. Relat. Models 1, 3 8) 355-368" PROPAGATION OF GEVREY REGULARITY FOR SOLUTIONS OF LANDAU EQUATIONS HUA CHEN, WEI-XI LI AND CHAO-JIANG XU Abstract. By using the energy-type
More informationGlobal Solutions for a Nonlinear Wave Equation with the p-laplacian Operator
Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator Hongjun Gao Institute of Applied Physics and Computational Mathematics 188 Beijing, China To Fu Ma Departamento de Matemática
More informationEXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM
EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using
More informationA CONNECTION BETWEEN A GENERAL CLASS OF SUPERPARABOLIC FUNCTIONS AND SUPERSOLUTIONS
A CONNECTION BETWEEN A GENERAL CLASS OF SUPERPARABOLIC FUNCTIONS AND SUPERSOLUTIONS RIIKKA KORTE, TUOMO KUUSI, AND MIKKO PARVIAINEN Abstract. We show to a general class of parabolic equations that every
More informationJournal of Differential Equations. On the coupling between a Navier Stokes equation and a total energy equation in spatial dimension N = 3
J. Differential Equations 46 9 4591 4617 Contents lists available at ScienceDirect Journal of Differential Equations www.elsevier.com/locate/jde On the coupling between a Navier Stokes equation and a total
More informationA TWO PARAMETERS AMBROSETTI PRODI PROBLEM*
PORTUGALIAE MATHEMATICA Vol. 53 Fasc. 3 1996 A TWO PARAMETERS AMBROSETTI PRODI PROBLEM* C. De Coster** and P. Habets 1 Introduction The study of the Ambrosetti Prodi problem has started with the paper
More informationRegularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains
Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains Ilaria FRAGALÀ Filippo GAZZOLA Dipartimento di Matematica del Politecnico - Piazza L. da Vinci - 20133
More informationASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS
ASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS Juan CASADO DIAZ ( 1 ) Adriana GARRONI ( 2 ) Abstract We consider a monotone operator of the form Au = div(a(x, Du)), with R N and
More informationNONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian
More informationLERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION
2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 73 81. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
More informationNONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 29, 327 338 NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS Shouchuan Hu Nikolas S. Papageorgiou
More informationA capacity approach to the Poincaré inequality and Sobolev imbeddings in variable exponent Sobolev spaces
A capacity approach to the Poincaré inequality and Sobolev imbeddings in variable exponent Sobolev spaces Petteri HARJULEHTO and Peter HÄSTÖ epartment of Mathematics P.O. Box 4 (Yliopistonkatu 5) FIN-00014
More informationPERIODIC SOLUTIONS OF NON-AUTONOMOUS SECOND ORDER SYSTEMS. 1. Introduction
ao DOI:.2478/s275-4-248- Math. Slovaca 64 (24), No. 4, 93 93 PERIODIC SOLUIONS OF NON-AUONOMOUS SECOND ORDER SYSEMS WIH (,)-LAPLACIAN Daniel Paşca* Chun-Lei ang** (Communicated by Christian Pötzsche )
More informationLORENTZ ESTIMATES FOR ASYMPTOTICALLY REGULAR FULLY NONLINEAR ELLIPTIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 27 27), No. 2, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LORENTZ ESTIMATES FOR ASYMPTOTICALLY REGULAR FULLY NONLINEAR
More informationBLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION EQUATION WITH HOMOGENEOUS NEUMANN BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 016 (016), No. 36, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST
More informationPERIODIC SOLUTIONS FOR A KIND OF LIÉNARD-TYPE p(t)-laplacian EQUATION. R. Ayazoglu (Mashiyev), I. Ekincioglu, G. Alisoy
Acta Universitatis Apulensis ISSN: 1582-5329 http://www.uab.ro/auajournal/ No. 47/216 pp. 61-72 doi: 1.17114/j.aua.216.47.5 PERIODIC SOLUTIONS FOR A KIND OF LIÉNARD-TYPE pt)-laplacian EQUATION R. Ayazoglu
More informationOn the discrete boundary value problem for anisotropic equation
On the discrete boundary value problem for anisotropic equation Marek Galewski, Szymon G l ab August 4, 0 Abstract In this paper we consider the discrete anisotropic boundary value problem using critical
More informationNonlinear elliptic systems with exponential nonlinearities
22-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 9, 22, pp 139 147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationAPPROXIMATE IDENTITIES AND YOUNG TYPE INEQUALITIES IN VARIABLE LEBESGUE ORLICZ SPACES L p( ) (log L) q( )
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 35, 200, 405 420 APPROXIMATE IDENTITIES AND YOUNG TYPE INEQUALITIES IN VARIABLE LEBESGUE ORLICZ SPACES L p( ) (log L) q( ) Fumi-Yuki Maeda, Yoshihiro
More informationarxiv: v1 [math.ap] 28 Mar 2014
GROUNDSTATES OF NONLINEAR CHOQUARD EQUATIONS: HARDY-LITTLEWOOD-SOBOLEV CRITICAL EXPONENT VITALY MOROZ AND JEAN VAN SCHAFTINGEN arxiv:1403.7414v1 [math.ap] 28 Mar 2014 Abstract. We consider nonlinear Choquard
More informationPERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 207 222 PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS Fumi-Yuki Maeda and Takayori Ono Hiroshima Institute of Technology, Miyake,
More informationGRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS
LE MATEMATICHE Vol. LI (1996) Fasc. II, pp. 335347 GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS CARLO SBORDONE Dedicated to Professor Francesco Guglielmino on his 7th birthday W
More informationVariational and Topological methods : Theory, Applications, Numerical Simulations, and Open Problems 6-9 June 2012, Northern Arizona University
Variational and Topological methods : Theory, Applications, Numerical Simulations, and Open Problems 6-9 June 22, Northern Arizona University Some methods using monotonicity for solving quasilinear parabolic
More informationBesov-type spaces with variable smoothness and integrability
Besov-type spaces with variable smoothness and integrability Douadi Drihem M sila University, Department of Mathematics, Laboratory of Functional Analysis and Geometry of Spaces December 2015 M sila, Algeria
More informationApplications of the compensated compactness method on hyperbolic conservation systems
Applications of the compensated compactness method on hyperbolic conservation systems Yunguang Lu Department of Mathematics National University of Colombia e-mail:ylu@unal.edu.co ALAMMI 2009 In this talk,
More informationSYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy
More informationWavelets and modular inequalities in variable L p spaces
Wavelets and modular inequalities in variable L p spaces Mitsuo Izuki July 14, 2007 Abstract The aim of this paper is to characterize variable L p spaces L p( ) (R n ) using wavelets with proper smoothness
More informationSome asymptotic properties of solutions for Burgers equation in L p (R)
ARMA manuscript No. (will be inserted by the editor) Some asymptotic properties of solutions for Burgers equation in L p (R) PAULO R. ZINGANO Abstract We discuss time asymptotic properties of solutions
More informationEXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE. Leszek Gasiński
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 409 418 EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE Leszek Gasiński Jagiellonian
More informationBulletin of the. Iranian Mathematical Society
ISSN: 1017-060X (Print) ISSN: 1735-8515 (Online) Bulletin of the Iranian Mathematical Society Vol. 42 (2016), No. 1, pp. 129 141. Title: On nonlocal elliptic system of p-kirchhoff-type in Author(s): L.
More informationarxiv: v2 [math.ap] 6 Sep 2007
ON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1, arxiv:0708.3067v2 [math.ap] 6 Sep 2007 A. CHESKIDOV AND R. SHVYDKOY ABSTRACT. We show that if a Leray-Hopf solution u to the
More informationEXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS
Electronic Journal of Differential Equations, Vol. 2014 (2014), o. 28, pp. 1 10. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS
More informationEIGENVALUE PROBLEMS INVOLVING THE FRACTIONAL p(x)-laplacian OPERATOR
Adv. Oper. Theory https://doi.org/0.5352/aot.809-420 ISSN: 2538-225X (electronic) https://projecteuclid.org/aot EIGENVALUE PROBLEMS INVOLVING THE FRACTIONAL p(x)-laplacian OPERATOR E. AZROUL, A. BENKIRANE,
More informationFunction spaces with variable exponents
Function spaces with variable exponents Henning Kempka September 22nd 2014 September 22nd 2014 Henning Kempka 1 / 50 http://www.tu-chemnitz.de/ Outline 1. Introduction & Motivation First motivation Second
More informationON PARABOLIC HARNACK INEQUALITY
ON PARABOLIC HARNACK INEQUALITY JIAXIN HU Abstract. We show that the parabolic Harnack inequality is equivalent to the near-diagonal lower bound of the Dirichlet heat kernel on any ball in a metric measure-energy
More informationNonlinear aspects of Calderón-Zygmund theory
Ancona, June 7 2011 Overture: The standard CZ theory Consider the model case u = f in R n Overture: The standard CZ theory Consider the model case u = f in R n Then f L q implies D 2 u L q 1 < q < with
More informationA MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS
A MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS A BRESSAN, G M COCLITE, AND W SHEN Abstract The paper is concerned with the optimal harvesting of a marine resource, described
More informationON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR TERM
Internat. J. Math. & Math. Sci. Vol. 22, No. 3 (999 587 595 S 6-72 9922587-2 Electronic Publishing House ON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR
More informationOn Estimates of Biharmonic Functions on Lipschitz and Convex Domains
The Journal of Geometric Analysis Volume 16, Number 4, 2006 On Estimates of Biharmonic Functions on Lipschitz and Convex Domains By Zhongwei Shen ABSTRACT. Using Maz ya type integral identities with power
More informationExistence and multiplicity results for Dirichlet boundary value problems involving the (p 1 (x), p 2 (x))-laplace operator
Note di atematica ISSN 23-2536, e-issn 590-0932 Note at. 37 (207) no., 69 85. doi:0.285/i5900932v37np69 Existence and multiplicity results for Dirichlet boundary value problems involving the (p (x), p
More informationStability for parabolic quasiminimizers. Y. Fujishima, J. Habermann, J. Kinnunen and M. Masson
Stability for parabolic quasiminimizers Y. Fujishima, J. Habermann, J. Kinnunen and M. Masson REPORT No. 1, 213/214, fall ISSN 113-467X ISRN IML-R- -1-13/14- -SE+fall STABILITY FOR PARABOLIC QUASIMINIMIZERS
More informationOn Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. Ben Schweizer 1
On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma Ben Schweizer 1 January 16, 2017 Abstract: We study connections between four different types of results that
More informationHeat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control
Outline Heat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control IMDEA-Matemáticas & Universidad Autónoma de Madrid Spain enrique.zuazua@uam.es Analysis and control
More informationOn some nonlinear parabolic equation involving variable exponents
On some nonlinear parabolic equation involving variable exponents Goro Akagi (Kobe University, Japan) Based on a joint work with Giulio Schimperna (Pavia Univ., Italy) Workshop DIMO-2013 Diffuse Interface
More informationON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1
ON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1, A. CHESKIDOV AND R. SHVYDKOY ABSTRACT. We show that if a Leray-Hopf solution u to the 3D Navier- Stokes equation belongs to
More informationRenormalized solutions for nonlinear partial differential equations with variable exponents and L 1 -data
Renormalized solutions for nonlinear partial differential equations with variable exponents and L 1 -data vorgelegt von Diplom-Mathematikerin Aleksandra Zimmermann geb. Zmorzynska Warschau von der Fakultät
More informationEXISTENCE OF WEAK SOLUTIONS FOR A NONUNIFORMLY ELLIPTIC NONLINEAR SYSTEM IN R N. 1. Introduction We study the nonuniformly elliptic, nonlinear system
Electronic Journal of Differential Equations, Vol. 20082008), No. 119, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp) EXISTENCE
More informationOverview of differential equations with non-standard growth
Overview of differential equations with non-standard growth Petteri Harjulehto a, Peter Hästö b,1, Út Văn Lê b,1,2, Matti Nuortio b,1,3 a Department of Mathematics and Statistics, P.O. Box 68, FI-00014
More informationRégularité des équations de Hamilton-Jacobi du premier ordre et applications aux jeux à champ moyen
Régularité des équations de Hamilton-Jacobi du premier ordre et applications aux jeux à champ moyen Daniela Tonon en collaboration avec P. Cardaliaguet et A. Porretta CEREMADE, Université Paris-Dauphine,
More information