Journa of Mathematica Research with Appications Jan.,, Vo. 3, No., pp. 53 6 DOI:.377/j.issn:95-65...7 Http://jmre.dut.edu.cn Genera Decay of Soutions in a Viscoeastic Equation with Noninear Locaized Damping Xiao Jun SONG, Rong ZENG,, Chun Lai MU. Coege of Mathematics and Information, China West Norma University, Sichuan 637, P. R. China;. Coege of Mathematics and Statistics, Chongqing University, Chongqing 433, P. R. China Abstract In this paper, we consider the foowing viscoeastic equation u tt u + t g(t s u(sds + a(xu t + u u r = with initia condition and Dirichet boundary condition.the decay property of the energy function cosey depends on the properties of the reaxation function g(t at infinity. In the previous works of [3, 7, ], it was required that the reaxation function g(t decay exponentiay or poynomiay as t +. In the recent work of Messaoudi [, 3], it was shown that the energy decays at a simiar rate of decay of the reaxation function, which is not necessariy dacaying in a poynomia or exponentia fashion. Motivated by [, 3], under some assumptions on g(x, a(x and r, and by introducing a new perturbed energy, we aso prove the simiar resuts for the above equation. Keywords genera decay; viscoeastic equation; reaxation function. MR( Subject Cassification 35L5; 35L7. Introduction In this paper we are concerned with the foowing equation with a tempora nonoca term u tt u + t u(x, t =, x, t, g(t s u(sds + a(xu t + u u r =, u(x, = u (x, u t (x, = u (x, x, where is a bounded domain in R n with smooth boundary, the reaxation function g(x is a positive nonincreasing function, and the coefficient a(x of the weak frictiona damping is supposed to be positive. The equation in (. describes the motion of a viscoeastic body. It is we known that the viscoeastic materias exhibit nature damping, which is due to the specia property of these materias to keep memory of their past history. These damping effects are represented by memory term such as in (. (see [9] and the references therein. Received March, ; Accepted January, Supported by the Fundamenta Research Funds for the Centra Universities (Grant No.CDJXS6. * Corresponding author E-mai address: zengrong6543@yahoo.com.cn (R. ZENG (.
54 X. J. SONG, R. ZENG and C. L. MU One important question is at which rate the energy of soutions of the damped equation approaches as time tends to. This probem has aready been investigated in severa papers (see [3, 7, 8, ] under different additiona assumptions for g. It is known to us that Cavacanti et a. in [7] firsty studied the above probem. Under the condition that a(x a > on ω, with ω satisfying some geometric restrictions and ξ g(t g (t ξ g(t, t, when g(sds is sufficienty sma, an exponentia rate of decay E(t Ce βt was obtained for some positive constants C, β, where E(t wi be specified in Theorem (.5. This work extended the resut of Zuazua [6], in which (. was considered with g = and the inear damping was ocaized. Later, Berrimi and Cavacanti in [3] considered u tt k u + t div[a(xg(t τ u(τ]dτ + b(xh(u t + f(u =, under simiar conditions on the reaxation function g and a(x + b(x δ > and improved the resut in [7]. They estabished an exponentia stabiity when g is decaying exponentiay and h is inear, and a poynomia stabiity when g is decaying poynomiay and h is noninear. In [3], Berrimi and Messaoudi studied the equation u tt u + t in a bounded domain. Under the condition that g(t s u(sds + a(x u t m u t + u r u =, (. g (t ξg(t, t for some positive constant ξ, the authors aso proved an exponentia decay under weaker conditons on both a and g, where a is aowed to vanish on any part of (incuding itsef. Then the geometric restriction imposed on by Cavacanti et a [7] can be dropped. Liu [] aso considered probem (. under condition g (t ξg p (t, t, p < 3/, where ξ is a positive constant. They showed the exponentia decay when p =, and poynomia decay when < p < 3/. This resut extended the work in [3] where ony exponentia decay was estabished. In [], Aabau-Boussouira et a. deveoped a unified method to derive decay estimates for the abstract integro-differentia evoution equation u (t + Au(t t β(t sau(sds = F ( u(t, t (,, in a Hibert space X, where A : D(A X X is an accretive sef-adjoint inear operator with dense domain, and F denotes the gradient of a Găteaux differentiabe functiona F : D( A R. Depending on the properties of convoution kerne β at infinity, they showed that the energy of soution decays exponentiay or poynomiay as t.
Genera decay of soutions in a viscoeastic equation with noninear ocaized damping 55 Recenty, in [] Messaoudi studied the equation u tt u + t where the reaxation function g is assumed as foows g(t s u(sds = (i g : R + R + is a nonincreasing differentiabe function such that g( >, (ii There exists a differentiabe function ξ satisfying g(sds = >. g (t ξ(tg(t, t, ξ (t ξ(t t k, ξ(t >, ξ (t, t >. He proved that the soution energy decays at the same rate of decay of the reaxation function, which is not necessariy poynomia or exponentia decay. Then in [3], the same author studied the foowing equation u tt u + t where the reaxation function g is assumed as foows g(t s u(sds = u u γ, (i g : R + R + is a nonincreasing differentiabe function such that g( >, (ii There exists a differentiabe function ξ satisfying (iii For the noninear term, assume that g(sds = >. g (t ξ(tg(t, t, ξ (t ξ(t k, ξ(t >, ξ (t, t >. < γ, n 3, γ >, n =,. n It was aso proved that the soution energy decays at the same rate of decay of the reaxation function, which is not necessariy poynomia or exponentia decay. Motivated by the above work of [,3], in this paper we aso concern with probems (. and (.. By using Lyapunov type technique for some perturbed energy, which was introduced in [,3], we show that the soution energy decays at a simiar rate of decay of the reaxation function, which is not necessariy the decay in an exponentia or poynomia fashion. Therefore, our resut aows a arger cass of reaxation functions and improves earier resuts in the iterature [3,7, ]. Our assumptions on the function g(x, a(x, and r are as foows. (A g : R + R + is a nonincreasing differentiabe function such that g( >, g(sds = >.
56 X. J. SONG, R. ZENG and C. L. MU (A There exists a differentiabe function ξ satisfying g (t ξ(tg(t, t, ξ (t ξ(t k, ξ(t >, ξ (t, t >. (A3 Assume that a(x is a nonnegative and bounded function such that (A4 For the noninear term, we assume a(x a >. < r <, n 3; r >, n =,. n Remark. There are many functions satisfying the assumptions (A and (A, and exampes have been given in [], such as for a, b > to be chosen propery. g (t = a( + t v, v <, g (t = ae b(t+p, < p, g 3 (t = ae bt, n =,, ( + t n Remark. Since ξ is nonincreasing, ξ(t ξ( = M. Remark.3 Condition (A is necessary to guarantee the hyperboicity of the system (.. We wi aso use the embedding H ( Lq ( for q n/n, if n 3 and q if n =, ; and L r ( L q (, for q < r and we wi use the same embedding constant denoted by C, i.e., u q C u, u q C u r. The existence of goba soution of probem (. can be easiy obtained by making use of the Faedo-Gaerkin method, and we refer to [5] for detais. Proposition.4 Let (u, u t H ( L (. Assume (A (A4 hod, then probem (. has a unique goba soution Our main resut is stated as foows. u C ([, ; H ( C ([, ; L (. Theorem.5 Let (u, u H ( L ( be given. Assume that (A (A4 hod, then for each t >, there exist stricty positive constants K and λ such that the soution of (. satisfies E(t Ke λ t t ξ(sds, t t, where E(t = ( t g(sds u + u t + (g u(t + r + u r+ r+ (.3
Genera decay of soutions in a viscoeastic equation with noninear ocaized damping 57 and (g v(t = t g(t s v(t v(s ds. Remark.6 Our method used in this paper is appicabe to the equation (. in a bounded domain with max {m, r} n, if n 3. Thus we can extend the resut in []. The rest of this paper is organized as foows. In next section, we present some Lemmas needed for our work. Section 3 contains the proof of our main resut.. Preiminaries In this section we wi prove some emmas. Lemma. If u is a soution of (., then energy E(t satisfies E (t. Proof By mutipying equation (. by u t and integrating over, then using integration by parts and hypotheses (A and (A, after some manipuation, we obtain. E (t = (g u(t g(t u a(x u t dx. (. Remark. It foows form Lemma. that the energy is uniformy bounded (by E( and deceasing in t, which aso impies that Then we define the perturbed energy functiona where ε and ε are positive constants and ψ(t = ξ(t φ(t = ξ(t u E(. (. F(t = E(t + ε ψ(t + ε φ(t, (.3 uu t dx, t u t g(t s(u(t u(sdsdx. (.4 Lemma.3 For u H (, we have ( t g(t s(u(t u(sds dx ( C (g u(t. Proof ( t g(t s(u(t u(sds dx = ( t g(t s g(t s(u(t u(sds dx. By appying Cauchy-Schwarz inequaity and Poincaré s inequaity, we easiy see that ( t g(t s(u(t u(sds dx ( t ( t g(t sds g(t s(u(t u(s ds dx ( C (g u(t.
58 X. J. SONG, R. ZENG and C. L. MU Lemma.4 For ε and ε sma enough, we have hods for two positive constant α and α. α F(t E(t α F(t (.5 Proof After straightforward computation, we can see that F(t E(t + ε ( ξ(t u dx + u t dx + ε ξ(t u t dx+ ε ( t ξ(t g(t s(u(t u(sds dx E(t + ( ε + ε M u t dx + ε MC u dx + ε MC ( (g u(t α E(t, (.6 and F(t E(t ε ξ(t u t dx ε ξ(tc u dx ε ξ(t u t dx ε ξ(tc ( (g u(t ( M (ε + ε u t dx + ( ε MC u dx+ ( ε MC ( (g u(t + r + u r+ r+ dx α E(t, (.7 for ε and ε sma enough. Lemma.5 Let u be the soution of probem (. derived in Proposition (.4. Then we have ψ (t [ + C (k + a ] ξ(t u t dx + ξ(t(g u(t 4 ξ(t u dx ξ(t u r+ dx. (.8 Proof By using Eq.(., we easiy see that t ψ (t =ξ (t uu t dx + ξ(t u t dx ξ(t u(t g(t s u(sdsdx ξ(t a(xuu t dx ξ(t u r+ dx. (.9 We now estimate the third term in the right hand side of (.9 as foows t u(t g(t s u(sdsdx u(t dx + u(t dx + ( t dx g(t s u(s ds ( t dx. g(t s( u(s u(t + u(t ds (.
Genera decay of soutions in a viscoeastic equation with noninear ocaized damping 59 Thanks to Young s inequaity, and the fact t g(sds g(sds =, we obtain that for any η >, ( t dx g(t s( u(s u(t + u(t ds ( t ( t dx g(t s u(s u(t ds + g(t s u(s u(t ds ( t dx+ g(t s( u(t ds ( t g(t s u(t ds dx ( t dx ( t dx ( + η g(t s( u(t ds + ( + η g(t s u(s u(t ds ( + η ( (g u(t + ( + η( u(t dx. (. Combining (. and (., and using uu t dx αc a(xuu t dx α a(x C we get ψ (t [ + 4α ( a + ξ (t ξ(t ]ξ(t u(t dx + u tdx, α >, (. 4α u(t dx + 4α a(x u tdx, α >, (.3 u t + ( + ( ξ(t(g u(t η [ ( + η( αc ( a + ξ (t ξ(t ]ξ(t u dx ξ(t u r+ dx [ + 4α ( a + k]ξ(t u t + ( + ( ξ(t(g u(t η [ ( + η( αc ( a + k]ξ(t u dx ξ(t u r+ dx. (.4 Choosing η = and α = 4C (k+ a yieds (.9. Lemma.6 Let u be the soution of probem (. derived in Proposition (.4. Then we have φ (t δ[ + ( + C r+ ( E( r ]ξ(t u(t dx + k δ ξ(t(g u(t [ t δ( + k + a where g( C ξ(t(g u(t + K δ = ] g(sds ξ(t u t dx, (.5 + (δ + ( + ( a + k + C (. (.6 Proof It is obtained after direct computations that ( φ t (t =ξ(t u(t g(t s( u(t u(sds dx ξ(t ( t ( t g(t s u(sds g(t s( u(t u(sds dx+
6 X. J. SONG, R. ZENG and C. L. MU t ξ(t a(xu t g(t s(u(t u(sdsdx+ t ξ(t u r u g(t s(u(t u(sdsdx t t ξ(t u t g (t s(u(t u(sdsdx ξ(t g(sds u t dx t ξ (t u t g(t s(u(t u(sdsdx. (.7 We now estimate the right-hand side terms of (.7. Appying Young s inequaity to the first term gives ( t u(t g(t s( u(t u(sds dx δ Simiary, the second term can be estimated as foows: ( t ( t g(t s u(sds g(t s( u(t u(sds dx δ δ t u dx+ (g u(t, δ >. (.8 g(t s u(s ds dx + t g(t s( u(t u(sds dx ( t dx+ g(t s( u(t u(s + u(t ds ( t dx g(t s( u(t u(sds (δ + ( t dx g(t s( u(t u(s ds + δ( u dx (δ + ( (g u(t + δ( As for the third term, we have t a(xu t g(t s(u(t u(sdsdx δ a The fourth term The fifth term u t t t u r u g(t s(u(t u(sdsdx δ u r+ dx+ C ( (g u(t δc r+ u r+ δc r+ ( E( + C Combining (.8 (. gives Lemma.6. u dx. (.9 u t C dx + a ( (g u(t. (. ( (g u(t r u + C ( (g u(t. (. g (t s(u(t u(sdsdx δ u t dx g( (g u(t. (.
Genera decay of soutions in a viscoeastic equation with noninear ocaized damping 6 3. Decay of soutions Now we prove our main resut Theorem.4. Proof Since g is positive, continuous and g( >, for any t, we have t g(sds t g(sds = g >, t t. By using (., (.9 and (.6, we obtain for t t, F (t [ ε {g δ( + k + a } ε ( + C (k + a ] ξ(t u t dx [ ε 4 ε δ{ + ( + C r+ ( E( r } ] ξ(t u dx+ ( ε g( C M(g u(t + (ε + ε k δ ξ(t(g u(t ε ξ(t u r+ dx. (3. At this point we choose δ so sma that g δ( + k > g, 4 δ[ + ( + C r+ ( E( r ] < 4( + C (k+ a g, (3. where δ is fixed. The choice of any two positive constants ε and ε satisfying g 4( + C (k+ a ε g < ε < ( + C (k+ a ε (3.3 wi make k = ε {g δ( + k + a } ε ( + C (k + a >, k = ε 4 ε δ[ + ( + C r+ ( E( r ] >. (3.4 We then take ε and ε such that (.6 and (3.3 remain vaid. Further, k 3 = ( ε g( C M (ε + ε k δ >. (3.5 Hence ( ε g( C M(g u(t + (ε + ε k δ ξ(t(g u(t k 3 ξ(t(g u(t. (3.6 Since ξ is nonincreasing, by using (.6, (3. and (3.6 we arrive at A simpe integration of (3.7 eads to Thus from (.6 and (3.8 it foows F (t β ξ(te(t β α ξ(tf(t. (3.7 F(t F(t e βα t t ξ(sds, t t. (3.8 E(t α F(t e βα t t ξ(sds = Ke λ t t ξ(sds, t t, (3.9
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