King Fahd University of Petroeum & Mineras DEPARTMENT OF MATHEMATICAL SCIENCES Technica Report Series TR 369 December 6 Genera decay of soutions of a viscoeastic equation Saim A. Messaoudi DHAHRAN 3161 SAUDI ARABIA www.kfupm.edu.sa/math/ E-mai: mathdept@kfupm.edu.sa
Genera decay of soutions of a viscoeastic equation Saim A. Messaoudi King Fahd University of Petroeum and Mineras Department of Mathematica Sciences Dhahran 3161, Saudi Arabia. E-mai : messaoud@kfupm.edu.sa Abstract In this paper we consider the semiinear viscoeastic equation t u + t g(t τ) u(τ)dτ =, in a bounded domain, and estabish a genera decay estimate for weak soutions. This resut generaizes and improves earier ones in the iterature. Keywords and phrases: genera decay, reaxation function. AMS Cassification : 35B35, 35B4, 35L. 1 Introduction In this paper we consider the foowing probem t (x, t) u(x, t)+ t g(t τ) u(x, τ)dτ =, in (, ) u(x, t) =, x, t u(x, ) = u (x), (x, ) = u 1 (x), x, (1.1) where is a bounded domain of IR n (n 1) with a smooth boundary and g is a positive nonincreasing function defined on IR +. Cavacanti et a. [5] studied (1.1) in the presence of a ocaized damping cooperating with the dissipation induced by the viscoeastic term. Under the condition ξ 1 g(t) g (t) ξ g(t), t, with g L 1 ((, )) sma enough, they obtained an exponentia rate of decay. Berrimi et a. [] improved Cavacanti s resut by showing that the viscoeastic dissipation aone is enough to stabiize the system. To achieve their goa, Berrimi et a. introduced a different functiona, which aowed them to weaken the conditions on g as we as on 1
the ocaized damping. This resut has been ater extended to a situation, where a source is competing with the viscoeastic dissipation, by Berrimi et a. []. Cavacanti et a. [6] considered t k u + t div[a(x)g(t τ) u(τ)]dτ + b(x)h( )+f(u) =, under simiar conditions on the reaxation function g and a(x) +b(x) δ > and improved the resut in [5]. 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. Though both resuts in [1] and [6] improve the earier one in [5], the approaches are different. 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]. An xistence and uniform decay rate resuts were estabished. A reated probem, in a bounded domain, of the form ρ t u t + t g(t τ) u(τ)dτ γ =, (1.) for ρ >, was aso studied by Cavacanti et a. [4]. A goba existence resut for γ, as we as an exponentia decay for γ >, has been estabished. This ast resut has been extended to a situation, where a source term is competing with the strong mechanism damping and the one induced by the viscosity, by Messaoudi and Tatar [8]. In their work, Messaoudi and Tatar combined the we depth method with the perturbation techniques to show that soutions with positive, but sma, initia energy exist gobay and decay to the rest state exponentiay. Furthermore, Messaoudi and Tatar [11], [1] considered (1.), for γ =, and estabished exponentia and poynomia decay resuts in the absence, as we as in the presence, of a source term. We aso mention the work of Kawashima and Shibata [7], in which a goba existence and exponentia stabiity of sma soutions to a noninear viscoeastic probem has been estabished. For nonexistence, Messaoudi [9] considered t u + t g(t τ) u(τ)dτ + a 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 >mand continue to exist if m γ. This bow-up resut has been pushed to some situations, where the initia energy is positive, by Messaoudi [1]. A simiar resut has been aso obtained by Wu [13] using a different method. In the present work we generaize our earier decay resut to soutions of (1.1). Precisey, we show that the soution energy decays at a simiar rate of decay of the reaxation function, which is not necessariy decaying in a poynomia or exponentia fashion. In fact, our resut aows a arger cass of reaxation functions. The paper is organized as foows. In Section, we present some notations and materia needed for our work and state a goba existence theorem, which can be proved foowing exacty the arguments of [5]. Section 3 contains the statement and the proof of our main resut.
Preiminaries In this section we present some materia needed in the proof of our main resut. Aso, for the sake of competeness we state, without a proof, the goba existence resut of [5] and [6]. We use the standard Lebesgue space L p () and the Soboev space H 1 () with their usua scaar products and norms. For the reaxation function g we assume (G1) g :IR + IR + is a differentiabe function such that g() >, 1 g(s)ds = >. (G) There exists a differentiabe function ξ satisfying g (t) ξ(t)g(t), t, ξ (t) ξ(t) k, ξ(t) >, ξ (t), t >. Remark.1.There are many functions satisfying (G1) and (G). Exampes of such functions are g(t) = a(1 + t) ν, ν < 1 g(t) = ae b(t+1)p, <p 1. for a and b to be chosen propery. Remark.. Since ξ is nonincreasing then ξ(t) ξ() = M Remark.3 Condition (G1) is necessary to guarantee the hyperboicity of the system (1.1). Proposition Let (u,u 1 ) H 1 () L () be given. Assume that g satisfies (G1). Then probem (1.1) has a unique goba soution u C [, );H 1 (), C [, );L (). (.1) We introduce the modified energy functiona E(t) := 1 1 t g(s)ds u(t) + 1 + 1 (g u)(t), (.) where (g v)(t) = t g(t τ) v(t) v(τ) dτ. (.3) 3
3 Decay of soutions In this section we state and prove our main resut. For this purpose we set F (t) :=E(t)+ε 1 Ψ(t)+ε χ(t), (3.1) where ε 1 and ε are positive constants and Ψ(t) : = ξ(t) u dx (3.) χ(t) : = ξ(t) t g(t τ)(u(t) u(τ))dτdx. Lemma 3.1 If u is a soution of (1.1), then the modified energy satisfies E (t) = 1 (g u)(t) 1 g(t) u(t) 1 (g u)(t). (3.3) Proof. By mutipying equation (1.1) by and integrating over, using integration by parts, hypotheses (G1) and (G) and some manipuations as in [9], we obtain (3.3) for reguar soutions. This inequaity remains vaid for weak soutions by a simpe density argument. Lemma 3.. For u H 1 (), we have t g(t τ)(u(t) u(τ))dτ dx (1 )Cp (g u)(t), where C p is the Poincaré constant. Proof. t g(t τ)(u(t) u(τ))dτ dx = t g(t τ) g(t τ)(u(t) u(τ))dτ dx. By appying Cauchy-Schwarz inequaity and Poincaré s inequaity, we easiy see that t t g(t τ)(u(t) u(τ))dτ g(t τ)dτ Lemma 3.3. For ε 1 and ε sma enough, we have t dx g(t τ)(u(t) u(τ)) dτ dx (1 )Cp (g u)(t). α 1 F (t) E(t) α F (t) (3.4) 4
hods for two positive constants α 1 and α. Proof. Straightforward computations, using Lemma 3., ead to F (t) E(t)+(ε 1 /) ξ(t) dx +(ε 1 /) ξ(t) u dx +(ε /) ξ(t) dx +(ε /) ξ(t) t g(t τ)(u(t) u(τ))dτ dx E(t)+[(ε 1 + ε )/] M dx +(ε 1 /) CpM u dx +(ε /) CpM(1 )(g u)(t) α E(t). Simiary, F (t) E(t) (ε 1 /) ξ(t) dx (ε 1 /) ξ(t)c p u dx (ε /) ξ(t) dx (ε /) ξ(t)cp(1 )(g u)(t) + 1 M (ε 1 + ε ) / dx +[ 1 M (ε 1/) Cp] u dx +[ 1 M (ε /) Cp(1 )](g u)(t) α 1 E(t), for ε 1 and ε 1 sma enough. Lemma 3.4 Under the assumptions (G1) and (G), the functiona Ψ(t) :=ξ(t) u dx satisfies, aong the soution of (1.1), Ψ (t) 1+ k Cp ξ(t) u t dx 4 ξ(t) u dx + (3.5) (3.6) (1 ) ξ(t)(g u)(t). (3.7) Proof. By using equation (1.1), we easiy see that Ψ (t) = ξ(t) (ut + u t )dx + ξ (t) u dx = ξ(t) u t dx ξ(t) u dx (3.8) +ξ(t) t u(t). g(t τ) u(τ)dτdx + ξ (t) u dx. We now estimate the third term in the RHS of (3.8) as foows: t u(t). g(t τ) u(τ)dτdx 1 u(t) dx + 1 t g(t τ) u(τ) dτ 1 u(t) dx + 1 t g(t τ)( u(τ) u(t) + u(t) )dτ 5 dx dx. (3.9)
We then use Lemma 3., Young s inequaity, and the fact that t g(τ)dτ g(τ)dτ =1, to obtain, for any η >, (1 + η) t g(t τ)( u(τ) u(t) + u(t) )dτ t g(t τ)( u(τ) u(t) dτ + t g(t τ)( u(τ) u(t) dτ t g(t τ) u(t) dτ dx +(1+ 1) η (1 + 1)(1 )(g u)(t)+(1+η)(1 η ) By combining (3.8)-(3.1) and using u dx αcp u dx + 1 4α we arrive at Ψ (t) 1+ 1 ξ (t) ξ(t) 4α ξ(t) dx (3.1) dx + t g(t τ) u(t) dτ dx t g(t τ) u(t) dτ dx t g(t τ)( u(τ) u(t) dτ dx u(t) dx. u t dx, α >, u t dx + 1 (1 + 1 )(1 )ξ(t)(g u)(t) η 1 1 (1 + η)(1 ) ξ (t) ξ(t) αc p ξ(t) u(t) dx (3.11) 1+ 1 4α k ξ(t) u t dx + 1 (1 + 1 )(1 )ξ(t)(g u)(t) η 1 1 (1 + η)(1 ) kαcp ξ(t) u(t) dx. By choosing η = /(1 ) and α = /4kCp, (3.7) is estabished. Lemma 3.5 Under the assumptions (G1) and (G), the functiona t χ(t) := ξ(t) g(t τ)(u(t) u(τ))dτdx satisfies, aong the soution of (1.1), χ (t) δξ(t) 1+(1 ) u(t) dx + {δ + 1 δ }(1 )+C p 4δ k ξ(t)(g u)(t) (3.1) + g() t 4δ C p ξ(t)( (g u)(t)+ δ(k +1) g(s)ds ξ(t) u t dx, δ >. 6
Proof. Direct computations, using (1.1), yied χ (t) = ξ(t) ξ(t) ξ (t) = ξ(t) t ξ(t) t t t u(t). ξ(t) ξ (t) t t t g(t τ)(u(t) u(τ))dτdx g (t τ)(u(t) u(τ))dτdx ξ(t) g(t τ)(u(t) u(τ))dτdx t t g(s)ds u t dx g(t τ)( u(t) u(τ))dτ dx (3.13) g(t τ) u(τ)dτ. g (t τ)(u(t) u(τ))dτdx ξ(t) t g(t τ)(u(t) u(τ))dτdx. g(t τ)( u(t) u(τ))dτ dx t g(s)ds u t dx Simiary to (3.8), we estimates the RHS terms of (3.13). So, by using Young s inequaity, the first term gives t u(t). g(t τ)( u(t) u(τ))dτ dx (3.14) δ u dx + 1 (g u)(t), δ >. 4δ Simiary, the second term can be estimated as foows t t g(t s) u(s)ds. g(t s)( u(t) u(s)) ds dx δ t g(t s) u(s)ds dx + 1 t g(t s)( u(t) u(s)) ds dx 4δ δ t g(t s)( u(t) u(s) + u(t) ) ds dx + 1 t g(t s)( u(t) u(s)) ds dx 4δ t δ + 1 g(t s) u(t) u(s) ds dx +δ (1 ) u dx 4δ δ + 1 4δ (1 )(g u)(t)+δ(1 ) u dx. 7 (3.15)
As for the third and the fourth terms we have and t g (t τ)(u(t) u(τ))dτdx δ dx g() 4δ C p(g u)(t). (3.16) t g(t τ)(u(t) u(τ))dτdx δ dx + C p (g u)(t). (3.17) 4δ By combining (3.13)-(3.17), the assertion of the emma is estabished. Theorem 3.6 Let (u,u 1 ) H 1 () L () be given. Assume that g and ξ satisfy (G1) and (G). Then, for each t >, there exist stricty positive constants K and λ such that the soution of (1.1) satisfies E(t) Ke λ t t ξ(s)ds, t t. (3.18) Proof Since g is positive and g() > then for any t > we have t g(s)ds t g(s)ds = g >, t t. (3.19) By using (3.1), (3.3), (3.7), (3.1) and (3.19), we obtain for t t, F (t) ε {g δ(1 + k)} ε 1 1+ k Cp ξ(t) u t dx +{ 1 ε g() 4δ C pm}(g u)(t) ε1 4 ε δ{1+(1 ) } ε1 (1 ) + + ε δ + 1 δ At this point we choose δ so sma that ξ(t) u (3.) C p (1 )+ε 4δ k ξ(t)(g u)(t). g δ(1 + k) > 1 g 4 δ 1+(1 ) < 1 4 1+ k C p g. Whence δ is fixed, the choice of any two positive constants ε 1 and ε satisfying g 4 1+ k C p ε < ε 1 < 8 g 1+ k C p ε (3.1)
wi make k 1 : = ε {g δ(1 + k)} ε 1 1+ k Cp > k : = ε 1 4 ε δ 1+(1 ) >. We then pick ε 1 and ε so sma that (3.4) and (3.1) remain vaid and, further, 1 k 3 := ε g() ε1 4δ C pm + ε {δ + 1 δ } Cp (1 ) ε 4δ k>. Hence { 1 ε g() 4δ C p M}(g u)(t) (3.) ε1 (1 ) + + ε δ + 1 C p (1 )+ε δ 4δ k ξ(t)(g u)(t) { 1 ε g() 4δ C pm} ε1 (1 ) + k 3 ξ(t)(g u)(t), t + ε δ + 1 δ ξ(t τ)g(t τ) u(τ) u(t) dτdx C p (1 )+ε 4δ k ξ(t)(g u)(t) since ξ is nonincreasing. Therefore, by using (3.4), (3.), and (3.), we arrive at F (t) β 1 ξ(t)e(t) β 1 α 1 ξ(t)f (t) t t. (3.3) A simpe integration of (3.3) eads to F (t) F (t )e β 1 α t 1 ξ(s)ds t, t t. (3.4) Thus (3.4), (3.4) yied E(t) α F (t )e β 1 α t t 1 ξ(s)ds λ ξ(s)ds t = Ke t, t t. (3.5) This competes the proof. Remark 3.1 This resut generaizes and improves the resuts of [1], [], [5] and [6]. In particuar, it aows some reaxation functions which satisfy g ag ρ, 1 ρ <. This improves eary works [] and [6], where it is assumed that 1 ρ < 3/. Remark 3. Note that the exponentia and the poynomia decay estimates, given in eary works [1], [], [5] and [6], are ony particuar cases of (3.5). More precisey, we obtain exponentia decay for ξ(t) a and poynomia decay for ξ(t) =a(1 + t) 1, where a> is a constant. Remark 3.3 Observe that our resut is proved without any condition on g and g unike what was assumed in (.4) of [6]. We ony need g to be differentiabe satisfying 9
(G1) and (G). Remark 3.4 Estimates (3.18) are aso true for t [,t ] by virtue of continuity and boundedness of E(t) and ξ(t). Remark 3.5 A simiar resut can be estabished for the semiinear probem t (x, t) u(x, t)+ t g(t τ) u(x, τ)dτ + b u p u(x, t) =, in (, ) u(x, t) =,x,t u(x, ) = u (x), (x, ) = u 1 (x), x, (3.6) where b> and p (n 1)/(n ), if n 3. Acknowedgment: The author woud ike to express his sincere thanks to King Fahd University of Petroeum and Mineras for its support. This work has been funded by KFUPM under Project # MS/DECAY/334. 4 References 1. Berrimi S. and Messaoudi S.A., Exponentia decay of soutions to a viscoeastic equation with noninear ocaized damping, Eect J. Diff. Eqns. Vo. 4 # 88 (4), 1-1.. Berrimi S. and Messaoudi S.A., Existence and decay of soutions of a viscoeastic equation with a noninear source, Noninear Anaysis 64 (6), 314-331 3. 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 (1) 1, 85-116. 4. 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. 4 (1), 143-153. 5. 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. () 44, 1-14. 6. Cavacanti M.M. and Oquendo H.P., Frictiona versus viscoeastic damping in a semiinear wave equation, SIAM J. Contro Optim. Vo. 4 # 4 (3), 131-134. 7. Kawashima S. and Shibata Y., Goba Existence and exponentia stabiity of sma soutions to noninear viscoeasticity, Comm. Math. Physics Vo. 148 (199), 189-8. 8. Messaoudi S.A. and Tatar N.-e., Goba existence asymptotic behavior for a Noninear Viscoeastic Probem, J. Math. Sci. Research Vo. 7 # 4 (3), 136-149. 1
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