EXPONENTIAL DECAY OF SOLUTIONS TO A VISCOELASTIC EQUATION WITH NONLINEAR LOCALIZED DAMPING

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1 Eectronic Journa of Differentia Equations, Vo. 24(24), No. 88, pp ISSN: URL: or ftp ejde.math.txstate.edu (ogin: ftp) EXPONENTIAL DECAY OF SOLUTIONS TO A VISCOELASTIC EQUATION WITH NONLINEAR LOCALIZED DAMPING SAID BERRIMI, SALIM A. MESSAOUDI Abstract. In this paper we consider the noninear viscoeastic equation t u tt u + g(t τ) u(τ) dτ + a(x) u t m u t + b u γ u =, in a bounded domain. Without imposing geometry restrictions on the boundary, we estabish an exponentia decay resut, under weaker conditions than those in [3]. 1. Introduction Cavacanti et a [3] studied the equation u tt u + t g(t τ) u(τ)dτ + a(x)u t + u γ u =, in (, ) u(x, t) =, x, t u(x, ) = u (x), u t (x, ) = u 1 (x), x, (1.1) where is a bounded domain of R n (n 1) with a smooth boundary, γ >, g is a positive function, and a : R + is a function, which may be nu on a part of. Under the condition that a(x) a > on ω, with ω satisfying some geometry restrictions and ξ 1 g(t) g (t) ξ 2 g(t), t, such that g L 1 ((, )) is sma enough, the authors obtained an exponentia rate of decay. This work extended the resut of Zuazua [19], in which he considered (1.1) with g = and the inear damping is ocaized. Cavacanti et a [4] considered the equation u tt k u + t div[a(x)g(t τ) u(τ)]dτ + b(x)h(u t ) + f(u) =, in (, ), under simiar conditions on the reaxation function g and a(x) + b(x) δ >, for a x. They improved the resut in [3] by estabishing exponentia stabiity for g decaying exponentiay and h inear and poynomia stabiity for g decaying poynomiay and h noninear. Their proof, based on the use of piecewise mutipiers, 2 Mathematics Subject Cassification. 35B35, 35L2, 35L7. Key words and phrases. Exponentia decay; goba existence; noninear ocaized damping. c 24 Texas State University - San Marcos. Submitted May 17, 24. Pubished June 29, 24. 1

2 2 S. BERRIMI, S. A. MESSAOUDI EJDE-24/88 is simiar to the one in [3]. Another probem, where the damping induced by the viscosity is acting on the domain and a part of the boundary, was aso discussed by Cavacanti et a [5] and existence and uniform decay rate resuts were estabished. In the same direction, Cavacanti et a [2] have aso studied, in a bounded domain, the equation u t ρ u tt u u tt + t g(t τ) u(τ)dτ γ u t =, with x, t >, ρ >. They proved a goba existence resut for γ and an exponentia decay for γ >. This ast resut has been extended to a situation, where a source term is competing with the strong damping mechanism and the one induced by the viscosity, by Messaoudi and Tatar [16]. There, the authors combined we known methods with perturbation techniques to show that a soution with positive but sma energy exist gobay and decay to the rest state exponentiay. Messaoudi [17] considered the equation u tt u + t g(t τ) u(τ)dτ + au t u t m = b u γ u, in (, ) and showed, under suitabe conditions on g, that soutions with negative energy bow up in finite time if γ > m, and continue to exist if m γ. We aso shoud mention the work of Kavashima and Shibata [9], in which a goba existence and exponentia stabiity of sma soutions to a noninear viscoeastic probem has been estabished. In the absence of the viscoeastic term (g = ), the probem has been extensivey studied and many resuts concerning goba existence and nonexistence have been proved. For instance, for the probem u tt u + au t u t m = b u γ u, in (, ) u(x, t) =, x, t u(x, ) = u (x), u t (x, ) = u 1 (x), x, (1.2) with m, γ, it is we known that, for a =, the source term bu u γ, (γ > ) causes finite time bow up of soutions with negative initia energy (see [1, 8]) and for b =, the damping term au t u t m assures goba existence for arbitrary initia data (see [7, 1]). The interaction between the damping and the source terms was first considered by Levine [11, 12] in the inear damping case (m = ). He showed that soutions with negative initia energy bow up in finite time. Georgiev and Todorova [6] extended Levine s resut to the noninear damping case (m > ). In their work, the authors introduced a different method and determined suitabe reations between m and γ, for which there is goba existence or aternativey finite time bow up. Precisey; they showed that soutions with negative energy continue to exist gobay in time if m γ and bow up in finite time if γ > m and the initia energy is sufficienty negative. Without imposing the condition that the initia energy is sufficienty negative, Messaoudi [15] extended the bow up resut of [6] to soutions with negative initia energy ony. For resuts of same nature, we refer the reader to Levine and Serrin [13] and Levine and Park [14], Vitiaro [18].

3 EJDE-24/88 EXPONENTIAL DECAY OF SOLUTIONS 3 In the present work, we are concerned with u tt u + t g(t τ) u(τ)dτ + a(x)u t u t m + u γ u =, in (, ) u(x, t) =, x, t u(x, ) = u (x), u t (x, ) = u 1 (x), x, (1.3) for m. We wi prove an exponentia decay resut under weaker conditions on both a and g. In fact we wi aow a to vanish on any part of (incuding itsef). As a consequence, the geometry restriction imposed on a part of by Cavacanti et a [3] is dropped. Athough this present work and [4] both improve [3], they have different nature and use different approaches. Our method of proof is based on the use of the perturbed energy technique. Our choice of the Lyaponov functiona made our proof easier than the one in [3, 4]. This paper is organized as foows. In Section 2, We present some notation and materia needed for our work and we state the goba existence theorem in [3]. Section 3 contains the statement and the proof of our main resut. 2. Preiminaries In this section, we sha prepare some materia needed in the proof of our resut and state, without proof, a goba existence resut, which may be proved by repeating the argument of [3]. We use the standard Lebesgue space L p () and Soboev space H 1 () with their usua scaar products and norms. The symbos and wi stand for the gradient and the Lapacian respectivey and the subscript t wi denote the time differentiation. For the reaxation function g(t) we assume (G1) g : R + R + is a bounded C 1 function such that g() > and 1 g(s)ds = >. (G2) There exists a positive constant ξ such that g (t) ξg(t), for t. Proposition 2.1. Let (u, u 1 ) H 1 () L 2 (). Assume that g satisfies (G1) and γ 2 n 2, n 3 γ, n = 1, 2. Then probem (1.3) has a unique goba soution, where L m+2 a u C([, ); H 1 ()) u t C([, ); L 2 ()) L m+2 a ( (, )), is the weighted Lebesgue space. (2.1) (2.2) Remark 2.2. Condition (2.1) is needed so that the noninearity is Lipschitz from H 1 () to L 2 (). Condition (G1) is necessary to guarantee the hyperboicity of the system (1.3). Now, we introduce the energy E(t) := 1 ( t ) 1 g(s)ds u(t) u t (g u)(t) + 2 γ + 2 u γ+2, (2.3)

4 4 S. BERRIMI, S. A. MESSAOUDI EJDE-24/88 where (g v)(t) = t g(t τ) v(t) v(τ) 2 2dτ. (2.4) Remark 2.3. Mutipying equation (1.3) by u t and integrating over, then using integration by parts and hypotheses (G1) and (G2), we obtain, after some manipuations, ( E (t) a(x) u t m+2 dx 1 2 (g u)(t) g(t) u(t) 2) a(x) u t m+2 dx + 1 (2.5) 2 (g u)(t). This impies that modified energy is uniformy bounded (by E()) and is decreasing in t. We wi aso use the embedding H 1 () L q () for 2 q 2n/(n 2) if n 3 or q 2 if n = 1, 2 and L r () L q (), for q < r. We wi use the same embedding constant denoted by C p ; i.e. v q C p v 2, v q C p v r. (2.6) 3. Exponentia decay Before we state and prove our main resut, we prove the foowing emma. Lemma 3.1. Let m 2/(n 2), for n 3. Then there exists a constant C depending on C p, a, E(), and m ony, such that the soution (2.2) satisfies a(x) u m+2 dx C ( u u γ+2 ) γ+2 (3.1) Proof. If m γ then we have two cases either u m+2 1, in which case a(x) u m+2 dx a u m+2 m+2 a u 2 m+2 Cp a 2 u 2 2; (3.2) or u m+2 > 1, in which case a(x) u m+2 dx a u m+2 m+2 a u γ+2 m+2 Cγ+2 p a u γ+2 γ+2. (3.3) If m > γ then a(x) u m+2 dx C m+2 p a u m+2 2 C m+2 p a u 2 2 (2E(t)) m/2 Cp m+2 (2E()) m/2 u 2 a 2 Combining (3.2), (3.3) with the above inequaity, we compete the proof. Theorem 3.2. Let (u, u 1 ) H 1 () L 2 (). Assume that g satisfies (G1) and (G2), such that max{m, γ} 2 n 2, n 3. Then there exist positive constants k and K, such that the soution given by (2.2) satisfies E(t) Ke kt for a t.

5 EJDE-24/88 EXPONENTIAL DECAY OF SOLUTIONS 5 Proof. We define the function F (t) := E(t) + ε 1 Ψ(t) + ε 2 χ(t) (3.4) where ε 1 and ε 2 are positive constants to be specified ater and Ψ(t) := uu t dx t χ(t) := u t g(t τ)(u(t) u(τ))dτ dx. It is straightforward to see that for ε 1 and ε 2 sma, we have hods for two positive constants α 1 and α 2. In fact F (t) E(t) + (ε 1 /2) u t 2 dx + (ε 1 /2) u 2 dx α 1 F (t) E(t) α 2 F (t), (3.5) ( t + (ε 2 /2) u t 2 dx + (ε 2 /2) g(t τ)(u(t) u(τ))dτ) E(t) + (ε 1 /2) u t 2 dx + (ε 1 /2)C p u 2 dx + (ε 2 /2) u t 2 dx + (ε 2 /2)C p (1 )(g u)(t) 1 α 1 E(t) where E(t) is the energy, and F (t) E(t) (ε 1 /2) u t 2 dx (ε 1 /2) u 2 dx (ε 2 /2) u t 2 dx (ε 2 /2)C p (1 )(g u)(t) 1 2 u(t) u t (g u)(t) + 2 γ + 2 u γ+2 ε 1 + ε 2 u t 2 dx ( ε )C p u 2 dx ( ε 2 2 )C p(1 )(g u)(t) 1 α 1 E(t) (3.6) for ε 1 and ε 2 sma enough. Using equation (1.3), we easiy see that Ψ (t) = (uu tt + u 2 t )dx = u 2 t dx u 2 dx + u(t) u γ+2 dx a(x) u t m u t udx t g(t τ) u(τ)dτ dx We now estimate the third term in the right-hand side of (3.7) as foows: t u(t). g(t τ) u(τ)dτ dx (3.7)

6 6 S. BERRIMI, S. A. MESSAOUDI EJDE-24/ u(t) 2 dx u(t) 2 dx ( t g(t τ) u(τ) dτ) ( t g(t τ)( u(τ) u(t) + u(t) )dτ). Using Cauchy-Schwarz and Young s inequaity, and t g(τ)dτ g(τ)dτ = 1, we obtain that for any η >, ( t g(t τ)( u(τ) u(t) + u(t) )dτ) + 2 ( t (1 + η) ) g(t τ)( u(τ) u(t) dτ + ( t g(t τ) u(t) dτ) ( t )( t ) g(t τ)( u(τ) u(t) dτ g(t τ) u(t) dτ dx + (1 + 1 η ) (1 + 1 η ) + (1 + η) ( t g(t τ) u(t) dτ) t (1 + η)(1 ) 2 ( t g(t τ)( u(τ) u(t) dτ) g(t τ)dτ t u(t) 2( t g(t τ)dτ) + (1 + 1 η )(1 ) u(t) 2 dx t g(t τ) u(τ) u(t) 2 dτ dx g(t τ) u(τ) u(t) 2 dτ dx. For the fifth term of the right-hand side of (3.7), we use Young s inequaity and Lemma 3.1 to get a(x) u t m u t udx δ a(x) u m+2 dx + c(δ) a(x) u t m+2 dx (3.8) c(δ) a(x) u t m+2 dx + δc{ u u γ+2 γ+2 } By combining (3.7) (3.8), we have Ψ (t) u 2 t dx u 2 dx u γ+2 dx + 1 u(t) 2 dx (1 + η)(1 )2 u(t) 2 dx + c(δ) a(x) u t m+2 dx + δc p u δc p u γ+2 γ+2 } (1 + 1 t η )(1 ) g(t τ) u(τ) u(t) 2 dτ dx u 2 t dx u 2 dx u γ+2 dx [1 + (1 + η)(1 )2 ] u(t) 2 dx

7 EJDE-24/88 EXPONENTIAL DECAY OF SOLUTIONS (1 + 1 η )(1 )(g u)(t) + c(δ) a(x) u t m+2 dx + δc{ u u γ+2 γ+2 } By choosing η = /(1 ) and δ = /4C, the above inequaity becomes Ψ (t) u 2 t dx u 2 dx 4 u γ+2 dx + 1 (g u)(t) c(δ) a(x) u t m+2 dx. Next we estimate t χ (t) = u tt g(t τ)(u(t) u(τ))dτ dx = ( + + t u t g (t τ)(u(t) u(τ))dτ dx ( u(t).( t t t g(t τ)( u(t) u(τ))dτ)dx g(t τ) u(τ)dτ).( a(x)u t (t) u γ u t t t g(t τ)(u(t) u(τ))dτ dx g(t τ)(u(t) u(τ))dτ dx g(s)ds) u 2 t dx g(t τ)( u(t) u(τ))dτ)dx (3.9) (3.1) t t u t g (t τ)(u(t) u(τ))dτ dx ( g(s)ds) u 2 t dx Simiary to (3.7), we estimates the right-hand side terms of the above inequaity. So for δ >, we have: For the first term, u(t).( t g(t τ)( u(t) u(τ))dτ)dx δ For the second term, ( t )( t g(t s) u(s)ds δ δ ) g(t s)( u(t) u(s))ds dx u 2 dx + 1 (g u)(t). 4δ (3.11) t g(t s) u(s)ds 2 dx + 1 t g(t s)( u(t) u(s))ds 2 dx 4δ ( t g(t s)( u(t) u(s) + u(t) )ds) + 1 4δ ( t g(t s)ds) t g(t s) u(t) u(s) 2 ds dx. ( t δ g(t s) u(t) u(s) ds) + 2δ(1 ) 2 u 2 dx + 1 (1 )(g u)(t) 4δ (2δ + 1 4δ )(1 )(g u)(t) + 2δ(1 )2 u 2 dx.

8 8 S. BERRIMI, S. A. MESSAOUDI EJDE-24/88 For the third term, t a(x)u t (t) g(t τ)(u(t) u(τ))dτ dx δ a For the fourth term, t u γ u g(t τ)(u(t) u(τ))dτ dx δ u 2(γ+1) dx + 1 4δ ( t u 2 t dx + C p(1 ) (g u)(t). 4δ ) (3.12) g(t τ)(u(t) u(τ))dτ We use (2.1) (2.3) and (2.5) to obtain u 2(γ+1) dx C p u 2(γ+1) 2 C p ( E() ) 2γ u 2 2 (3.13) By inserting (3.13) in (3.12), we get t u γ u g(t τ)(u(t) u(τ))dτ dx δc p ( E() For the fifth term, t u t g (t τ)(u(t) u(τ))dτ dx δ u t 2 dx + g() 4δ C p Combining (3.1) (3.14) yieds t ) 2γ u C p(1 ) (g u)(t). 4δ g (t s) u(t) u(s) 2 ds dx. (3.14) χ (t) δ{1 + 2(1 ) 2 + C p ( E() ) 2γ } u 2 [ δ(1 ) + C p(1 ) ] (g u)(t) 2δ 2δ + g() 4δ C p( (g u)(t)) + [ t δ(1 + a ) g(s)ds ] u 2 t dx. Since g() > then there exists t > such that t g(s)ds t Using (3.4), (3.9), (3.15), and (3.16), we obtain F (t) (1 ε 1 c(δ)) a(x) u t m+2 dx [ ] ε 2 {g δ(1 + a )} ε 1 u 2 4 t dx ε 1 4 [ ε 1 4 ε 2δ{1 + 2(1 ) 2 + C p ( E() ) 2γ } ] u [ 1 2 ε 1(1 ) ε 2 { g() 2ξ 4δ C p + (1 )C p + 1 2δξ 2δξ (3.15) g(s)ds = g >, t t. (3.16) u γ+2 dx 2δ(1 ) + } ] (g u)(t). ξ

9 EJDE-24/88 EXPONENTIAL DECAY OF SOLUTIONS 9 Now, we choose δ so sma that g δ(1 + a ) > 1 2 g 4 δ{1 + 2(1 )2 + C p ( E() ) 2γ } < 1 4 g. Whence δ is fixed, the choice of any two positive constants ε 1 and ε 2 satisfying 1 4 g ε 2 < ε 1 < 1 2 g ε 2 (3.17) wi make k 1 = ε 2 {g δ(1 + a )} ε 1 > k 2 = ε 1 4 ε 2δ{1 + 2(1 ) 2 + C p ( E() ) 2γ } >. We then pick ε 1 and ε 2 so sma that (3.5) and (3.17) remain vaid and 1 ε 1 c(δ) >, 1 2 ε 1(1 ) ε 2 { g() 2ξ 4δ C p + (1 )C p + 1 2δξ 2δξ 2δ(1 ) + } > ξ Therefore, we arrive at F (t) βe(t) for a t t. This inequaity and (3.5) yied F (t) βα 1 F (t), for a t t. A simpe integration eads to This inequaity and (3.5) yieds which competes the proof. F (t) F (t )e βα1t e βα1t, t t. E(t) α 2 F (t )e βα1t e βα1t, t t, Remark 3.3. Note that our resut is proved without imposing any restriction on the size of g L 1. Aso note that the function a may vanish on the whoe domain. In other words, contrary to [3], measure (ω) can be zero. As a consequence, no geometry restriction on the boundary has been assumed. Acknowedgment. The authors woud ike to express their sincere thanks to King Fahd University of Petroeum and Mineras for its support. This work has been funded by KFUPM under Project # MS/VISCO ELASTIC/27. References [1] Ba J., Remarks on bow up and nonexistence theorems for noninear evoutions equations, Quart. J. Math. Oxford 28 (1977), [2] Cavacanti M. M., V. N. Domingos Cavacanti and J. Ferreira; Existence and uniform decay for noninear viscoeastic equation with strong damping, Math. Meth. App. Sci. 24 (21), [3] Cavacanti M. M., V. N. Domingos Cavacanti and J. A. Soriano; Exponentia decay for the soution of semiinear viscoeastic wave equations with ocaized damping, Eect. J. Diff. Eqs. Vo. 22 (22) 44, [4] Cacacanti M. M. and Oquendo H. P.; Frictiona versus viscoeastic damping in a semiinear wave equation, SIAM J. Contro Optim. vo 42 # 4 (23), [5] Cavacanti M. M., V. N. Domingos Cavacanti, J. S. Prates Fiho and J. A. Soriano; Existence and uniform decay rates for viscoeastic probems with noninear boundary damping, Diff. Integ. Eqs. 14 (21) 1, [6] Georgiev, V. and G. Todorova, Existence of soutions of the wave equation with noninear damping and source terms, J. Diff. Eqs. 19 (1994),

10 1 S. BERRIMI, S. A. MESSAOUDI EJDE-24/88 [7] Haraux, A. and E. Zuazua, Decay estimates for some semiinear damped hyperboic probems, Arch. Rationa Mech. Ana. 15 (1988), [8] Kaantarov V. K. and O. A. Ladyzhenskaya, The occurrence of coapse for quasiinear equations of paraboic and hyperboic type, J. Soviet Math. 1 (1978), [9] Kawashima S. and Shibata Y., Goba Existence and exponentia stabiity of sma soutions to noninear viscoeasticity, Comm. Math. Physics, Vo. 148 # 1(1992), [1] Kopackova M., Remarks on bounded soutions of a semiinear dissipative hyperboic equation, Comment. Math. Univ. Caroin. 3 (1989), [11] Levine, H. A., Instabiity and nonexistence of goba soutions of noninear wave equation of the form P u tt = Au + F (u), Trans. Amer. Math. Soc. 192 (1974), [12] Levine H. A., Some additiona remarks on the nonexistence of goba soutions to noninear wave equation, SIAM J. Math. Ana. 5 (1974), [13] Levine H. A. and J. Serrin, A goba nonexistence theorem for quasiinear evoution equation with dissipation, Arch. Rationa Mech. Ana., 137 (1997), [14] Levine, H. A. and S. Ro Park, Goba existence and goba nonexistence of soutions of the Cauchy probem for a nonineary damped wave equation, J. Math. Ana. App. 228 (1998), [15] Messaoudi S. A., Bow up in a nonineary damped wave equation, Mathematische Nachrichten, 231 (21), 1-7. [16] Messaoudi S. A. and Tatar N-E., Goba existence asymptotic behavior for a Noninear Viscoeastic Probem, Math. Meth. Sci. Research J., Vo. 7 # 4 (23), [17] Messaoudi S. A., Bow up and goba existence in a noninear viscoeastic wave equation, Mathematische Nachrichten, vo. 26 (23), [18] Vitiaro E., Goba nonexistence theorems for a cass of evoution equations with dissipation, Arch. Rationa Mech. Ana., 149 (1999), [19] Zuazua E., Exponentia decay for the semiinear wave equation with ocay distributed damping, Comm. PDE. 15 (199), Said Berrimi Math Department, University of Setif, Setif, Ageria E-mai address: berrimi@yahoo.fr Saim A. Messaoudi Mathematica Sciences Department, KFUPM, Dhahran 31261, Saudi Arabia E-mai address: messaoud@kfupm.edu.sa