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(1)This article was downloaded by: [University of Perugia], [P. Pucci] On: 09 September 2011, At: 06:48 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK. Complex Variables and Elliptic Equations Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gcov20. Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces a. G. Autuori & P. Pucci. a. a. Dipartimento di Matematica e Informatica, Universitàà degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy Available online: 09 Sep 2011. To cite this article: G. Autuori & P. Pucci (2011): Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces, Complex Variables and Elliptic Equations, 56:7-9, 715-753 To link to this article: http://dx.doi.org/10.1080/17476931003786691. PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-andconditions This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan, sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material..

(2) Complex Variables and Elliptic Equations Vol. 56, Nos. 7–9, July–September 2011, 715–753. Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spacesy. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. G. Autuori and P. Pucci* Dipartimento di Matematica e Informatica, Universita` degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy Communicated by V. Radulescu (Received 8 March 2010; final version received 11 March 2010) We deal with the question of global and local asymptotic stabilities, as time tends to infinity, of solutions of dissipative anisotropic Kirchhoff systems, governed by the p(x)-Laplacian operator, in the framework of the variable exponent Sobolev spaces. Concrete applications are presented in special subcases of the external force f and the distributed damping Q involved in the systems. Keywords: dissipative anisotropic p(x)-Kirchhoff systems; strongly nonlinear potential energies; time-dependent nonlinear damping forces; local and global asymptotic stabilities; variable exponent Sobolev spaces AMS Subject Classifications: Primary; 35B40; 35L70; 35L80; Secondary; 35Q35; 68T40. 1. Introduction In this article, we present some asymptotic stability results for Kirchhoff systems, governed by the p(x)-Laplacian operator, in the anisotropic setting given by the variable exponent Lebesgue and Sobolev spaces. We first address our interest to the model in Rþ 0 ( utt MðIuðtÞÞDpðxÞ u þ juj pðxÞ2 u þ Qðt, x, u, ut Þ þ f ðt, x, uÞ ¼ 0, ðP 1 Þ uðt, xÞ ¼ 0, on Rþ 0 @, where Rn is a bounded domain and u ¼ (u1, . . . , uN) ¼ u(t, x) represents the vectorial displacement, with N 1. The elliptic nonhomogeneous p(x)-Laplacian operator is defined by DpðxÞ u ¼ divðjDuj pðxÞ2 DuÞ,. *Corresponding author. Email: pucci@dmi.unipg.it yDedicated to Professor V.V. Zhikov on the occasion of his 70th birthday. ISSN 1747–6933 print/ISSN 1747–6941 online ß 2011 Taylor & Francis DOI: 10.1080/17476931003786691 http://www.informaworld.com.

(3) 716. G. Autuori and P. Pucci. where div is the vectorial divergence and Du the Jacobian matrix of u. The term jujp(x)2u, where 0, plays the role of a perturbation. The functional Z jDuðt, xÞj pðxÞ dx IuðtÞ ¼ pðxÞ is the natural associated p(x)-Dirichlet energy integral. The dissipative Kirchhoff function M has the standard form. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. MðÞ ¼ a þ b 1 ,. a, b 0,. 1,. a þ b 4 0,. 41. if b 4 0:. When a40 and b 0, system (P 1) is said nondegenerate, otherwise, when a ¼ 0 and b40, it is called degenerate. Finally, (P 1) reduces to the usual well-known semilinear wave system, when a40, b ¼ 0 and ¼ 0. A very special case of (P 1), but with 1 as here, was first studied in [1]. For general comments and previous references we refer to [1]. On the dissipation Q acting on the body and on the source force f, we assume throughout the article N N f 2 CðRþ 0 R ! R Þ,. Q 2 CðRþ 0. N. N. Fðt, x, 0Þ ¼ 0, N. R R ! R Þ,. f ðt, x, uÞ ¼ Fu ðt, x, uÞ,. ðQðt, x, u, vÞ, vÞ 0. ðHÞ. N for all arguments t 2 Rþ 0 , x 2 and u, v 2 R , with (, ) denoting the inner product N N on R R . Here and in the sequel, the function p is fixed in the space Cþ ðÞ ¼ h 2 CðÞ : min hðxÞ 4 1 ,. x2. and the following notation will be adopted throughout the article h ¼ inf hðxÞ, x2. hþ ¼ sup hðxÞ: x2. The most interesting case occurs when p5pþ, that is in the so-called nonstandard growth condition of ( p, pþ) type, cf. [2]. In particular, in the applications, the function p is supposed to satisfy the usual request 15p pþ5n. In this article we need and assume the stronger condition 2n=ðn þ 2Þ p pþ 5 n,. ð1:1Þ. in order to get the necessary standard embeddings. For the study of the nonhomogeneous p(x)-Kirchhoff operator the natural functional spaces are the variable exponent Lebesgue and Sobolev spaces Lp()() and W1, p()(), which have been used in the past decades to model various phenomena concerning nonhomogeneous materials, see [3–12], as well as [13,14] for the Lavrentiev phenomenon discovered by Zhikov. In particular, a great interest has aroused around the so-called electrorheological fluids, which are anisotropic fluids, that is the intrinsic characteristics of their structure depend on the directions of the space. Moreover, the anisotropic Lebesgue and Sobolev spaces have been chosen in many contexts regarding thermo-convective flows of non-Newtonian fluids, as well as in image restoration problems. Finally, for existence theorems of solutions of.

(4) Complex Variables and Elliptic Equations. 717. nonlinear degenerate elliptic problems, we refer to [15–17], and for the anisotropic case to [11]. Our first stability result concerning (P 1) is Theorem 3.3, in which we prove that any global solution of the system is stable under fairly natural assumptions on the external force f and the nonlinear damping Q. The analysis is carried out by means of the natural energy associated with the systems, and we say that a global solution u is stable if and only if the energy functional Eu approaches zero as time goes to infinity, that is lim EuðtÞ ¼ 0 and. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. t!1. lim fkut ðt, Þk2 þ kDuðt, ÞkpðÞ g ¼ 0,. t!1. ð1:2Þ. where kkp() is the norm in the Lebesgue space Lp()(). In other words, the rest state (0, 0) is an attracting set in X L2(), where for simplicity of notation X ¼ pðÞ pðÞ ðÞ ¼ ½W1, ðÞN and L2() ¼ [L2()]N or, sometimes, L2() ¼[L2()]nN. W1, 0 0 From now on we drop the exponents N and nN in all the functional spaces involved in the treatment, as made before. Theorem 3.3 extends Theorem 3.1 of [18], to the case 40 and f ¼ f (t, x, u), and it is based on the proof arguments of Theorem 3.1 of [19] and of Theorem 1 of [20], given by Pucci and Serrin for dissipative nonlinear wave systems. The main tool of the proof technique, introduced by Pucci and Serrin already in [21], is the a priori existence of a suitable function k. Corollary 3.4 shows natural ways to construct k in concrete prototypes, generalizing to the anisotropic setting Corollary 3.4 of [22], see also the earlier Corollary 5.3 of [19]. Problems like (P 1) had been considered earlier for potential energies arising from restoring forces, whereas in this article we allow also the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. Theorem 3.10, established in [18] for ¼ 0 and f ¼ f(x, u), provides a local stability result for (P 1) under a growth condition on f, assumed only for u sufficiently small, but when (P 1) is nondegenerate. The purpose is then reached thanks to a deep qualitative analysis of the geometry of the problem, which allows us ~ 0 in to get stability, provided that the initial data belong to an appropriate region ~ 0 then the phase plane, see Figure 1. In particular, if the couple ðkuð0, ÞkqðÞ , Euð0ÞÞ 2 ~ 0 for all t 2 Rþ , along any global solution u of (P 1). This means ðkuðt, ÞkqðÞ , EuðtÞÞ 2 0 the trajectory of (ku(t, )kq(), Eu(t)) lies in the same region with increasing time, once the initial values (ku(0, )kq(), Eu(0)) belong to it. Here kkq() denotes the norm in the anisotropic Lebesgue space Lq()(), where q() is a variable exponent related to the growth of f in the u variable. The nondegeneracy of (P 1), assumed for the local stability, allows to overcome some technical difficulties due to the Kirchhoff structure of the problem. For further general comments we refer to the recent paper [23], where local stability was studied for a very special case of (P 1). Moreover, concerning the local stability, we also require that the first eigenvalue 0 of the p(x)-Laplacian operator, with homogeneous Dirichlet boundary conditions, defined by R jDuðxÞj pðxÞ dx pðxÞ ð1:3Þ 0 ¼ inf R pðxÞ juðxÞj u2X nf0g dx pðxÞ is positive. It is well-known that when p(x) p, then 040. On the other hand, if the function p is not constant, the situation is much more involved and in general 0 may.

(5) Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. 718. G. Autuori and P. Pucci. Figure 1. The phase plane (, E ).. be zero. Fan et al. in [4] give sufficient conditions to get 040. For example, 040, when p is monotone along a direction. A series of useful lemmas helps in understanding the nature of the problem and constitutes not only a necessary tool for the main proofs, but sometimes it also represents an interesting outcome of independent interest. In mechanical engineering, we often encounter structure composed of rigid and elastic components. The flexible parts are of course sensitive of disturbances and inserting an internal dissipation can lead to satisfactory results. Similar considerations motivated us to consider the more delicate problem in Rþ 0 8 pðxÞ2 u > < utt MðIuðtÞÞðDpðxÞ u þ gðtÞDpðxÞ ut Þ þ juj ðP 2 Þ þ Qðt, x, u, ut Þ þ f ðt, x, uÞ ¼ 0, > : þ uðt, xÞ ¼ 0 on R0 @, in which g 0 is in L1loc ðRþ 0 Þ. Of course, (P 2) includes the previous model (P 1) when g 0, that is when no higher dissipation occurs. In the Kirchhoff model case, the expression ag(t)Dp(x)ut, involved in the third term of (P 2), represents the internal material damping of Kelvin–Voigt type of the body structure. For a detailed physical discussion in the case p(x) 2 the reader is referred to [24,25], as well as the references therein. However, it is worth noting that, in addition to the distributed damping Q, an internal damping mechanism is always present, even if small, in real materials as long as the system vibrates, see the book of Gent [26, Chapter 4, Dynamical Mechanical Properties, p. 73]. When p(x) 2, a further physical discussion on the common use in nonlinear acoustics, as well as in several other natural and industrial applications, of the dissipation higher order term ag(t)Dut, similar to the classical stress tensor representing a Stokesian fluid, is given by Destrade and Saccomandi in [27]. For example, D’Ancona and Shibata consider in [28] a model describing nonlinear viscoelastic materials with short memory, which is a special case of (P 2), when N ¼ 1, g 1, Q 0 and f 0..

(6) Complex Variables and Elliptic Equations. 719. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. Problem (P 2) is studied by methods similar to the ones employed for (P 1), but of course new situations occur because of the presence of the internal damping ag(t)Dp(x)ut. In particular, a new integral term should be estimated, and additional results are established to take into account the effect of the intrinsic dissipation caused by ag(t)Dp(x)ut, see Lemmas 4.1 and 4.2. In Theorem 4.3, we deal with global asymptotic stability for (P 2), the proof being mainly based on Theorem 7.1 of [19], see also [18, Theorem 5.3]. A special feature characterizing (P 2) is that new global and local stability results can be established when the auxiliary function k and g are related in some way. More precisely, in Theorem 4.5 we provide a global stability result for (P 2), whenever kðtÞ const gðtÞ for all t sufficiently large,. ð1:4Þ. see also [22, Theorem 4.4] when p(x) 2, and the earlier similar result given in Theorem 7.2 of [19]. The importance of Theorem 4.5 relies on the fact that condition (1.4) allows to remove an assumption on the distributed damping Q, denoted below by (Q)-(b), which forces a control from below for Q, while the natural growth condition for Q from above, that is (Q)-(a), continues to be required. When g is in L1 ðRþ 0 Þ and (1.4) is replaced by jk0 j const g1=p k1=ð pþ Þ. 0. a.e. in Rþ 0,. ð1:5Þ. in Theorem 4.7 we prove again global stability for (P 2), but assuming the entire (Q). However, the assumption g 2 L1 ðRþ 0 Þ can be removed if p(x) p, that is in the standard case of constant exponents, as it is shown in Corollary 4.8. Finally, combining (1.4) with (1.5), a third result involving a relation between k and g is given in Theorem 4.9, once more without requiring (Q)-(b). These facts were first noted in Section 7 of the celebrated article [19] in the simpler case of strongly damped systems. In the proof of Theorems 4.5 and 4.9, a crucial role is played by Lemma 4.4, of independent interest even when p is constant. In Lemma 4.4 and its consequences the fairly natural relation pþ p is assumed, where p ðxÞ ¼. npðxÞ , n pðxÞ. x2. ð1:6Þ. is the critical Sobolev exponent of X ¼ [W1, p()()]N, well-defined by (1.1). Of course the condition pþ p is automatic, when p(x) p. Combining the techniques introduced by Pucci and Serrin in [20] for damped wave systems and extended in [18] to Kirchhoff problems, we establish in Theorem 4.11 a new local stability result for (P 2). Finally, Theorems 4.5, 4.7 and 4.9 on global asymptotic stability have the corresponding results for local asymptotic stability in Theorems 4.12, 4.14 and 4.16. In all these six theorems, never established in such generality, (P 2) is assumed nondegenerate, see also [29]. The main difficulties, which force this choice, rely on the presence of the Kirchhoff function M in front of g in the system. Improved results have been obtained for the simpler model in Rþ 0 8 pðxÞ2 u > < utt MðIuðtÞÞDpðxÞ u gðtÞDpðxÞ ut þ juj ðP 3 Þ þ Qðt, x, u, ut Þ þ f ðt, x, uÞ ¼ 0, > : þ uðt, xÞ ¼ 0 on R0 @,.

(7) Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. 720. G. Autuori and P. Pucci. where again g 0 is in L1loc ðRþ 0 Þ as for (P 2). Clearly (P 3) coincides with (P 2) when a40 and b ¼ 0. However, (P 3) has an intrinsic interest from an applicative point of view. Indeed, in several concrete situations, the strong internal damping g(t)Dp(x)ut occurs in physics, with g(t) const40. For the model (P 3) all the global stability theorems, even those involving a relation between k and g of the type (1.4) and (1.5), are proved also in the degenerate case, that is when a ¼ 0. While, for the corresponding local stability theorems, we still require that a040, see Section 5. The restriction pþ p51 is assumed throughout Sections 4 and 5, while it is not needed in Section 3 for (P 1). It is worth noting that, in the study of local stability for all the models (P 1)–(P 3), the variable exponent q 2 Cþ ðÞ, related to the growth on f in u, is assumed to satisfy the restriction pþ5q q p in , which is not required in the corresponding global stability theorems and which implies in particular pþ p . Finally, it is worth mentioning that Theorems 4.5–4.9 and Theorems 4.12–4.16, concerning, respectively, the global and local stability for (P 2), as well as the corresponding results of Section 5 for (P 3), are extremely simple in the relevant model case in which Q does not depend on t, usually treated in the literature. Indeed, þ þ 1 in this situation, as easy corollaries, we get that if either g 2 W1,1 loc ðR0 Þ \ L ðR0 Þ, þ þ 0 1 g ò 0 and jg (t)| ¼ o(g(t)) as t ! 1, or g 2 CBVðR0 Þ n L ðR0 Þ, it is enough to take k ¼ g and conclude that the rest state (0, 0) is a globally or locally attracting set. Last but not least, as a consequence of the main results, the validity of (1.2) follows at once, by simply taking k ¼ 1, when Q R t does not depend on t, g(t) g040 for t sufficiently large and lim inft!1 tp1 0 gðÞd 5 1, where p1 ¼ pþ/(1 þ pþ p). All these situations cover the trivial widely studied case in which g const40.. 2. Preliminaries and structural framework Hereafter p 2 Cþ ðÞ is fixed. The variable exponent Lebesgue space, denoted by Lp()() ¼ [Lp()()]N andR consisting of all the measurable vector-valued functions u : ! RN such that ju(x)jp(x)dx is finite, is endowed with the so-called Luxemburg norm ( ) Z uðxÞ pðxÞ dx 1 , kukpðÞ ¼ inf 4 0 : and is a separable and reflexive Banach space, cf. [10, Corollaries 2.12 and 2.7]. For basic properties of the variable exponent Lebesgue spaces we refer to [10]. Since here 05jj51, if q 2 Cþ ðÞ and p q in , then the embedding Lq()() ,! Lp()() is continuous, cf. [10, Theorem 2.8]. 0 Let Lp ()() be the conjugate space of Lp()(), obtained by conjugating the exponent pointwise that is, 1/p(x) þ 1/p0 (x) ¼ 1, [5, Theorem 1.14]. For any 0 u 2 Lp()() and v 2 Lp ()() the following Ho¨lder type inequality Z ðuðxÞ, vðxÞÞdx rp kukpðÞ kvkp0 ðÞ , rp ¼ 1 þ 1 ð2:1Þ p p0 is valid, where (, ) is the inner product on RN RN, [10, Theorem 2.1]..

(8) Complex Variables and Elliptic Equations. 721. An important role in manipulating the generalized Lebesgue–Sobolev spaces is played by the p()-modular of the Lp()() space, which is the convex continuous function p() : Lp()() ! R defined by Z juðxÞj pðxÞ dx: pðÞ ðuÞ ¼ . If (uj)j, u 2 L (¼1;41),. p(). (), then the following relations hold: kukp()51 (¼1;41) ,p()(u)51. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. p pþ kukpðÞ 1 ) kukpðÞ pðÞ ðuÞ kukpðÞ , pþ p pðÞ ðuÞ kukpðÞ , kukpðÞ 1 ) kukpðÞ. ð2:2Þ. and kuj ukp() ! 0 , p() (uj u) ! 0 , (uj)j converges to u in measure in and p()(uj) ! p()(u), since pþ51. In particular, p() is continuous in Lp()(). For a proof of these facts see [5, Theorem 1.4] and [10]. The variable exponent Sobolev space W1, p()() ¼ [W1, p()()]N, consisting of functions u 2 Lp()() whose distributional Jacobian matrix Du is in [Lp()()]nN, is endowed with the norm kuk1, pðÞ ¼ kukpðÞ þ kDukpðÞ : Thus W1, p()() is a separable and reflexive Banach space, cf. [10, Theorem 3.1]. pðÞ pðÞ ðÞ ¼ ½H1, ðÞN as the closure of C01 ðÞ ¼ ½C01 ðÞN in W1, p()(), Define H1, 0 0 1, pðÞ pðÞ ðÞN as the Sobolev space of the functions and X ¼ W0 ðÞ ¼ ½W1, 0 1, p() (), with zero boundary values, cf. [6]. As shown by Zhikov [13,14], the u2W smooth functions are in general not dense in W1, p()(), but if p 2 Cþ ðÞ is logarithmic Ho¨lder continuous, that is if there exists L40 such that for all x, y 2 , with 05jx yj 1/2, j pðxÞ pð yÞj . L , logðjx yjÞ. ð2:3Þ. pðÞ then H1, ðÞ ¼ X, namely the density property holds, see [6,9] and in particular 0 [7, Theorem 3.3]. Since is a bounded domain, if p 2 Cþ ðÞ satisfies (2.3), then the p()-Poincare´ inequality. kukpðÞ CkDukpðÞ is valid for all u 2 X, where C depends on p, jj, diam(), n and N, [7, Theorem 4.1], and so kuk ¼ kDukpðÞ is an equivalent norm on X, which will be adopted throughout the article. Moreover pðÞ ðÞ, kkÞ is a separable and reflexive Banach space. If pþ5n and (2.3) X ¼ ðW1, 0. holds, then the embedding X ,! Lp ()() is continuous, see [8, Theorem 1.1], where p is the critical Sobolev variable exponent, defined in (1.6) thanks to (1.1). Details, extensions and further references can be found in [5–10,24,26,30]. Hereafter, we assume that p 2 Cþ ðÞ satisfies ð1:1Þ and ð2:3Þ:.

(9) 722. G. Autuori and P. Pucci. For all h 2 CðÞ, such that 1 h p in , we denote by sh ¼ shðÞ the Sobolev constant of the continuous embedding X ,! Lh()(), that is. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. kukhðÞ sh kDukpðÞ. for all u 2 X;. ð2:4Þ. here sh depends on h, p, jj, n and N, see [8, Theorem 1.1] and also [10, Theorem 2.8]. Of course in (2.4) we can take h 1. Throughout the article, the usual Lebesgue [L2()]N and [L2()]nN are R spaces 2 equipped with the canonical norm k’k2 ¼ ( j’(x)j dx)1/2, while the elementary R bracket pairing h’, i ¼ (’(x), (x))dx is clearly well-defined for all ’, such that (’, ) 2 L1(). Both spaces are briefly denoted, if necessary, by L2(). Indeed, in order to simplify the notation, we drop the exponents N and nN in all the functional spaces involved in the treatment, as made above. By K we denote the main solution and test functions space for (P 1)–(P 3), that is K ¼ f 2 K 0 : E is locally bounded on Rþ 0 g, while K 0 is a functional space, which is specified according to the problem under consideration. The energy functional E along 2 K is defined pointwise by 1 EðtÞ ¼ kt ðt, Þk22 þ AðtÞ þ FðtÞ, 2. t 2 Rþ 0,. ðEÞ. where Z jðt, xÞj pðxÞ jDðt, xÞj pðxÞ dx, IðtÞ ¼ dx, pðxÞ pðxÞ Z MðIðtÞÞ ¼ aIðtÞ þ b½IðtÞ , FðtÞ ¼ Fðt, x, ðt, xÞÞdx, Z. AðtÞ ¼ MðIðtÞÞ þ . R1. ðAÞ. . N being Fðt, x, uÞ ¼ 0 ð f ðt, x, uÞ, uÞ d for all ðt, x, uÞ 2 Rþ by (H) and 0 R þ M() M() for all 2 R0 by the special form of M. The term A is the elliptic part of the system, I is the so-called Dirichlet energy integral at level p(x) along , while F is the potential energy of the field . In order to study the global asymptotic stability, we assume on f the subsequent condition (F ), where 0 0 is defined in (1.3). (F ) Suppose that there exist 2 [0, 0p/pþ) such that N ðaÞ ð f ðt, x, uÞ, uÞ ajuj pðxÞ in Rþ 0 R ,. a variable exponent q 2 Cþ ðÞ, with q p in , and a positive constant such that N for all ðt, x, uÞ 2 Rþ 0 R ðbÞ j f ðt, x, uÞj ð1 þ jujqðxÞ1 Þ: Moreover, if there exists y 2 such that q( y)4p ( y), then f verifies (a), (b) and also ðcÞ ð f ðt, x, uÞ, uÞ 1 jujqðxÞ 2 juj1=qðxÞ 3 juj p. ðxÞ. N for all ðt, x, uÞ 2 Rþ 0 R and appropriate constants 140, 2, 3 0. When 0 ¼ 0, we take also ¼ 0 in (F )-(a). Therefore, when either 0 ¼ 0 or a ¼ 0, that is in the latter case when the problem is degenerate, condition (F )-(a) reduces to.

(10) Complex Variables and Elliptic Equations. 723. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. the more familiar inequality ( f (t, x, u), u) 0, namely to the request that f is of restoring type. When f 0 then (F )-(b) holds for any fixed q 2 Cþ ðÞ, with p q p in , so that (F )-(c) is unnecessary. In [31], global existence for hyperbolic equations is proved in the standard setting of the Sobolev spaces with constant exponents, without imposing any bound on the exponent q of the source term f. This justifies the importance to consider for asymptotic stability also the case in which the relation q p in fails. We refer to [31] for a complete recent bibliography on wave equations with also nonlinear dampings. Consider the function ~ f ðt, x, uÞ ¼ V1 ðt, xÞjujqðxÞ2 u V2 ðt, xÞjuj pðxÞ2 u,. ð2:5Þ. with q~ 2 Cþ ðÞ, V1, V2 0, V1, V2 2 CðRþ 0 Þ and ¼ sup ½V1 ðt, xÞ þ V2 ðt, xÞ 5 1,. 2 ¼ sup V2 ðt, xÞ a,. Rþ 0 . ð2:6Þ. Rþ 0 . for some 2 [0, 0p/pþ) if 040 and ¼ 0 if 0 ¼ 0. If there exists y 2 for which ~ yÞ 4 p ð yÞ, we assume (2.6) and also that qð q~ p. in . and. 1 ¼ inf V1 ðt, xÞ 4 0: þ. ð2:7Þ. R0 . It is easy to see that (F )-(a) is fulfilled. In order to prove (F )-(b) and (F )-(c) ~ we distinguish two cases. If 1 q~ p in , it is enough to take q ¼ maxf p, qg, so that (F )-(b) also is verified for such choice of q and as in (2.6) and (F )-(c) is not ~ yÞ 4 p ð yÞ and q~ p in , taking required. While, if there exists y 2 for which qð q q~ in , we easily see that (F )-(b) is verified with as in (2.6) and (F )-(c) holds, again with q q~ in , 2 ¼ 3 ¼ supRþ0 V2 ðt, xÞ, and 140 assumed in (2.7). In turning from global to local asymptotic stability for (P 1)–(P 3), we do not use (F ), but (F )0 , in particular (F )-(a) is required only for u small. More precisely, we denote by (F )0 the condition (F ), in which (a) is replaced by ðaÞ0 a0 4 0, lim inf u!0. ð f ðt, x, uÞ, uÞ a , for some 2 ½0, 0 p =pþ Þ, juj pðxÞ. while (b) is assumed with the further restriction that pþ5q q p in , and (c) is dropped. Take f (t, x, u) ¼ V1(t, x)jujq(x)2u V2(t, x)jujp(x)2u V3(t, x)jujq(x)2u, where q 2 Cþ ðÞ is such that pþ5q q p in , while V1, V2 are as in (2.5), V3 2 CðRþ 0 Þ, and (2.6) is here replaced by sup fjV1 ðt, xÞ V3 ðt, xÞj þ V2 ðt, xÞg 5 1, Rþ 0 . sup V2 ðt, xÞ a, Rþ 0 . for some 2 [0, 0p/pþ). Whenever 040 and a40, the function f verifies (F )0 . Concerning the continuous damping function Q we also require the validity of (Q) There exist constant exponents m, r satisfying 2 m 5 r s,. s ¼ maxfq , p g,.

(11) 724. G. Autuori and P. Pucci. and nonnegative continuous functions d1 ¼ d1(t, x), d2 ¼ d2(t, x), such that for all arguments t, x, u, v, 0. 0. ðaÞ jQðt, x, u, vÞj d1 ðt, xÞ1=m ðQðt, x, u, vÞ, vÞ1=m þ d2 ðt, xÞ1=r ðQðt, x, u, vÞ, vÞ1=r , and the following functions 1 and 2 are well-defined kd2 ðt, Þks=ðsrÞ ,. 1 ðtÞ ¼ kd1 ðt, Þks=ðsmÞ , 2 ðtÞ ¼ kd2 ðt, Þk1 ,. if r 5 s, if r ¼ s:. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. Moreover, there are non-negative functions ¼ (t), ! ¼ !(), ¼ jvj, such that ðbÞ ðQðt, x, u, vÞ, vÞ ðtÞ!ðjvjÞ for all arguments t, x, u, v, where ! 2 CðRþ 0 Þ is such that !ð0Þ ¼ 0,. !ðÞ ¼ 2. !ðÞ 4 0 for 0 5 5 1,. for 1,. while 1} 2 L1loc ðRþ 0 Þ for some exponent +41. Hence for the local asymptotic stability results concerning (P 1)–(P 3) the exponents s in (Q)-(a) reduces simply to p . An interesting example of Q is given by Qðt, x, u, vÞ ¼ jvjm2 A1 ðt, x, uÞv þ jvjr2 A2 ðt, x, uÞv, N where A1, A2 are semidefinite positive continuous matrices defined in Rþ 0 R , and 2 m5r s, as in (Q). Denote by hi(t, x) the number infu 2 RN h~i ðt, x, uÞ, where h~i ðt, x, uÞ is the least eigenvalue of the symmetric part of the coefficient matrix Ai(t, x, u) and assume Hi ðt, xÞ ¼ supu 2 RN H~ i ðt, x, uÞ 5 1, where H~ i ðt, x, uÞ is the Euclidean norm of Ai(t, x, u). By assumption hi(t, x) 0. Suppose furthermore that there exist 1 1 and 2 1 such that. Hi i hi , i ¼ 1, 2,. d1 ¼ 1m1 H1 ,. d2 ¼ 2r1 H2. in Rþ 0 ,. with H1(t, ) 2 Ls/(sm)() and either H2(t, ) 2 Ls/(sr)() if r5s, or H2(t, ) 2 L1() if r ¼ s. Condition Hi ihi is the weak uniform definiteness of Ai. Of course, it is automatic in the scalar case N ¼ 1 or when Q is autonomous and also independent of x, that is Q ¼ Q(v). By the definition of hi we have for all ðt, xÞ 2 Rþ 0 hi ðt, xÞjvj2 ðAi ðt, x, uÞv, vÞ,. jAi ðt, x, uÞj Hi ðt, xÞ,. i ¼ 1, 2,. so that, by the so-called weak uniform definiteness of Ai, 0. jQðt, x, u, vÞj H1 ðt, xÞ1=m ðH1 ðt, xÞjvjm Þ1=m þ H2 ðt, xÞ1=r ðH2 ðt, xÞjvjr Þ1=r 0. 0. 0. H1 ðt, xÞ1=m ð1 h1 jvjm Þ1=m þ H2 ðt, xÞ1=r ð2 h2 jvjr Þ1=r : On the other hand, (Q(t, x, u, v), v) h1(t, x)jvjm þ h2(t, x)jvjr and, combining the last two relations, we get (Q)-(a). Indeed, jQ(t, x, u, v)j H1(t, x)jvjm1 þ.

(12) 725. Complex Variables and Elliptic Equations H2(t, x)jvjr1 yields 0. 0. jQðt, x, u, vÞj 11=m H1 ðt, xÞ1=m ðQðt, x, u, vÞ, vÞ1=m 0. 0. þ 21=r H2 ðt, xÞ1=r ðQðt, x, u, vÞ, vÞ1=r , which immediately gives (Q)-(a). Finally, (Q)-(b) holds, with ðtÞ ¼ inf fh1 ðt, xÞ þ h2 ðt, xÞg,. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. x2. !ðÞ ¼ minf r , 2 g,. provided that 1} 2 L1loc ðRþ 0 Þ, for some +41. For other special examples of Q we refer to [19,20,22,29]. The main results of the article are proved under the additional assumption that Ft u 0 in Rþ 0, along a global solution u 2 K. This request is trivially automatic, whenever either f N does not depend on t, or ft 0 in Rþ 0 R , as well as in many other cases, and for simplicity, we assume it in the definition of solution.. 3. The model (P 1): the p(x)-Laplacian In this section, we study the global and local asymptotic stability for the model (P 1) and take þ 1 2 K 0 ¼ CðRþ 0 ! X Þ \ C ðR0 ! L ðÞÞ:. As already said in Section 2, the energy functional associated with (P 1) is defined in (E), with A and F given in (A). In the first part of the section, without further mentioning, we also assume (F ) and (Q)-(a), while in studying local asymptotic stability, we assume (F )0 and (Q)-(a). We say that a solution of (P 1) is a function u 2 K satisfying the following two conditions: (A) Distribution identity Z tn t hut , i 0 ¼ hut , t i MðIuðÞÞhjDuj pðÞ2 Du, Di hjuj pðÞ2 u, i 0 o hQð, , u, ut Þ þ f ð, , uÞ, i d for all t 2 Rþ 0 and 2 K. (B) Conservation law ðiÞ Du :¼ hQðt, , u, ut Þ, ut i Ft u 2 L1loc ðRþ 0 Þ, Zt ðiiÞ Ft u 0 and t ° EuðtÞ þ DuðÞd is nonincreasing in Rþ 0: 0. The assumptions (F ), or (F )0 , and (Q)-(a) make the definition of solution meaningful thanks to (B)-(i). For a proof of this fact we refer to [19,20]. Moreover, condition (B)-(ii) in the distribution identity, together with (H), implies that the energy function Eu is nonincreasing in Rþ 0 : Before proving our main theorem about.

(13) 726. G. Autuori and P. Pucci. the global asymptotic stability of solutions for (P 1), we give some preliminary lemmas and hereafter, u will be a fixed solution of (P 1). The nonincreasing energy function Eu verifies in Rþ 0 AuðtÞ a 0 IuðtÞ, if 0 4 0, 1 EuðtÞ kut ðt, Þk22 þ 2 AuðtÞ, if 0 ¼ 0, ( 1 a 1 0 IuðtÞ, if 0 4 0, kut ðt, Þk22 þ b½IuðtÞ þ 2 aIuðtÞ, if ¼ 0,. LEMMA 3.1. ð3:1Þ. 0. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. so that, in particular, Eu 0 in. Rþ 0.. Moreover. kuk2 , kut k2 , kDukpðÞ , kukqðÞ , kukp ðÞ , MðIuÞ 2 L1 ðRþ 0 Þ,. ð3:2Þ. Du ¼ hQðt, x, u, ut Þ, ut i Ft u 2 L1 ðRþ 0 Þ:. ð3:3Þ. Proof N Case 1 040 By (F )-(a) we have F(t, x, u) ajujp(x)/p(x) in Rþ 0 R , so that along the solution u ¼ u(t, x) of (P 1) Z FuðtÞ a. juðt, xÞj pðxÞ dx a IuðtÞ: pðxÞ 0 . ð3:4Þ. Hence (3.1) follows at once. In order to prove conditions (3.2), first note that Eu is bounded above by Eu(0). Hence (3.1), (E) and the special form of M imply that also . ð3:5Þ Iu þ bðIuÞ 2 L1 ðRþ a 1 0 Þ, 0 being 50p/pþ 0. Therefore, Iu 2 L1 ðRþ 0 Þ, since also a þ b40, so that also MðIuÞ 2 L1 ðRþ 0 Þ. Since Iu(t) p()(Du(t, ))/pþ and. kDukpðÞ max ½pðÞ ðDuÞ1=p , ½pðÞ ðDuÞ1=pþ , ð3:6Þ þ 1 we also get kDukpðÞ 2 L1 ðRþ 0 Þ. Then kut k2 2 L ðR0 Þ by (3.1) and (3.5). 1, pðÞ 2 Since X ¼ W0 ðÞ is continuously embedded in L () by (1.1), that is (2.4). holds for h 2, we have also kuk2 2 L1 ðRþ 0 Þ. Furthermore, when p q p in , q() p () since the Sobolev embeddings X ,! L () and X ,! L () are continuous by (2.4), also kukq(), kukp ðÞ 2 L1 ðRþ 0 Þ. Let us now consider the case in which there exists y 2 such that q( y)4p ( y). Hence, by (F )-(c) we get for all 2 K Z1 ð f ðt, x, Þ, Þd Fðt, x, Þ ¼. Z. 0 1 0. ¼. 0. ð 1 jjqðxÞ qðxÞ1 2 jj1=qðxÞ 1=q ðxÞ 3 jj p. . 1 3. jjqðxÞ qðxÞ 2 jj1=qðxÞ jj p ðxÞ , qðxÞ p ðxÞ. ðxÞ p ðxÞ1. . Þd.

(14) Complex Variables and Elliptic Equations and, since 140, we then have along the solution u 2 K . qþ 3 1=qþ . FuðtÞ þ K maxfkuðt,Þk1=q ,kuðt,Þk g þ ðuðt,ÞÞ , qðÞ ðuðt, ÞÞ p ðÞ 1 1 1 p 0. 727. ð3:7Þ. 0. where K ¼ ð1=q þ 1=q0 Þqþ 2 maxfjj1=q , jj1=qþ g. Observe that Fu is bounded above by Eu(0) in virtue of (E) and also below by (3.4) and the fact that Iu is in þ þ 1 1 L1 ðRþ 0 Þ. Moreover, since kutk2, kDukpðÞ 2 L ðR0 Þ, we also get kuk1 2 L ðR0 Þ by (2.4) when h 1, as well as p. p. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. p ðÞ ðuðt, ÞÞ maxfkuðt, Þkp ðÞ kuðt, Þkp þðÞ g const,. þ 1 since again kukp ðÞ 2 L1 ðRþ 0 Þ by (2.4) with h ¼ p . Therefore qðÞ ðuÞ 2 L ðR0 Þ þ 1 by (3.7). Hence kukqðÞ 2 L ðR0 Þ by (2.2). This completes the proof of (3.2).. Case 2 0 ¼ 0 The situation is much simpler since the external force f is of restoring type. It follows that Fu(t) 0 for all t 2 Rþ 0 and (3.1) follows at once. Hence Eu 0 since all the three terms in the definition of Eu, with u 2 K, are nonnegative, and clearly bounded by Eu(0). Therefore, kut k2 2 L1 ðRþ 0 Þ and from the fact that

(15) pðÞ ðDuðt, ÞÞ pðÞ ðDuðt, ÞÞ a þb aIuðtÞ þ b½IuðtÞ EuðtÞ Euð0Þ, pþ pþ by (E) and (B)-(ii). Thus M(Iu) and p()(Du) are also bounded in Rþ 0 since a þ b40 and 1. Hence kDukpðÞ 2 L1 ðRþ 0 Þ by (3.6). From now on the proof can proceed as in the previous case word by word. Therefore (3.2) is valid also in the case 0 ¼ 0. PropertyR (3.3) follows at once, since Du 0 in Rþ 0 , by (B)-(ii) and (H). t g Hence 0 0 DuðÞd Euð0Þ EuðtÞ Euð0Þ, since Eu 0 in Rþ 0 by (3.1). By Lemma 3.1 there exists l 0 such that lim EuðtÞ ¼ l:. t!1. ð3:8Þ. Note that in the proof of the next result we do not use the assumption Ftu 0 in Rþ 0, required in (B)-(ii). LEMMA 3.2. If l40 in (3.8), then there exists

(16) ¼

(17) (l )40 such that LuðtÞ

(18) for all t 2 Rþ 0 , where. LuðtÞ ¼ kut ðt, Þk22 þ apðÞ ðDuðt, ÞÞ þ. b p1 þ. ð3:9Þ. pðÞ ðDuðt, ÞÞ þ pðÞ ðuðt, ÞÞ. þ h f ðt, , uÞ, uðt, Þi: Proof. Since Eu(t) l we have in Rþ 0 it follows that kut k22 þ apðÞ ðDuÞ þ b pðÞ ðDuÞ þ pðÞ ðuÞ ðl FuÞ,. ð3:10Þ. where ¼ min{2, p}41. Let 0 J1 ¼ ft 2 Rþ 0 : FuðtÞ l= g,. 0 J2 ¼ ft 2 Rþ 0 : FuðtÞ 4 l= g,. ð3:11Þ.

(19) 728. G. Autuori and P. Pucci. being 0 ¼ /( 1) the Ho¨lder conjugate of . For t 2 J1 kut ðt, Þk22 þ apðÞ ðDuðt, ÞÞ þ b pðÞ ðDuðt, ÞÞ þ pðÞ ðuðt, ÞÞ l, so that kut ðt, Þk22 þ apðÞ ðDuðt, ÞÞ þ. b pðÞ ðDuðt, ÞÞ þ pðÞ ðuðt, ÞÞ . p1 þ. l p1 þ. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. Before dividing the proof into two parts, we observe that in Rþ 0 n o þ jFuj ½kuk1 þ qðÞ ðuÞ kuk1 þ max kukqqðÞ , kukqqðÞ. :. ð3:12Þ. ð3:13Þ. by (F )-(b) and (2.1). Case 1 q p in Let us first consider the case in which 040. Lemma 3.1 of [4], shows that 0 4 0, where Z Z 0 jðxÞj pðxÞ dx jDðxÞj pðxÞ dx ð3:14Þ . . for all 2 X. Thus, by (F )-(b) and the fact that 0 0 p =pþ , Z Z pðxÞ dx a jDuðt, xÞj pðxÞ dx h f ðt, , uÞ, ui a juðt, xÞj 0 pþ a pðÞ ðDuðt, ÞÞ, 0 p. ð3:15Þ. since u 2 K. Hence, using (3.12) and (3.15), we have for all t 2 J1 . pþ b LuðtÞ a 1 pðÞ ðDuðt, ÞÞ þ kut ðt, Þk22 þ 1 pðÞ ðDuðt, ÞÞ þ pðÞ ðuðt, ÞÞ 0 p pþ ! . pþ l pþ b 2 þ kut ðt, Þk2 þ 1 pðÞ ðDuðt, ÞÞ 1 0 p p1 0 p pþ þ . pþ l 1 : 1 0 p pþ Next consider t 2 J2. By (3.13), (2.4) and (2.2) we get in Rþ 0 n o q þ jFuj C kDukpðÞ þ max kDukpðÞ , kDukqpðÞ ,. ð3:16Þ. for an appropriate constant C40, depending on , s1 , sq introduced in (2.4) and p. Hence for all t 2 J2 kDuðt, Þk , if kDuðt, ÞkpðÞ 1, l ð3:17Þ 5 FuðtÞ 2C kDuðt, ÞkpðÞ qþ if kDuðt, ÞkpðÞ 4 1, 0 pðÞ , that is (. kDuðt, ÞkpðÞ. 1=qþ ) l l min , ¼ C2 ðl Þ 4 0: 2C 0 2C 0. ð3:18Þ.

(20) 729. Complex Variables and Elliptic Equations. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. By (3.15) for all t 2 J2 . pþ b LuðtÞ a 1 pðÞ ðDuðt, ÞÞ þ 1 ½pðÞ ðDuðt, ÞÞ þ pðÞ ðuðt, ÞÞ 0 p pþ . pþ p pþ , kDuðt, ÞkpðÞ g minfkDuðt, ÞkpðÞ a 1 0 p b p pþ , kDuðt, ÞkpðÞ g : þ 1 ½minfkDuðt, ÞkpðÞ pþ Denoting by C3 ¼ C3(l ) the positive number minfC2p ðl Þ, C2pþ ðl Þg, then . pþ b LuðtÞ a 1 C3 þ 1 C3 , t 2 J2 : 0 p pþ Therefore, (3.9) holds with ( ) . pþ l b

(21) ¼

(22) ðl Þ ¼ 1 min 1 , aC3 þ 1 C3 , 0 p pþ pþ provided that either a 6¼ 0 or J2 6¼ ;, being a þ b40. Now, if a ¼ 0 and J2 ¼ ;, then (F )-(a) reduces to ( f (t, x, u), u) 0, and so (3.9) holds with

(23) ¼ l=p1 þ 4 0. When 0 ¼ 0, that is when also ¼ 0 in (F )-(a), then the proof simplifies and (3.9) holds with 1

(24) ¼

(25) ðl Þ ¼ minfl=p1 þ , aC3 g þ b C3 =pþ 4 0,. since l40 and a þ b40. Case 2 There exists y 2 such that q( y)4p ( y) As before we first suppose 040. Using (3.13)1, (F )-(c), Ho¨lder’s inequality and the fact that 140 by (F )-(c), we have for t 2 J2

(26) n o l 1=q 1=qþ ~ 5FuðtÞ . kuðt,Þk þ. max kuðt,Þk ,kuðt,Þk h fðt, , uðt,ÞÞ,uðt,Þi þ. 0 1 2 1 1 1 0. þ 3 p ðÞ ðuðt,ÞÞ , where 0 ¼ / 140 and by (2.4), when h 1,. 0. 0. ~ 2 ¼ ð1=q þ 1=q0 Þ 2 maxfjj1=q , jj1=qþ g.. Therefore,. n o 1=qþ h f ðt, , uÞ, ui þ c1 kDukpðÞ þ c2 max kDuk1=q , kDuk pðÞ pðÞ n o p þ p þ c3 max kDukpðÞ , kDukpðÞ 4 l= 0 0 , p. p. þ , s1=q g 0 and c3 ¼ 3 maxfsp , sp þ g 0. where c1 ¼ 1 s1 40, c2 ¼ ~ 2 maxfs1=q 1 1 Hence for t 2 J2 and h f (t, , u(t, )), u(t, )i 0, then. either h f ðt, , uðt, ÞÞ, uðt, Þi l=2 0 0. or. kDuðt, ÞkpðÞ c4 ,. ð3:19Þ.

(27) 730. G. Autuori and P. Pucci. where c4 ¼ c4(l, 0, )40 is an appropriate constant, arising when p. p. 1=qþ 0 þ c1 kDukpðÞ þ c2 maxfkDuk1=q pðÞ , kDukpðÞ g þ c3 maxfkDukpðÞ , kDukpðÞ g l=2 0 . at the time t. On the other hand, when t 2 J2 and h f (t, , u(t, )), u(t, )i50, then kDu(t, )kp() c5, where c5 c4 is an appropriate number arising from p. p. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. 1=qþ 0 þ c1 kDukpðÞ þ c2 maxfkDuk1=q pðÞ , kDukpðÞ g þ c3 maxfkDukpðÞ , kDukpðÞ g l= 0 :. Now, when 040 in (F )-(a), putting c6 ¼ minfc5p , c5pþ g, the conclusion (3.9) holds, with ) (. . pþ l pþ b b l , a 1 c6 þ 1 c6 , ac4 þ 1 c4 , 4 0,

(28) ¼ min 1 0 p p1 0 p 2 0 0 pþ pþ þ since l40, 2 [0, 0 p/pþ), c440, c640 and a þ b40. While if 0 ¼ 0, hence ¼ 0 in (F )-(b), then (3.9) holds, with ( ) l b b l

(29) ¼ min 1 , ac6 þ 1 c6 , ac4 þ 1 c4 , 4 0: 2 0 0 pþ pþ pþ g. This completes the proof.. We possess now all the necessary tools to present the main theorem concerning the global asymptotic stability for (P 1). THEOREM 3.3 Let also (Q)-(b) hold. Suppose there exists a function k satisfying either þ þ 1 k 2 CBVðRþ 0 ! R0 Þ and k6 2 L ðR0 Þ. Z þ k 2 W1,1 loc ðR0. !. Rþ 0 Þ,. k 6 0. and. or,. ðK1 Þ. t. jk0 ðÞjd. 0. lim Z. ¼ 0:. t. t!1. ðK2 Þ. kðÞd 0. Assume finally ,Z. t. lim inf AðkðtÞÞ. kðÞd 5 1,. t!1. ð3:20Þ. 0. where Z AðkðtÞÞ ¼ BðkðtÞÞ þ Z BðkðtÞÞ ¼ 0. t. 1} k} d. 1=} ,. 0. t. 1=m Z t 1=r r. 1 k d þ. 2 k d : m. 0. Then along any solution u of (P 1), property (1.2) holds.. ð3:21Þ.

(30) 731. Complex Variables and Elliptic Equations. Proof We follow the principal ideas of [19, Theorem 3.1] and [20, Theorem 1], also employed in [22,18]. Let us start with proving (1.2)1 by contradiction. Hence, assume that l40 in (3.8), define the Lyapunov function by VðtÞ ¼ kðtÞhu, ut i ¼ hut , i,. ¼ kðtÞu,. and divide the proof in two cases.. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. þ 1 Case (K1) First assume the simpler situation in which k 2 CBVðRþ 0 Þ \ C ðR0 Þ, 0 so that t ¼ k u þ kut and clearly 2 K. Thus, by the distribution identity (A), we get for any t T 0. VðÞ. t T. ¼. Z tn k0 hu, ut i þ 2kkut k22 k kut k22 þ MðIuðÞÞpðÞ ðDuð, ÞÞ þ pðÞ ðuð, ÞÞ T Zt o khQð, , u, ut Þ, uid: ð3:22Þ þ h f ð, , uÞ, ui d T. In estimating the right-hand side of (3.22), we first note that sup jhuðt, Þ, ut ðt, Þij sup kuðt, Þk2 kut ðt, Þk2 ¼ U 5 1, Rþ 0. ð3:23Þ. Rþ 0. þ by (1.1) and (3.2) of Lemma 3.1. Now, being certainly k 2 L1 loc ðR0 Þ, by Lemma 3.2 it follows Zt Zt. k kut k22 þ MðIuðÞÞpðÞ ðDuð, ÞÞ þ pðÞ ðuð, ÞÞ þ h f ð, , uÞ, ui d

(31) k d, T. T. ð3:24Þ and by Lemma 3.2 of [20] Z. t. kjhQð, , u, ut Þ, uijd "1 ðT ÞBðkðtÞÞ,. ð3:25Þ. T. where "Z. 1. "1 ðT Þ ¼ sup kuðt, Þks Rþ 0. 1=m0 Z DuðtÞdt þ. T. 1. 1=r 0 # DuðtÞdt ¼ oð1Þ. ð3:26Þ. T. as T ! 1, since kuðt, Þks 2 L1 ðRþ 0 Þ by (3.2) and the definition of s. Similarly, by Lemma 3.3 of [20], Z. Z. t. T. kkut k22. t. d . Z t 1=} 1} } k d þ "2 ðT ÞCðÞ k d ,. 0. where CðÞ ¼ !1=} , ! ¼ supf 2 =!ðÞ : "2 ðT Þ ¼ sup kut ðt, Þk2=} 2 Rþ 0. ð3:27Þ. 0. T. pffiffiffiffiffiffiffiffiffiffiffi =jjg, and. Z. 1 T. 1=}0 DuðtÞdt ¼ oð1Þ. ð3:28Þ.

(32) 732. G. Autuori and P. Pucci. as T ! 1. Thus, by (3.22) it follows Z t 1=} Zt Zt t jk0 jd þ 2 k d þ 2"ðT ÞCðÞ 1} k} d VðÞ T U T T 0 Zt

(33) k d þ "ðT ÞBðkðtÞÞ,. ð3:29Þ. T. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. where "(T ) ¼ max{"1(T ), "2(T )}. By (3.20) there is a sequence ti % 1 and a number ‘40 such that Z ti ð3:30Þ Aðkðti ÞÞ ‘ k d: 0. Choose ¼ (l ) ¼

(34) /4 and fix T40 sufficiently large so that "ðT Þ½2CðÞ þ 1‘

(35) =4, being "(T ) ¼ o(1) as T ! 1. Consequently, for ti T, Z Z ti

(36) ti Vðti Þ U jk0 jd þ SðT Þ k d, 4 T T where U is given in (3.23) and SðT Þ ¼ VðT Þ þ "ðT Þ½2CðÞ þ 1‘ we get. ð3:31Þ. ð3:32Þ RT 0. k d. Thus by (K1). lim Vðti Þ ¼ 1,. i!1. ð3:33Þ. þ since k0 2 L1 ðRþ 0 Þ being k 2 CBVðR0 Þ. On the other hand, by (3.23) and recalling that k is bounded, we have for all t 2 Rþ 0 jVðtÞj supRþ0 k kuðt, Þk2 kut ðt, Þk2 supRþ0 k U: ð3:34Þ. This contradiction completes the first part of the proof. þ 1 We treat now the general case k 2 CBVðRþ 0 Þ n C ðR0 Þ, following Lemma A þ þ in [21]. Let k 2 C1 ðR0 Þ and G R0 be an open subset such that k in Rþ 0 nG (i) 2k k (ii) Var k 2 Var k; 0 in G; Z (iii) k ds 1: G. Clearly k 2 CBVðRþ 0 Þ by (ii). We next prove that k satisfies (K1) and (3.20). Note that since k6 2 L1 ðRþ 0 Þ it is possible to find a value T1 such that Z. T1. k d 2:. ð3:35Þ. 0. Considering t T1, by (i), (ii) and (3.35) we obtain Z Zt Z Zt Z Zt 1 t k d k d k d k d k d 1 k d: 2 0 ½0,tnG G 0 0 0. ð3:36Þ.

(37) Complex Variables and Elliptic Equations. 733. Hence k satisfies (K1). Moreover, by (i) and (3.36) for all t T1 Zt Zt AðkðtÞÞ k d 4 AðkðtÞÞ k d, 0. 0. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. where k ° A(k) is defined in (3.21). This shows that k also satisfies (3.20). The general case is therefore reduced to the situation when k is smooth, and the proof is complete under (K1). Case (K2) We still obtain t ¼ k0 u þ kut, so that 2 K. Clearly Lemmas 3.1 and 3.2 continue to hold, as well as Lemmas 3.2 and 3.3 of [20], that is (3.25) and (3.27) are available. From now on we follow the proof of case (K1) until obtaining (3.32). By the definition of V we now get Z ti jk0 jd , jVðti Þj Ukðti Þ U kð0Þ þ 0. so that by (3.32)

(38) 4. Z. Z. ti. ti. jk0 jd þ SðT Þ þ Ukð0Þ: ð3:37Þ 0 0 R ti First note that k6 2 L1 ðRþ 0 Þ by (K2). Now, dividing (3.37) by 0 k d, we contradict (K2) as i ! 1. In conclusion, in both cases, we have proved that Eu(t) approaches zero as t ! 1. Finally, by (3.1) and the facts that a þ b40 and n o p pþ , kDuðt, ÞkpðÞ pþ IuðtÞ min kDuðt, ÞkpðÞ , ð3:38Þ k d 2U. then (1.2) holds.. g. The integral condition (3.20) prevents the damping term Q being either too small (underdamping) or too large (overdamping) as t ! 1 and was introduced by Pucci and Serrin in [21], see also [19,20]. COROLLARY 3.4 Let also (Q)-(b) hold. Suppose that any one of the following conditions is satisfied: þ 1 m1. 1 þ q 1 2 2 L1 ðRþ ðaÞ 2 CBVðRþ 0 Þ n L ðR0 Þ, 0 Þ; þ 1 ðbÞ 1=ð1mÞ 2 CBVðRþ 0 Þ n L ðR0 Þ, } ¼ m, 1. 1m = 1 , 2 1ðq 1Þ=ð1mÞ 2 L1 ðRþ 0 Þ; þ 1 ðcÞ 21=ð1q Þ 2 CBVðRþ 0 Þ n L ðR0 Þ, } ¼ q ,. 1q = 2 , 1 2ðm1Þ=ð1q Þ 2 L1 ðRþ 0 Þ; þ 1 ðdÞ ð 1 þ 1m Þ1=ð1mÞ 2 CBVðRþ 0 Þ n L ðR0 Þ, } ¼ m,. 2 ð 1 þ 1m Þðq 1Þ=ð1mÞ 2 L1 ðRþ 0 Þ; þ 1 ðeÞ ð 2 þ 1q Þ1=ð1q Þ 2 CBVðRþ 0 Þ n L ðR0 Þ, } ¼ q ,. 1 ð 2 þ 1q Þðm1Þ=ð1q Þ 2 L1 ðRþ 0 Þ: Then (1.2) holds along any solution u of (P 1)..

(39) 734. G. Autuori and P. Pucci. Proof It is enough to take in each case, respectively, k ¼ , k ¼ 1=ð1mÞ , 1 k ¼ 21=ð1q Þ , k ¼ ( 1 þ 1m)1/(1m) and k ¼ ( 2 þ 1q)1/(1q), and find that (K1) and (3.20) are satisfied. g. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. As already observed, for the local asymptotic stability we require on f the main structural assumption (F )0 of Section 2 and assume not only that (P 1) is nondegenerate, that is a40, but also that 040, see (F )0 -(a)0 . We recall that the definition of solution for problem (P 1) is well-posed whenever (F )0 and (Q)-(a) are in charge. From the rest of the section we assume (F )0 and (Q)-(a), without further mentioning. Note that in the proof of the next three lemmas we do not use the assumption Ftu 0 in Rþ 0 , required in (B)-(ii). LEMMA 3.5. There exist 2 ð, 0 p =pþ Þ and c40 such that ð f ðt, x, uÞ, uÞ ajuj pðxÞ cjujqðxÞ. N for all ðt, x, uÞ 2 Rþ 0 R . Moreover, if u is a solution of (P 1), then Z Z juðt, xÞj pðxÞ juðt, xÞjqðxÞ dx c dx, FuðtÞ a pðxÞ qðxÞ 1 a 1 EuðtÞ kut ðt, Þk22 þ IuðtÞ 2 2 0 n o n o p pþ þ , kuðt, ÞkqðÞ , kuðt, ÞkqqðÞ c~ max kuðt, ÞkqqðÞ , þ a~ min kuðt, ÞkqðÞ . n o a pþ c 1 , 4 0, s~ q ¼ max sqpþ , sqp , c~ ¼ a~ ¼ ~ 2pþ sq 0 p q. ð3:39Þ. ð3:40Þ. for all t 2 Rþ 0 , where sq is given in (2.4). Proof Fix 2 ð, 0 p =pþ Þ. By (F )0 -(a)0 we have ( f (t, x, u), u) ajujp(x) for all N ðt, x, uÞ 2 Rþ 0 R , with juj5 , provided 2 (0, 1] is sufficiently small. On the 0 other hand, (F ) -(b) and juj imply ( f (t, x, u), u) (juj1q(x) þ 1) jujq(x) ( 1q(x) þ 1)jujq(x), so that ( f (t, x, u), u) cjujq(x) with c ¼ ( 1q þ 1) N for all ðt, x, uÞ 2 Rþ 0 R such that juj . Hence (3.39) holds, with c as large as we wish. Integrating (3.39), we obtain at once the first part of (3.40) along the solution u. Hence, by the definitions of Eu and of 0, together with (2.2), we have in Rþ 0 . 1 a 1 EuðtÞ kut ðt, Þk22 þ IuðtÞ 2 2 0 . n o a pþ p pþ 1 , kDuðt, ÞkpðÞ min kDuðt, ÞkpðÞ þ 2pþ 0 p n o c þ max kuðt, ÞkqqðÞ , kuðt, ÞkqqðÞ , q and so also the second inequality of (3.40) follows at once by application of (2.4) with h ¼ q. g Since pþ5q by (F )0 -(b), the set ~ 0 ¼ fð, E Þ 2 R2 : 0 5 z0 , ðÞ E 5 E~ 0 g, .

(40) 735. Complex Variables and Elliptic Equations ~ pþ c ~ q , where ðÞ ¼ 2a 1=ðq pþ Þ a~ z0 ¼ c~. and. . 1 E~ 0 ¼ a~ 2 z pþ , q 0. ð3:41Þ. with a˜ and c~ given in (3.40), see Figure 1, is well-defined. Without loss of generality, ~ c~ 1, by taking c sufficiently large, if necessary. Here and in we also assume that a= the rest of the section u is a fixed solution of (P 1) and (t) ¼ ku(t, )kq(). LEMMA 3.6. ~ 0 , then for all t 2 Rþ If ðð0Þ, Euð0ÞÞ 2 0. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. ~0 ððtÞ, EuðtÞÞ 2 Proof. and 2EuðtÞ . kut ðt, Þk22. . þa 1 IuðtÞ: 0. ð3:42Þ. By (3.38), (3.40) and again (2.4), with h ¼ q, we have n o n o p pþ þ , kuðt, ÞkqðÞ , kuðt, ÞkqqðÞ EuðtÞ 2a~ min kuðt, ÞkqðÞ c~ max kuðt, ÞkqqðÞ pþ q ~ ~ 2aðtÞ cðtÞ , if 0 ðtÞ 1, ¼ p qþ ~ ~ 2aðtÞ cðtÞ , if ðtÞ 4 1:. Now, if there would exist t such that (t) ¼ z0, then, since z0 1, we would get ~ q0 ¼ E~ 0 4 Euð0Þ EuðtÞ 2az ~ 0pþ cz ~ q0 , ~ 0pþ cz 2az which is impossible. Therefore (t) 6¼ z0 for all t 2 Rþ 0 . Hence by the continuity of Þ ½0, z Þ, being (0)5z . In particular, the case (t)41 can we have ðRþ 0 0 0 never occur. Consequently, we have proved that along any solution u 2 K pþ q ~ ~ cðtÞ 0 E~ 0 4 Euð0Þ EuðtÞ 2aðtÞ. for all t 2 Rþ 0,. ð3:43Þ. ~ since 0 (t)5z0 1 for all t 2 Rþ 0 . Hence ððtÞ, EuðtÞÞ 2 0 and (3.40) gives . 1 a pþ q ~ ~ 1 EuðtÞ kut ðt, Þk22 þ cðtÞ : ð3:44Þ IuðtÞ þ aðtÞ 2 2 0 ~ pþ c ~ q 0 in [0, z0) we get (3.42) at once. Since a. g. Remark Lemma 3.6 is easily visualized using the two-dimensional phase plane (, E ) shown in Figure 1. In particular, by (3.43) any point ((t), Eu(t)) on the ~ 0 is trajectory of a solution u 2 K must lie above the curve ¼ (). The region shaded in Figure 1, with z0 defined in (3.41). As we shall see, if the initial data ~ 0, kut(0, )k2 and kDu(0, )kp() are sufficiently small, then ðð0Þ, Euð0ÞÞ 2 limt!1 Eu(t) ¼ 0 and (1.2) holds, that is the rest state (0, 0) is a locally attracting set in X L2(). It is possible to obtain similar results as in Lemmas 3.5 and 3.6, in terms of kDu(t, )kp(), rather than ku(t, )kq(). LEMMA 3.7. ~ 0 , then for all t 2 Rþ If ðð0Þ, Euð0ÞÞ 2 0 apðÞ ðDuðt, ÞÞ þ. b pðÞ ðDuðt, ÞÞ þh f ðt, , uÞ, ui. p1 þ. . a pþ b 1 pðÞ ðDuðt, ÞÞ þ 1 pðÞ ðDuðt, ÞÞ : 2 0 p pþ.

(41) 736 Proof. G. Autuori and P. Pucci First of all h f ðt, , uÞ, ui apðÞ ðDuðt, ÞÞ cqðÞ ðDuðt, ÞÞ apþ pðÞ ðDuðt, ÞÞ cqðÞ ðuðt, ÞÞ, 0 p. by (3.39) and (3.15), so that. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. apðÞ ðDuðt,ÞÞ þ. b pðÞ ðDuðt,ÞÞ þh fðt, , uÞ,ui. p1 þ. . pþ b a 1 pðÞ ðDuðt,ÞÞ þ 1 pðÞ ðDuðt, ÞÞ cqðÞ ðuðt,ÞÞ: 0 p pþ. ð3:45Þ. On the other hand, by (2.2) and (2.4), with h ¼ q, we have n o 1 p pþ , kDuðt, ÞkpðÞ pðÞ ðDuðt, ÞÞ min kDuðt, ÞkpðÞ minfðtÞ p , ðtÞ pþ g, s~ q and in turn pðÞ ðDuðt, ÞÞ . ðtÞ pþ s~ q. for all t 2 Rþ 0,. ð3:46Þ. ~ q is defined in (3.40). Similarly, being 5z0 in Rþ 0 by Lemma 3.6, where s qðÞ ðuðt, ÞÞ maxfðtÞq , ðtÞqþ g ¼ ðtÞq. for all t 2 Rþ 0:. ð3:47Þ. Hence, putting (3.46) and (3.47) in (3.45) we get apðÞ ðDuðt, ÞÞ þ. b p1 þ. pðÞ ðDuðt, ÞÞ þh f ðt, , uÞ, ui. . a pþ b 1 pðÞ ðDuðt, ÞÞ þ 1 pðÞ ðDuðt, ÞÞ 2 0 p pþ . a pþ þ 1 ðtÞ pþ cðtÞq 2~sq 0 p . a pþ b 1 4 pðÞ ðDuðt, ÞÞ þ 1 pðÞ ðDuðt, ÞÞ 2 0 p pþ pþ ~ þ aðtÞ cðtÞq ,. ð3:48Þ. where a˜ is defined in (3.40). The quantity a˜(t)pþ c(t)q 0, since ðRþ 0 Þ ½0, z0 Þ, so that from (3.48) the lemma is proved. g LEMMA 3.8. ~ 0 , then (3.2) and (3.3) continue to hold. If ðð0Þ, Euð0ÞÞ 2 . Proof The fact that kutk2, M(Iu) and kDukp() are in L1 ðRþ 0 Þ now follows at once Þ by (1.1). The latter part of (3.2)1 is by (3.42) and (3.6); moreover kuk2 2 L1 ðRþ 0 a consequence of the continuity of the Sobolev embeddings X ,! Lq()() and. X ,! Lp ()(), being in particular p5q p in by (F )0 -(b). Property (3.3) can be proved exactly as in Lemma 3.1. g.

(42) Complex Variables and Elliptic Equations. 737. Also in the proof of the next lemma we do not use that Ftu 0 in Rþ 0 , required in (B)-(ii).. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. LEMMA 3.9 If l40 in (3.8), then there exists a positive number

(43) ¼

(44) (l ) such that (3.9) is true. Proof Let us denote by Lu the same operator introduced in (3.9). By Lemma 3.7, we get in Rþ 0 . a pþ 1 LuðtÞ kut ðt, Þk22 þ pðÞ ðDuðt, ÞÞ 2 0 p b þ 1 pðÞ ðDuðt, ÞÞ þ pðÞ ðuðt, ÞÞ p þ. 1 pþ n 1 kut ðt, Þk22 þ apðÞ ðDuðt, ÞÞ 2 0 p o b ð3:49Þ þ 1 pðÞ ðDuðt, ÞÞ þ pðÞ ðuðt, ÞÞ : pþ As in the proof of Lemma 3.2 we have (3.10) and now divide Rþ 0 into the sets J1 and J2 given in (3.11). Hence, by (3.12), in J1 we obtain . pþ l LuðtÞ 1 : ð3:50Þ 0 p 2p1 þ Condition (3.13) is still valid and in turn we get once more (3.16)–(3.18) in J2. Moreover, by (3.49) for all t 2 J2 . a pþ b 1 LuðtÞ pðÞ ðDuðt, ÞÞ þ 1 pðÞ ðDuðt, ÞÞ þ pðÞ ðuðt, ÞÞ 2 0 p pþ . n o a pþ p pþ 1 , kDuðt, ÞkpðÞ min kDuðt, ÞkpðÞ 2 0 p n o b p pþ þ 1 min kDuðt, ÞkpðÞ , kDuðt, ÞkpðÞ , pþ so that LuðtÞ . . a pþ b 1 C3 þ 1 C3 , 2 0 p pþ. ð3:51Þ. where C3 ¼ minfC2p ðl Þ, C2pþ ðl Þg and C2(l ) is defined in (3.18). Hence, combining (3.50) with (3.51), we obtain (3.9) with ( ) 8 > l b > > min 1 , aC3 þ 1 C3 , if J2 6¼ ;, 1 pþ < p p þ þ 1

(45) ¼

(46) ðl Þ ¼ 2 0 p > l > > if J2 ¼ ;: : 1 , pþ This completes the proof. The next theorem is our main local asymptotic stability result for (P 1).. g.

(47) 738. G. Autuori and P. Pucci. THEOREM 3.10 Let also (Q)-(b) hold and k be an auxiliary function satisfying (3.20) and either (K1) or (K2). If the initial data kDu(0, )kp(), kut(0, )k2 are sufficiently small, then (1.2) continues to hold.. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. Proof Let us start with proving that if the initial data kut(0, )k2 and kDu(0, )kp() ~ 0 . Indeed, (0) ¼ ku(0, )kq()5z0 1 are sufficiently small, then ðð0Þ, Euð0ÞÞ 2 if kDu(0, )kp() is small enough by the continuity of the embedding X ,! Lq()(), while the definition of Eu, (3.16) and (2.2) give . a þ b þ sp 1 2 þ 2C kDuð0, ÞkpðÞ : Euð0Þ kut ð0, Þk2 þ 2 p This shows that Euð0Þ 5 E~ 0 for sufficiently small data. Finally, since 0 (0)5z0 pþ q ~ ~ it follows that að0Þ cð0Þ 0 and so Eu(0) 0 by (3.40). Now we prove that Eu(t) ! 0 as t ! 1 and proceed by contradiction, assuming l40 in (3.8) and showing that all the necessary estimates used in the proof of Theorem 3.3 are still valid under (F )0 , with pþ5q q p in , in place of (F ), ~ 0 . We consider separately the cases (K1) and (K2). provided that ðð0Þ, Euð0ÞÞ 2 þ 1 Case (K1) Let us first suppose k 2 CBVðRþ 0 Þ \ C ðR0 Þ. Define the Lyapunov function V(t) ¼ k(t)hu, uti as in Theorem 3.3, so that (3.22) is still true. Of course, Lemmas 3.2 and 3.3 of [20] give, respectively, (3.25) and (3.27) as before. By virtue of (F )0 , with pþ5q and q p in , and (Q)-(a), inequality (3.24) is a consequence of Lemma 3.9, while (3.23) is now valid by Lemma 3.8 and also by the request that ~ 0 . Hence we obtain (3.29) and the proof can now follow word by ðð0Þ, Euð0ÞÞ 2 word the proof of Theorem 3.3 in the regular case of (K1), as well as in the general þ 1 case k 2 CBVðRþ 0 Þ n C ðR0 Þ.. Case (K2) The proof is the same as in Theorem 3.3, with the only exception that Lemmas 3.5–3.9 are used in place of Lemmas 3.1 and 3.2. Condition (1.2)2 is a consequence of (3.38), (3.42) and the fact that a40. g. 4. The model (P 2): the p(x)-Laplacian with higher dissipation In this section we take K 0 ¼ C1 ðRþ 0 ! X Þ,. pþ p 5 1:. These further restrictions depend on the fact that (P 2) is more involved because of the internal damping gDp()ut. Without further mentioning, in the first part of the section we assume (F ) and (Q)-(a), while in the latter, as studying local asymptotic stability, (F )0 and (Q)-(a). For any 2 K, the function E defined in (E), with A and F given in (A), is again the natural energy function associated with (P 2). By a solution of (P 2) we mean a function u 2 K satisfying the following two conditions (A) Distribution identity for all t 2 Rþ 0 and 2 K Zt t hut , i 0 ¼ fhut , t i MðIuðÞÞhjDuj pðxÞ2 Du gðÞjDut j pðxÞ2 Dut , Di 0. hjuj pðÞ2 u þ Qð, , u, ut Þ þ f ð, , uÞ, igd:.

(48) Complex Variables and Elliptic Equations. 739. (B) Conservation law ðiÞ Du :¼ hQðt, , u, ut Þ, ut i þ gðtÞMðIuðtÞÞpðÞ ðDut ðt, ÞÞ Ft u 2 L1loc ðRþ 0 Þ, Zt DuðÞd is nonincreasing in Rþ ðiiÞ Ft u 0 and t ° EuðtÞ þ 0: 0. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. Note that again Eu is nonincreasing in Rþ 0 by (B)-(ii), (H) and the fact that g is nonnegative. Furthermore, the definition of solution is clearly meaningful whenever (Q)-(a) and either (F ) or (F )0 hold. The function t ° Kgu(t) :¼ g(t)M(Iu(t))p()(Dut(t, )) in (B)-(ii) is the internal material damping of Kelvin–Voigt type, in the simple case a ¼ 1 and b ¼ 0. The study of (P 2) is fairly delicate, because of the terms involving the extra damping gDp()ut, so that the analysis of Section 3 need to be changed. However, Lemmas 3.1 and 3.2 continue to hold also for (P 2). Hereafter u is a fixed solution of (P 2), so that, in particular, (3.8) is valid for some l 0. LEMMA 4.1. The Kirchhoff damped function Kg u :¼ gMðIuÞpðÞ ðDut Þ 2 L1 ðRþ 0 Þ:. Furthermore, gpðÞ ðDut Þ 2 L1 ðRþ 0 Þ, whenever either (P 2) is nondegenerate, that is a40, or infRþ0 MðIuÞ 4 0. Proof The argument follows the proof of (3.3) in Lemma 3.1, with the exception that now Du consists of three nonnegative terms, each of which is in L1 ðRþ 0 Þ. þ 1 g In particular, Kg u 2 L ðR0 Þ. The last part of the lemma follows at once. þ LEMMA 4.2 If k 2 L1 loc ðR0 Þ, then there exists T, sufficiently large, such that for all t T. Z. Z. t. jDut ð, xÞj pðxÞ1 jDuð, xÞjdx d "3 ðT ÞCðkðtÞÞ,. kðÞ gðÞMðIuðÞÞ . T. Z. t. 1=p1 Z t 1=p2 p2 gðÞkðÞ d þ gðÞkðÞ d , p1. CðkðtÞÞ ¼ T. p1 ¼ Z where "3 ðT Þ ¼ K. 1. ð4:1Þ. T. pþ , 1 þ pþ p. p2 ¼. p , 1 þ p pþ. ð p 1Þ=pþ Kg ðtÞdt ¼ oð1Þ as T ! 1, and. T. 1=p2 1 K ¼ rp sup kDuðt, ÞkpðÞ max M1=p , 1 , M1 t2Rþ 0. M1 ¼ sup MðIuðtÞÞ, t2Rþ 0. with rp is defined in (2.1). By (3.2) clearly K51. By Lemma 4.1 we first take T so large that RProof 1 K ðtÞdt 1. By Ho¨lder’s inequality, see (2.1), g T Z. jDut ðt, xÞj pðxÞ1 jDuðt, xÞjdx rp k jDut ðt, Þj pðÞ1 kp0 ðÞ k jDuðt, Þj kpðÞ : .

(49) 740. G. Autuori and P. Pucci. By Lemma 2.1 of [32] and (2.2) ( k jDut j. pðÞ1. kp0 ðÞ . p 1 kDut kpðÞ pðÞ ðDut Þð p 1Þ=pþ ,. if kDut kpðÞ 1,. pþ 1 kDut kpðÞ. if kDut kpðÞ 4 1:. ð pþ 1Þ=p. pðÞ ðDut Þ. ,. Let D1 ¼ { 2 [T, t] : kDut(, )kp() 1} and D1 ¼ [T, t] n D1. Hence, by Ho¨lder’s inequality, Z. 0. kðÞ gðÞMðIuðÞÞ½pðÞ ðDut ð, ÞÞ1=p1 d. Downloaded by [University of Perugia], [P. Pucci] at 06:48 09 September 2011. D1. Z. 1=p01 Z D1. . gðÞMðIuðÞÞ½kðÞ p1 d. gðÞMðIuðÞÞpðÞ ðDut ð, ÞÞd. . 1=p1. D1. Z. 1 M1=p 1. 1=p01 Z. 1. 1=p1. t p1. gðÞMðIuðÞÞpðÞ ðDut ð, ÞÞ d. gðÞ½kðÞ d. T. ,. T. where p141 is given in (4.1); similarly Z. 0. kðÞ gðÞMðIuðÞÞ½pðÞ ðDut ð, ÞÞ1=p2 d D1. Z. 1=p02 Z. p2. gðÞMðIuðÞÞ½kðÞ d. gðÞMðIuðÞÞpðÞ ðDut ð, ÞÞd. D1. 2 M1=p 1. 1=p2. D1. Z. 1=p02 Z. 1. t. gðÞMðIuðÞÞpðÞ ðDut ð, ÞÞ d T. gðÞ½kðÞ p2 d. 1=p2 ,. T. being p2 ¼ p/(1 þ p pþ)41 by the main assumption pþ p51 of this section. Moreover, ( pþ 1)/p ( p 1)/pþ. Combining all these facts and the choice of T, g we get exactly (4.1). Finally, "3(T ) ! 0 as T ! 1 by Lemma 4.1. Remark When g 0, that is problem (P 2) coincides with (P 1), the assumption pþ p51 need no longer to be required, being C(k) 0, that is Lemma 4.2 is no more needed. Clearly, the assumption pþ p51 is automatic, whenever p(x) p. The following theorem is the principal result on the global stability of solutions for the strong damped system (P 2). THEOREM 4.3 Assume also (Q)-(b). If there exists a function k, satisfying either (K1) or (K2), and such that Z t. lim inf AðkðtÞÞ þ CðkðtÞÞ k d 5 1, t!1. ð4:2Þ. 0. where t ° A(k(t)) is given in (3.21) and t ° C(k(t)) in (4.1), then along any solution u of (P 2) property (1.2) holds. Proof As in the proof of Theorem 3.3, we first prove that limt!1 Eu(t) ¼ 0 by contradiction, assuming l40 in (3.8)..