Research Article General Decay for the Degenerate Equation with a Memory Condition at the Boundary

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1 Absrac and Applied Analysis Volume 3, Aricle ID 686, 8 pages hp://dx.doi.org/.55/3/686 Research Aricle General Decay for he Degenerae Equaion wih a Memory Condiion a he Boundary Su-Young Shin and Jum-Ran Kang Deparmen of Mahemaics, Dong-A Universiy, Saha-Ku, Busan 64-74, Republic of Korea Correspondence should be addressed o Jum-Ran Kang; poinegg@hanmail.ne Received December ; Acceped 5 March 3 Academic Edior: Abdelaziz Rhandi Copyrigh 3 S.-Y. Shin and J.-R. Kang. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. We consider a degenerae equaion wih a memory condiion a he boundary. For a wider class of relaxaion funcions, we esablish a more general decay resul, from which he usual exponenial and polynomial decay raes are only special cases.. Inroducion The main purpose of his paper is o invesigae he asympoic behavior of he soluions of he degenerae equaion wih a memory condiion a he boundary K (x) u +Δ u+f(u) = in Q= (, ), () u= u ] = on Γ (, ), () u + g ( s) B u (s) ds = on (, ), (3) u ] + g ( s) B u (s) ds = on (, ), (4) u (, x) =u (x), u (, x) =u (x) in, (5) where is a bounded domain of R n wih a smooh boundary Γ and le us assume ha Γ, can be divided ino wo nonnull pars Γ = Γ and Γ = and K C () and K(x) for all x which saisfies some appropriae condiions. ] is he uni ouward normal o Γ and τ=( ], ] ) he corresponding uni angen vecor. Here, he relaxaion funcions g i (i =,) are posiive and nondecreasing, he funcion f C (R) and B u=δu+( μ)b u, B u= Δu ] +( μ) B u τ, B u=] ] u xy ] u yy ] u xx, B u=(] ] )u xy + ] ] (u yy u xx ), and he consan μ, <μ</, represens Poisson s raio. From he physical poin of view, we know ha he memory effec described in inegral equaions (3) and(4) canbe caused by he ineracion wih anoher viscoelasic elemen. In fac, he boundary condiions (3) and(4) mean ha is composed of a maerial which is clamped in a rigid body in Γ and is clamped in a body wih viscoelasic properies in he complemenary par of is boundary named.problems relaed o () (5) are ineresing no only from he poin of view of PDE general heory, bu also due o is applicaions in mechanics. The exisence of global soluions and exponenial decay o he degenerae equaion wih = Γ has been invesigaed by several auhors. See Cavalcani e al. [] and Menezes e al. []. For insance, when K(x) is equal o, () describes he ransverse deflecion u(x, ) of beams. There exiss a large body of lieraure regarding viscoelasic problems wih he memory erm acing in he domain or a he boundary (6)

2 Absrac and Applied Analysis (see [3 7]). Sanos e al. [8]sudiedheasympoicbehavior of he soluions of a nonlinear wave equaion of Kirchhoff ype wih boundary condiion of memory ype. Cavalcani e al. [9] proved he uniform decay raes of soluions o a degenerae sysem wih a memory condiion a he boundary. Sanos and Junior [] invesigaedhesabiliyofsoluions for Kirchhoff plae equaions wih a boundary memory condiion. Rivera e al. [] showed he asympoic behavior o a von Karman plae wih boundary memory condiions. Park and Kang [] sudied he exponenial decay for he Kirchhoff plae equaions wih nonlinear dissipaion and boundary memory condiion. They proved ha he energy decays uniformly exponenially or algebraically wih he same rae of decay as he relaxaion funcions. In he presen work, we generalize he earlier decay resuls of he soluion of () (5). More precisely, we show ha he energy decays a heraesimilaroherelaxaionfuncions,whichareno necessarily decaying like polynomial or exponenial funcions. In fac, our resul allows a larger class of relaxaion funcions. Recenly, Messaoudi and Soufyane [3], Musafa and Messaoudi [4], and Sanos and Soufyane [5] proved he general decay for he wave equaion, Timoshenko sysem, and von Karman plae sysem wih viscoelasic boundary condiions, respecively. The organizaion of his paper is as follows. In Secion, we presen some noaions and maerial needed for our work and sae he exisence resul o sysem () (5). In Secion 3, we prove he general decay of he soluions o he degenerae equaion wih a memory condiion a he boundary.. Preliminaries In his secion, we inroduce some noaions and esablish he exisence of soluions of he problem () (5). Noe ha, because of condiion (), he soluion of sysem () (5)musbelongohefollowingspace: W={V H () : V = V ] = on Γ }. (7) Le us define he bilinear form a(, ) as follows: a (u, V) = {u xx V xx +u yy V yy +μ(u xx V yy +u yy V xx ) +( μ)u xy V xy }dxdy. Since Γ =,weknowhaa(u, u) is equivalen o he H () norm on W;hais, (8) c u H () a(u, u) C u H (), (9) and here and in he sequel, we denoe by c and C generic posiive consans. Simple calculaion, based on he inegraion by pars formula, yields (Δ u, V) =a(u, V) +(B u, V) Γ (B u, V ] ). () Γ We assume ha here exiss x R n such ha Γ ={x Γ:] (x) (x x ) }, () = {x Γ:] (x) (x x ) >}. () If we denoe he compacness of by m(x) = x x,condiion () implies ha here exiss a small posiive consan δ such ha <δ m(x) ](x),forallx. Nex, we will use (3)and(4)oesimaehevaluesB and B on.denoingby (g φ)() = g ( s) φ (s) ds (3) he convoluion produc operaor and differeniaing (3)and (4), we arrive a he following Volerra equaions: B u+ B u+ g () g B u= g () u, g () g B u= u g () ]. Applying he Volerra inverse operaor, we ge B u= g () {u +k u }, B u= g () { u ] +k u ] }, where he resolven kernels saisfy (4) (5) k i + g i () g i k i = g i () g i, i =,. (6) Denoing ha τ =/g () and τ =/g (),wehave B u=τ {u +k () u k () u +k u}, B u= τ { u ] +k () u ] k () u ] +k u ] }. (7) Therefore, we use (7) insead of he boundary condiions (3) and (4). Le us denoe ha (g V) () := g ( s) V() V (s) ds. (8) The following lemma saes an imporan propery of he convoluion operaor. Lemma. For g, V C ([, ) : R),onehas (g V) V = g () V () + g V d d [g V ( g (s) ds) V ]. (9)

3 Absrac and Applied Analysis 3 The proof of his lemma follows by differeniaing he erm g V. Lemma (see [6]). Suppose ha f L (), g H / ( ) and h H 3/ ( );hen,anysoluionof a (V,w) = fw dx + gw dγ + h w ] dγ, saisfies V H 4 () and also Δ V =f, V = V ] = on Γ, B V =h, B V = g on. We formulae he following assumpions. (A) Le f C (R) saisfy w W, () () f (s) s, s R. () Addiionally, we suppose ha f is superlinear; ha is, f (s) s (+η)f(s), z F(z) = f (s) ds, s R, (3) for some η>wih he following growh condiion: f (x) f(y) c(+ x ρ + y ρ ) x y, for some c>and ρ such ha (n )ρ n. x,y R (4) (A) K C (); H () L () wih K(x), forall x, and saisfy he following condiion K m in. (5) The well-posedness of sysem () (5) is given by he following heorem. Theorem 3 (see [7]). Consider assumpions (A)-(A) and le k i C (R + ) be such ha k i, k i,k i (i =,). (6) If u W H 4 (), u W, saisfying he compaibiliy condiion u B u = τ ], B u =τ u on, (7) hen here is only one soluion u of he sysem () (5) saisfying u L (, T : W H 4 ()), u L (, T : W), u L (, T : L ()). (8) 3. General Decay In his secion, we show ha he soluion of sysem () (5) may have a general decay no necessarily of exponenial or polynomial ype. For his we consider ha he resolven kernels saisfy he following hypohesis. (H) k i : R + R + is wice differeniable funcion such ha k i () >, lim k i () =, k i (), (9) and here exiss a nonincreasing coninuous funcion ξ i : R + R + saisfying k i () ξ i () k i (), i =,,. (3) The following ideniy will be used laer. Lemma 4 (see [6]). For every V H 4 () and for every μ R,onehas (m V) Δ Vdx =a(v, V) + Γ m ] [V xx + V yy +μv xxv yy +( μ)v xy ]dγ + [(B V)m V (B V) Γ ] (m V)]dΓ. (3) Le us inroduce he energy funcion E () = K (x) u dx + a (u, u) + F (u) dx + τ (k () u k u)dγ + τ u (k () k u ] )dγ. (3) Now, we esablish some inequaliies for he srong soluion of sysem () (5). Lemma 5. The energy funcional E saisfies, along he soluion of () (5),heesimae E () τ ( Γ u k () u k () u +k u)dγ τ u k ( () u k () u +k u ] )dγ. (33)

4 4 Absrac and Applied Analysis Proof. Muliplying () byu, inegraing over, andusing (), we ge d d { K u dx + a (u, u) + F (u) dx} = (B u) u dγ + (B u) u ] dγ. (34) Subsiuing he boundary erms by (7) andusinglemma and he Young inequaliy, our conclusion follows. Le us consider he following binary operaor: (k u)() := k ( s)(u() u(s)) ds. (35) Thenapplying heholder inequaliyfor α we have (k u)() [ k (s) ( α) ds] ( k α u)(). (36) Le us define he funcional ψ () = [m u+( n θ)u]ku dx. (37) The following lemma plays an imporan role in he consrucion of he desired funcional. Lemma 6. Supposehaheiniialdaa(u,u ) (H 4 () W) W, saisfying he compaibiliy condiion (7). Then, he soluion of sysem () (5) saisfies ψ () m ]K u dγ θ K u n dx ( + θ ελ )a(u, u) ( nη θ ηθ) F (u) dx + τ ε { Γ u +k () u +k () u + k u }dγ + τ ε u { + +k () u +k () k u }dγ u ( ελ δ ) m ] [u xx +u yy + μu xxu yy +( μ)u xy ]dγ. (38) Proof. Differeniaing ψ using ()andlemma 4,wege ψ () = [m u +( n θ)u ]Ku dx + [m u + ( n θ)u]ku dx = m ]K Γ u dγ θ K u dx K m u n dx ( + θ)a(u, u) +n F (u) dx ( n θ) f (u) udx Γm ] [u xx +u yy +μu xxu yy +( μ)u xy ]dγ (B u) [(m u) +( n Γ θ)u]dγ + (B u) [ Γ ] (m u) +(n θ) u ] ]dγ. (39) Le us nex examine he inegrals over Γ in (39). Since u= u/ ] =on Γ,wehaveB u=b u=on Γ and since (m u) = (m ]) Δu, ] u xx +u yy + μu xxu yy +( μ)u xy = (Δu) on Γ, (4) u xx u yy u xy = on Γ. (4) Therefore, from (39) and(4), we have ψ () m ]K Γ u dγ θ K u dx K m u n dx (+ θ)a(u, u) +n F (u) dx ( n θ) f (u) udx + Γ m ](Δu) dγ m ] [u xx +u yy +μu xxu yy +( μ)u xy ]dγ (B u) [(m u) +( n θ)u]dγ + (B u) [ ] (m u) +(n θ) u ] ]dγ. (4)

5 Absrac and Applied Analysis 5 Using he Young inequaliy, we ge (B u) [(m u) +( n θ)u]dγ ε B u dγ + ε ( m u +( n θ) u )dγ, (43) B u= τ { u ] +k () u ] k () u ] k u ] }, (47) our conclusion follows. Le us inroduce he Lyapunov funcional L () =NE() +ψ(), (48) (B u) [ ] (m u) +(n θ) u ] ]dγ ε B u dγ +ε ( ] (m u) +( n u θ) )dγ, (44) wih N>. Now, we are in a posiion o show he main resul of his paper. Theorem 7. Le (u,u ) W L (). Suppose ha he resolven kernels k, k saisfy he condiion (H). Then, here exis consans ω, C > such ha, for some large enough, he soluion of () (5) saisfies where ε is a posiive consan. Since he bilinear form a(u, u) is sricly coercive on W, using he race heory, we obain ( m u +( n θ) u )dγ + ( ] (m u) +( n u θ) )dγ λ a (u, u) + λ δ m ] [u xx +u yy + μu xxu yy +( μ)u xy ]dγ, (45) where λ is a consan depending on, μ, θ and n.subsiuing inequaliies (43) (45) ino(4) and aking ino accoun ha m ] on Γ,aswellas(3)and(5), we have ψ () m ]K Γ u dγ θ K u dx (+ n θ ελ )a(u, u) ( nη θ ηθ) F (u) dx + ε ( B u + B u )dγ ( ελ δ ) m ] [u xx +u yy + μu xxu yy +( μ)u xy ]dγ. (46) Since he boundary condiions (7)can be wrien as B u=τ {u +k () u k () u k u}, E () CE() e ω ξ(s)ds,, if u = u ] = on. (49) Oherwise, E () C(E() + k (s) e ω s ξ(τ)dτ ds) e ω ξ(s)ds, (5) for all,where ξ () = min {ξ (),ξ ()}, k () = k () u dγ + k () u dγ. (5) Proof. Applying inequaliy (36)wihα=/in Lemma 6 and from Lemma 5,weobain L () θk u dx τ N { Γ u k () u k () u +k u}dγ τ N u k { () u +k u ] }dγ k () u (+ n θ ελ )a(u, u) ( nη θ ηθ) F (u) dx+ τ { ε Γ u +k () u +k () u k () k u}dγ

6 6 Absrac and Applied Analysis + τ ε u { +k () u +k () k () k u ] }dγ + m ]K Γ u dγ ( ελ ) δ u m ] [u xx +u yy +μu xxu yy +( μ)u xy ]dγ. We ake θ and ε so small such ha (5) which gives c ξ () E () +cξ() k () u dγ +cξ() k () u dγ ce (), ], (56) ξ () L () +ce () c ξ () E () +cξ() k () u dγ +cξ() k () u dγ, ]. (57) nη θ ηθ>, +n θ ελ >, ελ >. δ (53) Using he fac ha ξ is a nonincreasing coninuous funcion as ξ and ξ are nonincreasing, and so ξ is differeniable, wih ξ (),fora.e.,henweinferha (ξl +ce) () ξ() L () +ce () Since K L () and hen choosing N large enough, we obain L () c E () +c k () u dγ + c k () u dγ c k udγ c k u ] dγ,. On he oher hand, we can choose N even larger so ha (54) L () E(). (55) If ξ() = min{ξ (), ξ ()},,hen,using(3) and(33), we have ξ () L () c ξ () E () +cξ() k () u dγ +cξ() k () u dγ cξ () k udγ cξ () k u ] dγ c ξ () E () +cξ() k () u dγ +cξ() k () u dγ +c k udγ+c k u ] dγ Since using (55), c ξ () E () +cξ() k () u dγ we obain, for some posiive consan ω, +cξ() k () u dγ, ]. (58) F=ξL +ce E, (59) F () ωξ() F () +c k () u dγ +c k () u dγ, ]. Case. If u = u / ] =on,hen(6)reduceso (6) F () ωξ() F (),. (6) A simple inegraion over (,)yields ω F () F( )e ξ(s)ds,. (6) By using (33) and(59), we hen obain for some posiive consan C E () CE() e ω ξ(s)ds,. (63) Thus, esimae (49) isproved. Case.If(u,( u / ])) = (, ) on,hen(6)gives F () ωξ() F () +ck (),, (64)

7 Absrac and Applied Analysis 7 where k () = k () u dγ + k () u dγ. (65) In his case, we inroduce ξ(s)ds ω G () := F () ce k (s) e ω ξ(τ)dτ ds. (66) A simple differeniaion of G, using(64), leads o G () =F ω ξ(s)ds () +ωξ() ce k (s) e ω s ξ(τ)dτ ds ck () ωξ() G (),. Again, a simple inegraion over (,)yields s (67) ω G () G( )e ξ(s)ds,, (68) which implies, for all, F () (F ( ) +c k (s) e ω s ξ(τ)dτ ω ds) e ξ(s)ds. (69) By using (59), we deduce ha E () C(E() + k (s) e ω s ξ(τ)dτ ω ds) e ξ(s)ds, (7). Consequenly, by he boundedness of ξ, (5) is esablished. Remark 8. Noe ha he exponenial and polynomial decay esimaes are only paricular cases of (49) and(5). More precisely, we have exponenial decay for ξ () c and ξ () c and polynomial decay for ξ () = c ( + ) and ξ () c, where c and c are posiive consans. Example 9. As in [4], we give some examples o illusrae he energy decay raes given by (49). () If k () = k () = ae b(+)p, <p,hen,fori=,, k i () ξ()k i (),whereξ() = bp( + )p.for suiably chosen posiive consans a and b, k i saisfies (H) and (49)gives E () ce ωb(+)p. (7) () If k () = a /( + ) q, q >, and k () = a e b(+)p, < p,hen,fori =,, k i () ξ()k i (),whereξ() = q( + ).Then c E () (+) ωq. (7) The aforemenioned wo examples are included in he following more general one. (3) For any nonincreasing funcions k i (), i =,, which saisfy (H), ξ i = k /k are also nonincreasing differeniable funcions, and cξ () ξ (), forsome <c,and(49)gives Acknowledgmen E () c[k ()] ω. (73) This research was suppored by Basic Science Research Program hrough he Naional Research Foundaion of Korea (NRF) funded by he Minisry of Educaion, Science and Technology (Gran no. RAA363). References [] M. M. Cavalcani, V. N. Domingos Cavalcani, J. S. Praes Filho, and J. A. Soriano, Exisence and uniform decay of soluions of a degenerae equaion wih nonlinear boundary damping and boundary memory source erm, Nonlinear Analysis: Theory, Mehods & Applicaions,vol.38,no.3,pp.8 94,999. [] S. D. B. Menezes, E. A. de Oliveira, D. C. Pereira, and J. Ferreira, Exisence, uniqueness and uniform decay for he nonlinear beam degenerae equaion wih weak damping, Applied Mahemaics and Compuaion,vol.54,no.,pp ,4. [3] M.Aassila,M.M.Cavalcani,andV.N.DomingosCavalcani, Exisence and uniform decay of he wave equaion wih nonlinear boundary damping and boundary memory source erm, Calculus of Variaions and Parial Differenial Equaions, vol.5,no.,pp.55 8,. [4]M.Aassila,M.M.Cavalcani,andJ.A.Soriano, Asympoic sabiliy and energy decay raes for soluions of he wave equaion wih memory in a sar-shaped domain, SIAM Journal on Conrol and Opimizaion,vol.38,no.5,pp.58 6,. [5] F. Alabau-Boussouira, P. Cannarsa, and D. Sforza, Decay esimaes for second order evoluion equaions wih memory, Journal of Funcional Analysis,vol.54,no.5,pp.34 37,8. [6] F. Alabau-Boussouira, J. Prüss, and R. Zacher, Exponenial and polynomial sabiliy of a wave equaion for boundary memory damping wih singular kernels, Compes Rendus Mahemaique,vol.347,no.5-6,pp.77 8,9. [7] M.M.Cavalcani,V.N.DomingosCavalcani,andJ.Ferreira, Exisence and uniform decay for a non-linear viscoelasic equaion wih srong damping, Mahemaical Mehods in he Applied Sciences,vol.4,no.4,pp.43 53,. [8] M. M. Cavalcani, V. N. Domingos Cavalcani, and P. Marinez, General decay rae esimaes for viscoelasic dissipaive sysems, Nonlinear Analysis: Theory, Mehods & Applicaions,vol. 68,no.,pp.77 93,8. [9] M. M. Cavalcani, V. N. Domingos Cavalcani, J. S. Praes, and J. A. Soriano, Exisence and uniform decay raes for viscoelasic problems wih nonlinear boundary damping, Differenial and Inegral Equaions,vol.4,no.,pp.85 6,. [] M. M. Cavalcani and H. P. Oquendo, Fricional versus viscoelasic damping in a semilinear wave equaion, SIAM Journal on Conrol and Opimizaion,vol.4,no.4,pp.3 34,3. [] A. Guesmia and S. A. Messaoudi, General energy decay esimaes of Timoshenko sysems wih fricional versus viscoelasic damping, Mahemaical Mehods in he Applied Sciences, vol. 3, no. 6, pp., 9.

8 8 Absrac and Applied Analysis [] X. S. Han and M. X. Wang, Global exisence and uniform decay for a nonlinear viscoelasic equaion wih damping, Nonlinear Analysis:Theory,Mehods&Applicaions,vol.7,no.9,pp , 9. [3]J.R.Kang, Energy decay raes for von Kármán sysem wih memory and boundary feedback, Applied Mahemaics and Compuaion,vol.8,no.8,pp ,. [4] S. A. Messaoudi, General decay of soluions of a viscoelasic equaion, Mahemaical Analysis and Applicaions, vol. 34, no., pp , 8. [5] S. A. Messaoudi, General decay of he soluion energy in a viscoelasic equaion wih a nonlinear source, Nonlinear Analysis: Theory, Mehods & Applicaions, vol.69,no.8,pp , 8. [6] S. A. Messaoudi and M. I. Musafa, On convexiy for energy decay raes of a viscoelasic equaion wih boundary feedback, Nonlinear Analysis: Theory, Mehods & Applicaions,vol.7,no. 9-, pp ,. [7] J. Prüss, Decay properies for he soluions of a parial differenial equaion wih memory, Archiv der Mahemaik, vol.9, no., pp , 9. [8] M.L.Sanos,J.Ferreira,D.C.Pereira,andC.A.Raposo, Global exisence and sabiliy for wave equaion of Kirchhoff ype wih memory condiion a he boundary, Nonlinear Analysis: Theory, Mehods & Applicaions, vol. 54, no. 5, pp , 3. [9] M. M. Cavalcani, V. N. Domingos Cavalcani, and M. L. Sanos, Exisence and uniform decay raes of soluions o a degenerae sysem wih memory condiions a he boundary, Applied Mahemaics and Compuaion, vol.5,no.,pp , 4. [] M. L. Sanos and F. Junior, A boundary condiion wih memory forkirchhoffplaesequaions, Applied Mahemaics and Compuaion,vol.48,no.,pp ,4. [] J.E.M.Rivera,H.P.Oquendo,andM.L.Sanos, Asympoic behavior o a von Kármán plae wih boundary memory condiions, Nonlinear Analysis: Theory, Mehods & Applicaions,vol. 6,no.7,pp.83 5,5. [] J. Y. Park and J. R. Kang, Uniform decay for hyperbolic differenial inclusion wih memory condiion a he boundary, Numerical Funcional Analysis and Opimizaion,vol.7,no.7-8, pp , 6. [3] S. A. Messaoudi and A. Soufyane, General decay of soluions of a wave equaion wih a boundary conrol of memory ype, Nonlinear Analysis: Real World Applicaions, vol.,no.4,pp ,. [4] M. I. Musafa and S. A. Messaoudi, Energy decay raes for a Timoshenko sysem wih viscoelasic boundary condiions, Applied Mahemaics and Compuaion, vol.8,no.8,pp ,. [5] M. L. Sanos and A. Soufyane, General decay o a von Kármán plae sysem wih memory boundary condiions, Differenial and Inegral Equaions, vol. 4, no. -, pp. 69 8,. [6] J. E. Lagnese, Boundary Sabilizaion of Thin Plaes,SIAM. Sociey for Indusrial and Applied Mahemaics, Philadelphia, Pa, USA, 989. [7] J. Y. Park and J. R. Kang, Exisence, uniqueness and uniform decay for he non-linear degenerae equaion wih memory condiion a he boundary, Applied Mahemaics and Compuaion, vol., no., pp , 8.

9 Advances in Operaions Research Volume 4 Advances in Decision Sciences Volume 4 Applied Mahemaics Algebra Volume 4 Probabiliy and Saisics Volume 4 The Scienific World Journal Volume 4 Inernaional Differenial Equaions Volume 4 Volume 4 Submi your manuscrips a Inernaional Advances in Combinaorics Mahemaical Physics Volume 4 Complex Analysis Volume 4 Inernaional Mahemaics and Mahemaical Sciences Mahemaical Problems in Engineering Mahemaics Volume 4 Volume 4 Volume 4 Volume 4 Discree Mahemaics Volume 4 Discree Dynamics in Naure and Sociey Funcion Spaces Absrac and Applied Analysis Volume 4 Volume 4 Volume 4 Inernaional Sochasic Analysis Opimizaion Volume 4 Volume 4