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(1)This article was downloaded by: [University of Perugia] On: 19 March 2012, At: 09:59 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: An International Journal Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gcov20. Blow up at infinity of solutions of polyharmonic Kirchhoff systems a. G. Autuori , F. Colasuonno. a b. & P. Pucci. a. a. Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, Perugia 06123, Italy b. Dipartimento di Matematica, Università degli Studi di Bari ‘A. Moro’, Via Orabona 4, Bari 70125, Italy Available online: 02 Aug 2011. To cite this article: G. Autuori, F. Colasuonno & P. Pucci (2012): Blow up at infinity of solutions of polyharmonic Kirchhoff systems, Complex Variables and Elliptic Equations: An International Journal, 57:2-4, 379-395 To link to this article: http://dx.doi.org/10.1080/17476933.2011.592584. 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, redistribution, reselling, 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. 57, Nos. 2–4, February–April 2012, 379–395. Blow up at infinity of solutions of polyharmonic Kirchhoff systems G. Autuoria, F. Colasuonnoab and P. Puccia* a. Dipartimento di Matematica e Informatica, Universita` degli Studi di Perugia, Via Vanvitelli 1, Perugia 06123, Italy; bDipartimento di Matematica, Universita` degli Studi di Bari ‘A. Moro’, Via Orabona 4, Bari 70125, Italy. Downloaded by [University of Perugia] at 09:59 19 March 2012. Communicated by A. Pankov (Received 8 April 2011; final version received 26 May 2011) This article concerns the blow up at infinity of global solutions of strongly damped polyharmonic Kirchhoff systems, involving lower order terms, a time dependent nonlinear dissipative function Q and a driving force f, under homogeneous Dirichlet boundary conditions. Some applications are presented in special subcases of f and Q. Keywords: blow up at infinity; polyharmonic Kirchhoff systems; strong damping terms; source forces AMS Subject Classifications: Primary: Secondary: 35G60; 35Q99. 35L75;. 35L35;. 35B99;. 1. Introduction In this article we study the blow up at infinity of global solutions of strongly damped polyharmonic Kirchhoff systems in Rþ 0 of the type 8 L 2 utt þ M kDL uðt, Þk2 ðDÞ u þ N kDL1 uðt, Þk22 ðDÞL1 u þ u > > > < þ %ðtÞK kDL uðt, Þk22 ðDÞL ut þ Qðt, x, u, ut Þ ¼ f ðt, x, uÞ, ðPÞ > > > D uðt, xÞ : ¼ 0 for each multi-index , with jj L 1, Rþ 0 @. where the function u ¼ (u1, . . . , ud) ¼ u(t, x) is the vectorial displacement, d 1, L 1, n Rþ 0 ¼ ½0, 1Þ, is a bounded domain of R and is a non–negative parameter. The operator DL is defined for any fixed L ¼ 0, 1, 2, . . . by if L ¼ 2j, Dj u, j ¼ 0, 1, 2, . . . : ð1:1Þ DL u ¼ DDj u, if L ¼ 2j þ 1, The main Kirchhoff function M has the form MðÞ ¼ a þ b 1 ,. a, b 0,. a þ b 4 0,. *Corresponding author. Email: pucci@dmi.unipg.it ISSN 1747–6933 print/ISSN 1747–6941 online ß 2012 Taylor & Francis http://dx.doi.org/10.1080/17476933.2011.592584 http://www.tandfonline.com. 4 1 if b 4 0,. ð1:2Þ.

(3) 380. G. Autuori et al.. where we take ¼ 1 if b ¼ 0, so that M() M() for all 2 Rþ 0 , with MðÞ ¼ R MðzÞdz; while N satisfies 0 (N ) N 2 L1loc ðRþ 0 Þ is non–negative and there exists 2 [1, ] such that Z NðÞ NðÞ, 2 Rþ , where NðÞ ¼ NðzÞdz: 0 0. The Kirchhoff function. K 2 L1loc ðRþ 0Þ. Kð0Þ 0. is such that. and KðÞ 4 0. for all 4 0,. ð1:3Þ. Downloaded by [University of Perugia] at 09:59 19 March 2012. and is related to M by the assumption ðKÞ for every 4 0 there is c 4 0 such that MðÞ c KðÞ for all , R where KðÞ ¼ 0 KðzÞdz. In particular, relation (K) is verified when K 1, while the trivial case K 0 has been treated in the main Sections 3 and 6 of [1], in which we gave some explicit a priori estimates for the lifespan of local solutions, that is no global solutions exist. Conversely, in this article we could expect the solutions to be global in time and we analyse their behaviour at infinity. Similar results for related problems under dynamical boundary conditions can be found, e.g. in [2–4]. The nonlinear external damping Q represents the most common suppressions of the vibrations of an elastic structure of passive viscous type and absorbs vibration d d d energy. The function Q 2 CðRþ 0 R R ! R Þ satisfies ðQðt, x, u, vÞ, vÞ 0. for all t, x, u, v:. ð1:4Þ. L. The term %(t)(D) ut represents the internal material damping of Kelvin–Voigt type, which strictly depends on the body structure. Actually, an internal dissipative mechanism is always present, even if small, in real materials as long as the system vibrates, see, e.g. [5]. Usually, in standard literature, % is a positive constant, see [6–9], while for (P) we assume only the mild request % 2 C1 ðRþ 0 Þ and. %, %0 0:. ð1:5Þ. Clearly condition (1.5) allows % to be identically zero. This case was already treated in the main Sections 3 and 6 of [1], so that in this article we consider only the more difficult case in which %(t)40 for t sufficiently large. Thanks to the presence of this additional damping, we could expect that global solutions of (P) exist. d d The function f 2 CðRþ 0 R ! R Þ is an external source force derivable from a potential F, that is f ðt, x, uÞ ¼ Fu ðt, x, uÞ,. Fðt, x, 0Þ ¼ 0:. ð1:6Þ. The first result for (P) is Theorem 3.2, in which the blow up at infinity of global solutions of the system is proved, when Eu(0) is bounded from above by a critical value E1, see Figure 1. In Theorem 3.2 and its consequences, the L1–norm of % could also be infinity, and anyhow the results here obtained are new even when k%k151. Several applications of Theorem 3.2 are given, especially in Theorem 4.3, when ku(0, )kq40 and Eu(0) E0, see (4.10)–(4.11) for the exact meaning of 0 and E0, where kkq is the Lq–norm and q is a parameter related to the growth of f in u. In particular, in the limit case Eu(0) ¼ E0, we cover also the case Q 0 not allowed e.g. in [8] when L ¼ 1. This is possible either when f significantly depends on t or.

(4) Downloaded by [University of Perugia] at 09:59 19 March 2012. Complex Variables and Elliptic Equations. 381. Figure 1. The phase plane.. assuming %(0)40 and K satisfying an additional growth condition, so that, in the latter case, the presence of an internal dissipation balances the absence of an external damping. The main reasons to consider model (P) lie on the fact that in the recent article [1] the non-continuation of maximal solutions of (P) has been considered, when %ðtÞK kDL uðt, Þk22 0 in Rþ 0 . In this case, under suitable assumptions on f and Q, Theorem 3.1 of [1] shows that no global solutions u of (P) exist, provided that the initial energy Eu(0) associated to the system is appropriately bounded above by E2, with E24E1, see Figure 1. This is possible thanks to a certain interaction between the external force and the damping term. Therefore, a fairly natural question was to 2 understand what happens if the additional damping %ðtÞK kDL uðt, Þk2 ðDÞL ut is included in the system. Indeed, a strong action of dissipative terms could make easier the existence of global solutions, since they play the role of stabilizing terms and their smoothing effect makes more difficult the blow up. In any case, the function % makes the analysis more delicate even when it helps in obtaining the stability of global solutions. For the stability problem of damped Kirchhoff systems, we refer to [10] for the general case and to [6] for special cases of (P), when L ¼ 1 and N 0. In Section 7 of [1] a reverse situation has also been considered, in a very special case in which (1.5) is replaced by the complementary condition % 2 C1 ðRþ 0 Þ,. %, %0 0 in Rþ 0,. ð1:7Þ. the external damping Q is of a special type, that is Q(t, x, u, v) ¼ d1(t)v, and K 1. More precisely, in [1, Theorems 7.1–7.3], under additional hypotheses on the initial data, we provide some a priori estimates for the lifespan of maximal solutions of (P), which imply the non-continuation of local solutions. This means that, in the case of linear external damping, Theorem 3.2 makes sense when %0 40. For a detailed discussion see the Remarks after the proofs of Theorems 3.2 and 4.3..

(5) 382. G. Autuori et al.. Downloaded by [University of Perugia] at 09:59 19 March 2012. In [11] D0 Ancona and Shibata study the global existence of analytic solutions of problems describing non–linear viscoelastic materials with short memory, which are a special case of (P) when L ¼ 1, N 0, % 1, Q 0 and f 0. We refer to [12] for a wide list of references. On the other hand, in [7], Ono proved the existence of a global solution u of a subcase of (P), with Q 0, % 1 and K 1, assuming Eu(0) limited above and ku(0, )kq small enough, obtaining ku(t, )kq ! 0 as t ! 1, see also [13,14]. This article is organized as follows. In Section 2 we give a series of notations and preliminary results, while Section 3 is dedicated to the main Theorem 3.2, concerning the blow up at infinity of solutions of (P). Finally, in Section 4 we present some applications of Theorem 3.2 in special subcases of f and Q. Throughout this article we assume (1.2)–(1.6) and the structure conditions described on M, N and K, without further mentioning.. 2. Preliminaries In this section we collect a series of notations and preliminaries used throughout this article. Since we are in the vectorial setting, we consider maps assuming values in Rd, endowed with the P Euclidean norm jjd. With ¼ ð1 , . . . , , n Þ 2 Nn0 we denote a multi–index, jj ¼ ni¼1 i its length and for each vector valued map : ! Rd its -order derivative is @jj 1 @jj d D ¼ , . . . , 1 : @x1 1 . . . @xn n @x1 . . . @xn n The vectorial space [L2()]d is endowed with the norm k k½L2 ðÞd ¼ k j jd k2 , where kk2 denotes the standard norm of the Lebesgue scalar space L2(). Similarly, for any L ¼ 1, 2, . . . the vectorialP space [HL()]d is the classical Sobolev space, endowed with the norm k k½HL ðÞd ¼ ð jjL k jD jd k22 Þ1=2 . The space ½HL0 ðÞd is the completion L,2 d ðÞd be the completion of of ½C1 0 ðÞ , with respect to the norm k k½HL ðÞd . Let ½D d 1 ½C0 ðÞ under the norm !1=2 X 2 k k½DL,2 ðÞd ¼ k jD jd k2 : jj¼L. Since is a bounded domain, there exists a positive constant C ¼ CðL, Þ such that for all 2 ½HL0 ðÞd k k2½HL ðÞd ¼. d X i¼1. k i k2HL ðÞ C. d X X. kD i k22 ¼ Ck k2½DL,2 ðÞd ,. i¼1 jj¼L. by Poincare´’s inequality, see, for example, Corollary 6.31 of [15], with m ¼ L, p ¼ 2. Hence the norms k k½HL ðÞd and k k½DL,2 ðÞd are equivalent, and in particular ½HL0 ðÞd ¼ ½DL,2 ðÞd . By using (1.1) and integrating by parts, we get k k½DL,2 ðÞd ¼ k jDL js k2 for all d 2 ½C1 0 ðÞ , where s ¼ d if L is even, while s ¼ nd if L is odd. Finally, by a density argument, it is easy to see that the equality also holds for all 2 ½HL0 ðÞd . In the.

(6) Complex Variables and Elliptic Equations. 383. sequel we endow the space ½HL0 ðÞd with the inner product Z ð’, ÞL ¼ ðDL ’, DL Þs dx for all ’, 2 ½HL0 ðÞd , . Downloaded by [University of Perugia] at 09:59 19 March 2012. where the symbol (, )s denotes the Euclidean inner product of Rd if L is even or Rnd if L is odd. This inner product clearly generates the norm kjDL jsk2 adopted throughout this article for the space ½HL0 ðÞd . From R now on, we delete the subscript s in (, )s. The elementary bracket pairing h’, i ¼ (’(x), (x))dx is well defined for all ’, such that (’, ) 2 L1(). For simplicity in notation we drop the exponents d and nd in all the functional spaces involved in the treatment, thus HL0 ðÞ denotes ½HL0 ðÞd , and L2() ¼ (L2(), kk2) is used in all the dimensions 1, d and nd. The main solution and test function space is L X ¼ C1 ðRþ 0 ! H0 ðÞÞ:. For any 2 X the functional A includes the elliptic part of the system, and is given by ðAÞ. A ðtÞ ¼. 1 MðkDL ðt, Þk22 Þ þ NðkDL1 ðt, Þk22 Þ þ k ðt, Þk22 , 2. so that A is the Fre´chet potential, with respect to , of the operator A , defined pointwise for all ðt, xÞ 2 Rþ 0 by A ðt, xÞ ¼ MðkDL ðt, Þk22 ÞðDÞL ðt, xÞ þ NðkDL1 ðt, Þk22 ÞðDÞL1 ðt, xÞ þ ðt, xÞ: Clearly, A 0 in Rþ 0 , M and N being non-negative and 0. Moreover, for all u, we have 2 X and t 2 Rþ 0 hAuðt, Þ, ðt, Þi ¼ MðkDL uðt, Þk22 ÞhDL uðt, Þ, DL ðt, Þi þ huðt, Þ, ðt, Þi þ NðkDL1 uðt, Þk22 ÞhDL1 uðt, Þ, DL1 ðt, Þi: In particular, hA ðt, Þ, ðt, Þi ¼ MðkDL ðt, Þk22 ÞkDL ðt, Þk22 þ k ðt, Þk22 þ NðkDL1 ðt, Þk22 ÞkDL1 ðt, Þk22 , so that hA ðt, Þ, ðt, Þi 2A ðtÞ for all ðt, Þ 2 Rþ 0 X. ð2:1Þ. by (1.2) and (N ). Concerning the external force f we require the following structural assumptions ðF1 Þ. Fðt, , ðt, ÞÞ, ð f ðt, , ðt, ÞÞ, ðt, ÞÞ 2 L1 ðÞ for all t 2 Rþ 0 , 2 X; hf ðt, , ðt, ÞÞ, ðt, Þi 2 L1loc ðRþ 0 Þ for all 2 X:.

(7) 384. G. Autuori et al.. The potential energy of the field 2 X is given by Z F ðtÞ ¼ F ðt, Þ ¼ Fðt, x, ðt, xÞÞdx, . and it is well defined by (F1). Denoting by Ft the partial derivative with respect to t L of F ¼ F(t, w) for ðt, wÞ 2 Rþ 0 H0 ðÞ, we assume the following monotonicity condition L Ft 0 in Rþ 0 H0 ðÞ:. ðF2 Þ. The natural total energy of the field 2 X, associated to (P) is 1 E ðtÞ ¼ k t ðt, Þk22 þ A ðtÞ F ðtÞ, 2. Downloaded by [University of Perugia] at 09:59 19 March 2012. ðEÞ. and it is well defined in X by (F1). We say that a solution of (P) is a function u 2 X satisfying the following properties (A) and (B): (A) Distribution Identity Z tn. t hut ð, Þ, ð, Þi 0 ¼ hut , t i hAuð, Þ, ð, Þi %ðÞKðkDL uð, Þk22 Þ 0. o hDL ut ð, Þ, DL ð, Þi hQð, , u, ut Þ f ð, , uÞ, ð, Þi d for all t 2 Rþ 0 and 2 X; (B) Energy Conservation DuðtÞ ¼ hQðt, , uðt, Þ, ut ðt, ÞÞ, ut ðt, Þi þ %ðtÞKðkDL uð, Þk22 ÞkDL ut ðt, Þk22 þ Ft uðtÞ ðiÞ ðiiÞ. DuðtÞ 2 L1loc ðRþ 0 Þ, Zt DuðÞd for all t 2 Rþ EuðtÞ Euð0Þ 0: 0. The Distribution Identity is meaningful provided that hf ðt, , uÞ, i 2 L1loc ðRþ 0 Þ and hQðt, , u, ut Þ, i 2 L1loc ðRþ Þ, along the field. 2 X. The first condition is valid 0 whenever (F1) is in charge, while the latter is assumed. These restrictions are satisfied in the special cases treated in the theorems, as well as in the applications. The other terms in the Distribution Identity are well defined thanks to the choice of the space X. The importance in considering solutions verifying the weak conservation law (B)–(ii) was first given in [16, Remark 4, p. 199].. 3. The main theorem In this section we present our first blow up result for (P). Assume (F3) There exists a number q such that q 4 2. if 1 n 2L. and 2 5 q 2 L ¼. 2n n 2L. if n 4 2L. ð3:1Þ.

(8) Complex Variables and Elliptic Equations. 385. and for all F 4 0 and 2 X for which inft 2 Rþ0 F ðtÞ F, there exist c1 ¼ c1 ðF, Þ 4 0 and "0 ¼ "0 ðF, Þ 4 0 such that ðiÞ F ðtÞ c1 k ðt, Þkqq. for all t 2 Rþ 0,. and for all " 2 (0, "0] there exists c2 ¼ c2 ðF, , "Þ 4 0 such that ðiiÞ. hf ðt, , ðt, ÞÞ, ðt, Þi ðq "ÞF ðtÞ c2 k ðt, Þkqq for all t 2 Rþ 0: From now on, given a solution u 2 X of (P), we put for convenience 2 w1 ¼ infþ AuðtÞ, w2 ¼ infþ FuðtÞ, E1 ¼ 1 w1 , q t 2 R0 t 2 R0. ð3:2Þ. Downloaded by [University of Perugia] at 09:59 19 March 2012. and throughout the section we assume (F1)–(F3). LEMMA 3.1 E140.. If u 2 X is a solution of (P) with Eu(0)5E1, then w140, w240 and. Proof Let u 2 X be a solution of (P) in Rþ 0 , as in the statement. Clearly, w1 0 by (A). Furthermore, by (E), (B)–(ii) and the fact that Du is non-negative, Fu(t) w1 Eu(0) for all t 2 Rþ 0 , so that w2 w1 Eu(0)42w1/q 0, being Eu(0)5E1. Hence w240. By (F3), in correspondence to F ¼ w2 4 0, ¼ u 2 X, there exists "0 ¼ "0(w2, u)40 and c1 ¼ c1(w2, u)40 such that for all t 2 Rþ 0 kuðt, Þkq c~1 4 0. and. kDL uðt, Þk2 c~1 =S q ,. ð3:3Þ. 1=q. where c~1 ¼ ðw2 =c1 Þ 4 0 and S q ¼ S q(d, L, ) is the constant of the Sobolev embedding HL0 ðÞ ,! Lq ðÞ. Moreover, by (1.2) we have.

(9) 2AuðtÞ a þ bkDL uðt, Þk22ð1Þ kDL uðt, Þk22 a1 kDL uðt, Þk22 , ð3:4Þ where a1 ¼ a þ bðc~1 =S q Þ2ð1Þ 4 0. In particular, w1 . a1 inf kDL uðt, Þk22 4 0, 2 t 2 Rþ0 g. as claimed, and in turn E140 by (3.1).. Furthermore, we assume that Q verifies the following condition (Q) Along any solution u 2 X of (P), there exist t 0, q140, m41, 0, +41 with m þ +5q, non-negative functions 1, 2 2 L1 loc ðJÞ, and positive functions 0 , k 2 W1,1 loc ðJÞ, J ¼ ½t, 1Þ, with k 0, such that for all t 2 J 0 1=m0 þ 2 ðtÞ1=} DuðtÞ1=} kuðt, Þkq , QðtÞ q1 1 ðtÞ1=m kuðt, Þk =m q DuðtÞ 1=ðm1Þ þ 1=ð}1Þ k= , 1 2 oð ðtÞÞ, if k%k1 5 1; 0 ðtÞ ¼ oð ðtÞ=%ðtÞÞ, if k%k1 ¼ 1;. ð3:5Þ as t ! 1,. where Q(t) ¼ hQ(t, , u(t, ), ut(t, )), u(t, )i. Without loss of generality, taking t even larger if necessary, we assume that %ðtÞ 4 0..

(10) 386. G. Autuori et al.. We refer to Section 4 for specific examples of functions Q verifying condition (Q). For simplicity of notation let us introduce the negative numbers. q q ð3:6Þ 1 ¼ 1 þ , 2 ¼ 1 : m m } Actually, 1 250, since (+ m)q q +, being 0, and +5q in (Q). THEOREM 3.2. Let u 2 X be a solution of (P) such that. Downloaded by [University of Perugia] at 09:59 19 March 2012. Euð0Þ 5 E1 : If condition (Q) holds along u and for all c040 Rt exp c0 t ðÞ½maxfkðÞ, ðÞg1 d ¼1 lim t!1 maxfkðtÞ, ðtÞg Rt exp c0 t ðÞ½maxfkðÞ, %ðÞ ðÞg1 d ¼1 lim t!1 maxfkðtÞ, %ðtÞ ðtÞg. ð3:7Þ. if k%k1 5 1, ð3:8Þ if k%k1 ¼ 1,. then lim kuðt, Þkq ¼ 1:. t!1. ð3:9Þ. Proof Let u 2 X be a solution of (P) satisfying (3.7). Take ¼ u in the Distribution Identity (A), thus o dn hut ðt, Þ, uðt, Þi þ %ðtÞKðkDL uðt, Þk22 Þ=2 dt ¼ kut ðt, Þk22 hAuðt, Þ, uðt, Þi þ hf ðt, , uÞ, uðt, Þi %0 ðtÞ KðkDL uðt, Þk22 Þ QðtÞ, þ 2. ð3:10Þ. where Q(t) ¼ hQ(t, , u(t, ), ut(t, )), u(t, )i as defined in (Q). Since u satisfies (3.7), then w140, w240 and E140 by Lemma 3.1. Hence, in correspondence to F ¼ w2 and ¼ u, there exists "0 ¼ "0(w2, u)40 such that inequalities (i) and (ii) of (F3) hold true. Without loss of generality, we take "040 so small that "0 w1 ðq 2Þw1 q½Euð0Þþ ¼ qfE1 ½Euð0Þþ g,. ð3:11Þ. which is possible by (3.7). We remark that (3.11) forces "0 q 2. Fix " 2 (0, "0). By (2.1) and (F3)–(ii) hAuðt, Þ, uðt, Þi hf ðt, , uðt, ÞÞ, uðt, Þi 2AuðtÞ ðq "ÞFuðtÞ c2 kuðt, Þkqq : Since q 24", then c3 ¼ "(q " 2)/2q40. Moreover, Fu(t) Au(t) Eu(t) by (E), so that 2AuðtÞ ðq "ÞFuðtÞ ðq "ÞEuðtÞ ðq " 2ÞAuðtÞ q" ðq "ÞEuð0Þ ðq " 2Þ 1 AuðtÞ q q" w1 ðq " 2Þ q.

(11) 387. Complex Variables and Elliptic Equations ðq "Þf½Euð0Þþ E1 þ "w1 =qg 2c3 AuðtÞ 5 c3 MðkDL uðt, Þk22 Þ,. by (3.7), (3.11) and (A). Combining the last inequalities together with (3.10) we get o dn hut ðt, Þ, uðt, Þi þ %ðtÞKðkDL uðt, Þk22 Þ=2 dt ð3:12Þ kut ðt, Þk22 þ c2 kuðt, Þkqq þ c3 MðkDL uðt, Þk22 Þ QðtÞ,. Downloaded by [University of Perugia] at 09:59 19 March 2012. where we have also used the non-negativity of %0 and K. Let us estimate the main dynamic quantity Q(t) using (Q). By (3.5) n 1=m0 QðtÞ q1 1 ðtÞ1=ðm1Þ DuðtÞ kuðt, Þk1þ =m q o 1=}0 þ 2 ðtÞ1=ð}1Þ DuðtÞ kuðt, Þkq ( ¼ q1. þ. h. h. 1 ðtÞ1=ðm1Þ DuðtÞ. 2 ðtÞ. 1=ð}1Þ. DuðtÞ. 1=m0. 1=}0. i kuðt, Þkq=m kuðt, Þkq 1 q. kuðt, Þkq=} q. i kuðt, Þkq 2. ). ( " # 1=ðm1Þ 2 ‘ q 1 ðtÞ q1 DuðtÞ þ kuðt, Þkq kuðt, Þkq 1 ‘ 2 " # ) 1=ð}1Þ 2 ‘ q 2 2 ðtÞ þ DuðtÞ þ kuðt, Þkq kuðt, Þkq , ‘ 2 for all t 2 J, where in the last step we have applied Young’s inequality, with ‘40 to be fixed later. Consequently, being 1 250, it follows that n o. QðtÞ q2 ‘~ 1 ðtÞ1=ðm1Þ þ 2 ðtÞ1=ð}1Þ DuðtÞ þ ‘ kuðt, Þkqq kuðt, Þkq 2 n o ~ ðtÞ þ ‘ kuðt, Þkqq , q3 ‘kðtÞDuðtÞ= ð3:13Þ 0 0 where q2 ¼ q1 maxf1, ðc~1 Þ1 2 g, ‘~ ¼ maxfð2=‘ Þm =m , ð2=‘ Þ} =} g, q3 ¼ q2 c~1 2 4 0 and in the last step we have used (3.5)2. Introduce the function Zt HðtÞ ¼ H0 þ DuðÞd ð3:14Þ. 0. t 2 Rþ 0,. where H0 is any number in the interval (0,E1 Eu(0)]. Of course, H is for all well defined by (B)–(i) and non-decreasing, being Du ¼ H0 0. Define for all t 2 Rþ 0 the main auxiliary function o n ð3:15Þ ZðtÞ ¼

(12) kðtÞHðtÞ þ ðtÞ hut , ui þ %ðtÞKðkDL uðt, Þk22 Þ=2 ,.

(13) 388. G. Autuori et al.. where

(14) 40 is a constant to be fixed later. Clearly, Z 2 W1,1 loc ðJÞ by Corollary 8.10 of [17] and a.e. in J, o n Z0 ¼

(15) k0 H þ

(16) kH0 þ 0 hut , ui þ %ðtÞKðkDL uk22 Þ=2 o dn þ ð3:16Þ hut , ui þ %ðtÞKðkDL uk22 Þ=2 : dt Hence, putting (3.12) and (3.13) into (3.16), and reminding that Du ¼ H0 , we get o n Z0

(17) k0 H þ

(18) kH0 þ 0 hut , ui þ %ðtÞKðkDL uk22 Þ=2. Downloaded by [University of Perugia] at 09:59 19 March 2012. þ. n o kut ðt, Þk22 þ c2 kuðt, Þkqq þ c3 MðkDL uðt, Þk22 Þ QðtÞ. k

(19) q3 ‘~ H0 þ þ. 0. o n hut , ui þ %ðtÞKðkDL uk22 Þ=2. n o kut k22 þ ðc2 q3 ‘ Þkukqq þ c3 MðkDL uk22 Þ ,. where in the last step we have also used the fact that

(20) k0 H 0. Next, by the Cauchy and Young inequalities, and the definition of X, we have that The embedding jhuðt, Þ, ut ðt, Þij kut ðt, Þk2 kuðt, Þk2 kut ðt, Þk22 þ kuðt, Þk22 . Lq() ,! L2() is continuous by (3.1), so that ku(t, )k2 jj(q 2)/2qku(t, )kq, and by (3.3), this gives kuðt, Þk22 jj12=q kuðt, Þk2q c4 kuðt, Þkqq , q2 , being q42 2. Hence where c4 ¼ jj1=q =c~1. ð3:17Þ. jhuðt, Þ, ut ðt, Þij kut ðt, Þk22 þ c4 kuðt, Þkqq :. ð3:18Þ. 0. 0. j j, and so a.e. in J, by (K) n Z0 k

(21) q3 ‘~ H0 þ ð1 j 0 j= Þkut k22 þ ðc2 q3 ‘ c4 j 0 j= Þkukqq o þ ðc3 c %ðtÞj 0 j=2 ÞKðkDL uk22 Þ :. Clearly . Fix ‘40 so small that 2q3‘5c2 and consider a time T1 in the interval J large enough to have 2j 0 j/ min{1, (c2 q3‘)/c4} and %j 0 j/ c3c in J1 ¼ [T1, 1), since 0 (t) ¼ o( (t)) and %(t) 0 (t) ¼ o( (t)) as t ! 1, by (3.5)3, both when k%k151 and k%k1 ¼ 1. Then take

(22) 40 so large that

(23) q3 ‘~ and Z(T1)40. In conclusion, we have shown that . ð3:19Þ Z0 ðtÞ B0 ðtÞ kut ðt, Þk22 þ kuðt, Þkqq þ KðkDL uðt, Þk22 Þ 0, for a.a. t 2 J1, where 2B0 ¼ min{1, c2 q3‘, c3c }. Hence, Z(t) Z(T1)40 for all t 2 J1, being Z(T1)40. Now observe that, by (B)–(ii), (E), (F3)–(i) and the choice of H0, for all t 2 Rþ 0 we have HðtÞ H0 þ Euð0Þ EuðtÞ H0 þ Euð0Þ AuðtÞ þ FuðtÞ 5 FuðtÞ c1 kuðt, Þkqq ,. ð3:20Þ.

(24) 389. Complex Variables and Elliptic Equations. since Au(t) w14E1 by (3.1). Consequently, by (3.15), (3.18), (3.20) and (1.5) it follows that for all t 2 J1 n o ZðtÞ

(25) c1 kðtÞkuðt, Þkqq þ ðtÞ kut ðt, Þk22 þ c4 kuðt, Þkqq þ %ðtÞKðkDL uðt, Þk22 Þ=2 .

(26) c1 kðtÞkuðt, Þkqq. . q kut ðt, Þk22 c4 kuðt, Þkq KðkDL uðt, Þk22 Þ þ þ þ %ðtÞ ðtÞ , 2 %ðT1 Þ %ðT1 Þ. so that, putting B ¼ max{1/%(T1),

(27) c1 þ c4/%(T1), 1/2}, we get n o ZðtÞ B maxfkðtÞ, %ðtÞ ðtÞg kut ðt, Þk22 þ kuðt, Þkqq þ KðkDL uðt, Þk22 Þ :. ð3:21Þ. Therefore, combining (3.19) with (3.21), we immediately obtain. Downloaded by [University of Perugia] at 09:59 19 March 2012. Z1 Z0 c0 ½maxfk, % g1 ,. ð3:22Þ. 0. where c0 ¼ B /B. On the other hand, by (B)–(ii), (E), (3.1) and the definition of w1, for all t 2 Rþ 0 H0 HðtÞ H0 þ Euð0Þ EuðtÞ E1 . 1 kut ðt, Þk22 þ MðkDL uðt, Þk22 Þ þ FuðtÞ: 2. In particular, kut ðt, Þk22 þ MðkDL uðt, Þk22 Þ 2½E1 H0 þ FuðtÞ 2½E1 þ FuðtÞ, and in turn, using (F3)–(i), we get kut ðt, Þk22 þ KðkDL uðt, Þk22 Þ C ½E1 þ FuðtÞ CK kuðt, Þkqq ,. ð3:23Þ. where C ¼ 2 maxf1, c1 g, c is the constant given in (K) corresponding to ¼ ðc~1 =S q Þ2 determined in (3.3), and finally CK ¼ C maxfE1 =c~1 , c1 g. Hence, putting (3.23) into (3.21), we get for all t 2 J1 ZðtÞ C maxfkðtÞ, %ðtÞ ðtÞgkuðt, Þkqq. for all t 2 J1 ,. ð3:24Þ. where C ¼ B(1 þ CK). Case k%k151. Relations (3.22) and (3.24) simplify. In particular, t 2 J1 Zt ZðtÞ ZðT1 Þ exp c% ðÞ½maxfkðÞ, ðÞg1 d , ð3:25Þ T1. where c% ¼ c0/max{1, k%k1}. Therefore, by (3.24), for all t T1, R

(28) t exp c% T1 ðÞ½maxfkðÞ, ðÞg1 d , kuðt, Þkqq Z0 maxfkðtÞ, ðtÞg where Z0 ¼ Z(T1)/C%40 and C% ¼ C max{1, k%k1}. Thus for all t 2 J1 J R

(29) t exp c% T1 ðÞ½maxfkðÞ, ðÞg1 d maxfkðtÞ, ðtÞg Rt exp c% t ðÞ½maxfkðÞ, ðÞg1 d ¼ E0 , maxfkðtÞ, ðtÞg. ð3:26Þ.

(30) 390. G. Autuori et al.. RT where E0 ¼ expðc% t 1 ðÞ½maxfkðÞ, ðÞg1 dÞ 4 0. Therefore, passing to the limit as t ! 1 in (3.26), from (3.8), valid in particular for c0 ¼ c%40, we obtain (3.9), and the proof is complete when k%k151. Case k%k1 ¼ 1. Without loss of generality we can suppose that % 1 in J1, assuming T1 even larger if necessary, being limt!1%(t) ¼ k%k1 ¼ 1 by (1.5). Hence, by (3.22) Zt ðÞ½maxfkðÞ, %ðÞ ðÞg1 d : ð3:27Þ ZðtÞ ZðT1 Þ exp c0 T1. Downloaded by [University of Perugia] at 09:59 19 March 2012. Combining (3.27) with (3.24), we get for all t T1 R

(31) t exp c0 T1 ðÞ½maxfkðÞ, %ðÞ ðÞg1 d , kuðt, Þkqq Z0 maxfkðtÞ, %ðtÞ ðtÞg where Z0 ¼ Z(T1)/C40. Property (3.9) follows at once by (3.8)2. g Remark In Theorem 3.2 we have %ðtÞK kDL uðt, Þk22 4 0 for t t by (1.3), (1.5) and (3.3). We point out that when %ðtÞK kDL uðt, Þk22 0 in Rþ 0 , global solutions of (P) do not exist when Eu(0) is bounded above by the value E2 given in Figure 1, as shown in Theorem 3.1 of [1].. 4. Applications In this section we give some applications of Theorem 3.2, in special subcases of the external damping and the source force. We start by giving some prototypes for f and Q, satisfying (F1)–(F3), (Q) and (3.8). Let us consider a continuous function Q verifying inequality (1.4) as in Section 1. Moreover, assume that there exists t 1 such that Qðt, x, u, vÞ ¼ d1 ðt, xÞjuj jvjm2 v þ d2 ðt, x, uÞjvj}2 v m 4 1 0 } 4 1, with m þ } 5 q. ð4:1Þ. }1 for all (t, x, u, v) 2 J Rd Rd, where J ¼ ½t, 1Þ, d1 2 CðRþ 0 ! L ðÞÞ, þ 1 d2 2 CðR0 ! L ðÞÞ are non-negative and +1 ¼ q/(q m). Define 1 ðtÞ ¼ kd1 ðt, Þk}1 and 2 ðtÞ ¼ jjðq}Þ=q supðx, Þ 2 Rd d2 ðt, x, Þ. Without loss of generality, we take t so large that %ðtÞ 4 0. Under condition (4.1), along any solution u of (P), we have n o 0 1=m0 þ 2 ðtÞ1=} DuðtÞ1=} kuðt, Þkq ð4:2Þ jQðtÞj 1 ðtÞ1=m kuðt, Þk =m q DuðtÞ. for all (t, x) 2 J . If k%k151, assume that for each t t 1 ðtÞ1=ðm1Þ þ 2 ðtÞ1=ð}1Þ Kð1 þ tÞs=ðm1Þ ,. ð4:3Þ. with K 1 and 0 s m 1, while, if k%k1 ¼ 1, assume that for each t t 1 ðtÞ1=ðm1Þ þ 2 ðtÞ1=ð}1Þ Const: %ðtÞ,. ð4:4Þ.

(32) Complex Variables and Elliptic Equations. 391. Zt 1 d exp c0 ¼ 1: t!1 %ðtÞ t %ðÞ. ð4:5Þ. and that for all c040. Downloaded by [University of Perugia] at 09:59 19 March 2012. lim. When the nonlinear driving term f satisfies conditions (1.6), (F1) and (F2), and Q is of the type given in (4.1) and (4.3)–(4.5), then conditions (Q) and (3.8) hold, as it is shown below. Case k%k151. It is enough to take kðtÞ ¼ Kð1 þ tÞs=ðm1Þ , K 1 and (t) ¼ 1 for all t 2 J, if 0 s 5 m 1; while kðtÞ ¼ K and (t) ¼ (1 þ t)1, for all t 2 J, if 0 0 (t) ¼ o( (t)) as s ¼ m 1. In both the situations , k 2 W1,1 loc ðJÞ are positive, k 0, t ! 1, k in J, being K 1, (3.5)2 is verified in J and (3.8)1 holds. For a proof of these facts we refer to [1, Proposition 4.1]. Case k%k1 ¼ 1. Put k(t) ¼ Const. %(t) and (t) ¼ 1 for all t 2 J, so that k 0 0 (t) ¼ o( (t)/%(t)) definitively, being %0 40. Here , k 2 W1,1 loc ðJÞ are positive, k 0, as t ! 1, (3.5)2 is verified in J by (4.4) and (3.8)2 holds by (4.5), see [3, Corollary 5.3]. Following Example (3.2) of [18], we consider the force function. q 4 2. f ðt, x, uÞ ¼ gðt, xÞjujp2 u þ cðxÞjujq2 u, 2n if 1 n 2L and 2 5 q 2 L ¼ n 2L. 1 p q, ð4:6Þ if n 4 2L,. where c 2 L1() is a non-negative function such that c1 ¼ kck140; while þ g 2 CðRþ 0 Þ, differentiable with respect to t, with gt 2 CðR0 Þ, satisfies 1 0 gðt, xÞ, gt ðt, xÞ hðxÞ in Rþ 0 , for some h 2 L ðÞ, q=ðq pÞ, if p 5 q , gðt, Þ 2 L ðÞ in Rþ 0 , where ¼ 1, if p=q.. ð4:7Þ. When f is as in (4.6)–(4.7), then (1.6), (F1)–(F2) and (F3)–(i) hold, with c1 ¼ c1/q, see [1, Proposition 5.1]. Following the main lines of investigation given in [1], we now give some qualitative results based on the geometric features of (P), assuming from now on (4.1) and (4.3)–(4.7), and denoting by (t) ¼ ku(t, )kq, in correspondence to a solution u of (P). Finally, we put s ¼ b if b40 or s ¼ a if b ¼ 0. The following inequality holds EuðtÞ ’ððtÞÞ ¼. s c1 ðtÞq ðtÞ2 2 q ð2S q Þ. for all t 2 Rþ 0,. ð4:8Þ. where S q ¼ S q(d, L, ) is the constant of the Sobolev embedding HL0 ðÞ ,! Lq ðÞ, being AuðtÞ . s ðtÞ2 ð2S 2q Þ. and. F uðtÞ . c1 ðtÞq : q. ð4:9Þ. It is easy to see that ’ attains its maximum at 0 ¼ a01=ðq2Þ ,. where a0 ¼. 2s : c1 ð2S 2q Þ. ð4:10Þ.

(33) 392. G. Autuori et al.. Moreover, ’ is strictly decreasing for 0, with ’() ! 1 as ! 1. Put 2 ’ð0 Þ ¼ 1 w0 ¼ E0 4 0, q. where w0 ¼. s2 0 4 0, ð2S 2q Þ. ð4:11Þ. 0 ¼ fð, EÞ 2 R2 : 4 0 , E 5 E0 g: LEMMA 4.1 [1, Lemma 5.1] Let u 2 X be a solution of (P) such that Eu(0)5E0. Then 0 =2 ðRþ 0 Þ and w1 6¼ w0. Moreover, the following conditions are equivalent:. Downloaded by [University of Perugia] at 09:59 19 March 2012. (1) w14w0; (2) ðRþ 0 Þ ð0 , 1Þ. Finally, if either (i) or (ii) holds, then E05E1. In particular, if (0)40 and Eu(0)5E0, then ((t), Eu(t)) 2 0 for all t 2 Rþ 0, properties (i), (ii) hold and E05E1. Lemma 5.1 of [1] is given for t 2 [0, T), T40, but here the same result can be obtained without changes, with T ¼ 1. In order to handle the delicate limit case Eu(0) ¼ E0 we assume, when necessary, that Q is of the type given in (4.1) globally, that is for all (t, x, u, v) in d d Rþ 0 R R Qðt, x, u, vÞ ¼ d1 ðt, xÞjuj jvjm2 v þ d2 ðt, x, uÞjvj}2 v,. ð4:12Þ. where d1, d2 and all the exponents are as in (4.1), and define also 1 and 2 as in (4.1). Hence relation (4.2) holds for all ðt, xÞ 2 Rþ 0 . Moreover consider now the following hypothesis. (D) There exists t*40 such that one of the following properties hold: (i) 2 X and hQ(t, , , t), ti 0 in [0, t*] implies either (t, ) 0 or t(t, ) 0 in [0, t*] and Q verifies (4.12); þ (ii) there exists a positive function g0 : Rþ 0 ! R such that gt(t, x) g0(t) for each (t, x) 2 [0, t*] ; (iii) %(0)40 and K() þ 1, 0, , 0, þ 40, 41. The case (D)–(iii) is trivially verified when % and K are positive constants. LEMMA 4.2. Assume (D). Let u 2 X be a solution of (P) such that ð0Þ 4 0. and Euð0Þ ¼ E0 :. ð4:13Þ. Then ðRþ 0 Þ ð0 , 1Þ. Furthermore ((t), Eu(t)) 2 0 for all t40, that is ðtÞ 4 0. and. EuðtÞ 5 E0. for all t 4 0:. In particular w14w0 and E05E1. Proof Let u be as in the statement. First we show that (t) 6¼ 0 for all t 2 Rþ 0. Proceed by contradiction and suppose that there exists t0 2 Rþ 0 such that (t0) ¼ 0. Then, by (4.8) and the assumption Eu(0) ¼ E0, it follows that E0 ¼ Euð0Þ Euðt0 Þ ’ððt0 ÞÞ ¼ E0 :.

(34) Complex Variables and Elliptic Equations. 393. Hence E0 ¼ Eu(t0), so that, by (B)–(ii) and the fact that Du 0, we get Z t0 E0 ¼ Euðt0 Þ Euð0Þ DuðtÞdt Euð0Þ ¼ E0 : 0. R t0. Downloaded by [University of Perugia] at 09:59 19 March 2012. Therefore 0 DuðtÞdt ¼ 0 and in turn Du 0 in [0, t0]. Consequently, by the definition of Du, we have that hQ(t, , u(t, ), ut(t, )), ut(t, )i ¼ 0, Ftu(t) ¼ 0 and %ðtÞKðkDL uðt, Þk22 ÞkDL ut ðt, Þk22 ¼ 0 for all t 2 [0, s0], where s0 ¼ min{t*, t0} and t* is the number given in (D). Let us now distinguish three cases. Case (D)–(i). Since hQ(t, , u(t, ), ut(t, )), ut(t, )i ¼ 0 for all t 2 [0, s0], we get that either u(t, ) ¼ 0 or ut(t, ) ¼ 0 for all t 2 [0, s0]. The first event cannot occur since (0) ¼ ku(0, )kq4040 by assumption. In the latter, u is clearly constant with respect to t in [0, s0], and so u(t, x) ¼ u(0, x) for each t 2 [0, s0]. Taking (t, x) ¼ u(0, x) in the Distribution Identity (A), for each t 2 [0, s0] we have that Rt thAuð0, Þ, uð0, Þi ¼ 0 hf ð, , uð0, ÞÞ, uð0, Þid, being hQ(t, , u(0, ), 0), u(0, )i ¼ 0, since Du 0 in [0, s0] and (4.2) holds in Rþ 0 by (4.12). Thus hAu(0, ), u(0, )i ¼ h f (t, , u(0, )), u(0, )i for each t 2 [0, s0], and in particular hAuð0, Þ, uð0, Þi ¼ hf ð0, , uð0, ÞÞ, uð0, Þi: Now 2Au(0) qFu(0) by (2.1) and (F3)–(ii). On the other hand, E0 ¼ Eu(0) ¼ Au(0) Fu(0) by (E), since ut(0, ) ¼ 0. By (4.9) and (4.13) we have Au(0)4w040, and so 2 2 E0 1 Auð0Þ 4 1 w0 ¼ E0 , q q which is an obvious contradiction. Case (D)–(ii). We have Z 0 ¼ Ft uðtÞ ¼. gt ðt, xÞ . juðt, xÞjp g0 ðtÞ kuðt, Þkpp 0 dx p p. for each t 2 [0, s0]. Therefore, ku(t, )kp 0 and so u 0 in [0, s0] . But this occurrence is impossible, since (0) ¼ ku(0, )kq4040, so that we reach a contradiction. Case (D)–(iii). By (1.5) we have 0 ¼ %ðtÞKðkDL uðt, Þk22 ÞkDL ut ðt, Þk22 n o %ð0Þ þ kDL uðt, Þk2ð 1Þ kDL ut ðt, Þk22 0, 2. ð4:14Þ. gkDL ut ðt, Þk22 ¼ 0, being %(0)40. for all t 2 [0, s0], and so f þ kDL uðt, Þk2ð 1Þ 2 We distinguish two cases. If 40, then kDLut(t, )k2 ¼ 0 for all t 2 [0, s0]. In this L case, clearly kut(t, )k2 SkDL ut(t, )k2, being u 2 X ¼ C1 ðRþ 0 ! H0 ðÞÞ, where S is the Sobolev embedding constant. Thus kut(t, )k2 ¼ 0 and consequently ut(t, x) ¼ 0 for all t 2 [0, s0] and x 2 . Therefore u is constant with respect to t in [0, s0], but this case cannot occur as shown above in case (i)..

(35) 394. G. Autuori et al.. Downloaded by [University of Perugia] at 09:59 19 March 2012. On the other hand, if ¼ 0, then kDL uðt, Þk2ð 1Þ kDL ut ðt, Þk22 ¼ 0 for all t 2 [0, s0], 2 being 40. Now d kDL uðt, Þk2 2jhDL ut ðt, Þ, DL uðt, Þij 2kDL ut ðt, Þk2 kDL uðt, Þk2 , 2 dt hence kDLu(t, )k2 ¼ Const. 0. If kDLu(t, )k2 ¼ Const.40, then kDLut(t, )k2 ¼ 0 for all t 2 [0, s0] and we obtain the contradiction following the argument above. Otherwise, kDLu(t, )k2 0, that is u(t, ) ¼ 0 for all t 2 [0, s0]. Again this occurrence is impossible, being (0) ¼ ku(0, )kq40. Consequently, in all the three cases, (t)40 for all t 2 Rþ 0 by the continuity of t ° (t), being (0)40 by (4.13). From now on we can proceed as in Lemma 5.2 of [1], since theremaining part of the proof does not involve the intrinsic damping %ðtÞK kDL uðt, Þk22 ðDÞL ut , but only the same energy functional Eu. g If in addition f satisfies 1 p5q. and c ¼ ess inf cðxÞ 4 0,. ð4:15Þ. then also (F3)–(ii) is verified with "0 2 (0, q p] and c2 ¼ c"=q for all " 2 (0, "0]. In the next theorem we also assume (4.15), so that all the structural assumptions of Theorem 3.2 are fulfilled. THEOREM 4.3. Let u 2 X be a solution of (P) such that either. (i) (0)40 and Eu(0)5E0, or (ii) (0)40, Eu(0) ¼ E0 and (D) holds. Then limt!1ku(t, )kq ¼ 1. Proof Let u 2 X be a solution of (P) as in the statement. If (i) holds, then by Lemma 4.1 we have that Eu(0)5E05E1. On the other hand, if (ii) holds, then Eu(0)5E1 by Lemma 4.2. In both cases the claim follows directly applying Theorem 3.2. g Remark In Theorem 4.3 conditions %40 definitively and K not trivial are crucial. Indeed, if %ðtÞK kDL uðt, Þk22 0 in Rþ 0 , global solutions may not exist by Theorem 5.1 of [1]. þ 0 Consider now that Q(t, x, u, v) ¼ d1(t)v, where d1 2 C1 ðRþ 0 ! R0 Þ and d1 0. Clearly Q satisfies (4.1), with ¼ 0, m ¼ 2, d1(t, x) ¼ d1(t) and d2 0. Assume furthermore that % Const.40 and K 1. In this situation, Theorems 7.2 and 7.3 of [1] say that there are no solutions u of (P) defined in the whole space Rþ 0 , when, in our notation, Eu(0) E0, condition (D) holds in the limit case Eu(0) ¼ E0 and hu(0, ), ut(0, )i is bounded from below by an appropriate constant, depending only on the initial data of the problem. This means that Theorem 4.3 can be applied only when hu(0, ), ut(0, )i is sufficiently small. In [7] Ono considered a subcase of (P), with Q 0, % 1 and K 1. For this model, in Theorem 3 of [7], he proved the existence of a global solution u, assuming Eu(0) limited above and (0) ¼ ku(0, )kq small enough. Furthermore he showed that (t) ¼ ku(t, )kq ! 0 as t ! 1. Under the assumptions of Theorem 4.3 of this article, condition (0)4040 actually implies that (t)40 for all t 2 Rþ 0..

(36) Complex Variables and Elliptic Equations. 395. Acknowledgements This work was performed while Francesca Colasuonno was visiting the Universita` degli Studi di Perugia for a scientific collaboration with Professor Patrizia Pucci, supported by a grant from the Universita` degli Studi di Bari. This research was supported by the Project Metodi Variazionali ed Equazioni Differenziali alle Derivate Parziali Non Lineari.. Downloaded by [University of Perugia] at 09:59 19 March 2012. References [1] G. Autuori, F. Colasuonno and P. Pucci, Lifespan estimates for solutions of polyharmonic Kirchhoff systems, to appear in Math. Models Methods Appl. Sci. [2] G. Autuori and P. Pucci, Kirchhoff systems with dynamic boundary conditions, Nonlinear Anal. 73 (2010), pp. 1952–1965. [3] G. Autuori and P. Pucci, Kirchhoff systems with nonlinear source and boundary damping terms, Commun. Pure Appl. Anal. 9 (2010), pp. 1161–1188. [4] S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Adv. Diff. Eq. 13 (2008), pp. 1051–1074. [5] A.N. Gent, Engineering with Rubber: How to Design Rubber Components, 2nd ed., H. Gardner Publ., Ohio, 2001. [6] G.C. Gorain, Exponential energy decay estimates for the solutions of n-dimensional Kirchhoff type wave equation, Appl. Math. Comput. 177 (2006), pp. 235–242. [7] K. Ono, On Global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci. 20 (1997), pp. 151–177. [8] S.T. Wu and L.Y. Tsai, On a system of nonlinear wave equations of Kirchhoff type with a strong dissipation, Tamkang J. Math. 38 (2007), pp. 1–20. [9] Z. Yang and D. Qiu, Energy decaying and blow–up of solution for a Kirchhoff equation with strong damping, J. Math. Res. Exposition 29 (2009), pp. 707–715. [10] G. Autuori, Kirchhoff systems: Asymptotic stability, Global Nonexistence and blow up, Ph.D. Thesis, University of Florence, 2010. [11] P. D’Ancona and Y. Shibata, On global solvability of non–linear viscoelastic equations in the analytic category, Math. Meth. Appl. Sciences 17 (1994), pp. 477–486. [12] G. Autuori, P. Pucci and M.C. Salvatori, Global nonexistence for nonlinear Kirchhoff Systems, Arch. Rational Mech. Anal. 196 (2010), pp. 489–516. [13] K. Ono, On the blowup problem for nonlinear Kirchhoff equations with nonlinear dissipative terms, J. Math. Tokushima Univ. 34 (2000), pp. 1–8. [14] K. Ono, Global existence, asymptotic behaviour, and global non-existence of solutions for damped non-linear wave equations of Kirchhoff type in the whole space, Math. Methods Appl. Sci. 23 (2000), pp. 535–560. [15] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, 2nd ed., Vol. 140, Academic Press, Amsterdam, 2003. [16] P. Pucci and J. Serrin, Asymptotic stability for non–autonomous dissipative wave systems, Comm. Pure Appl. Math. 49 (1996), pp. 177–216. [17] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2010. [18] P. Pucci and J. Serrin, Local asymptotic stability for dissipative wave systems, Israel J. Math. 104 (1998), pp. 29–50..

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